Homomorphic Sensing Manolis C. T sakiris Liangzu Peng - - PowerPoint PPT Presentation

homomorphic sensing
SMART_READER_LITE
LIVE PREVIEW

Homomorphic Sensing Manolis C. T sakiris Liangzu Peng - - PowerPoint PPT Presentation

Homomorphic Sensing Manolis C. T sakiris Liangzu Peng ShanghaiT ech University <latexit


slide-1
SLIDE 1

Homomorphic Sensing

Manolis C. T

sakiris Liangzu Peng ShanghaiT

ech University

slide-2
SLIDE 2

Shuffled Linear Regression

[1] A. Pananjady, M. J. Wainwright, T . D. Courtade, “Linear regression with shuffled data: statistical and computational limits of permutation recovery”, IEEE T ransactions on Information Theory, 2018. [2] J. Unikrishnan, S. Haghighatsoar, M. Vetterli, “Unlabeled sensing with random linear measurements”, IEEE T ransactions on Information Theory, 2018. [3] M. Slawski and E. Ben-David, “Linear regression with sparsely permuted data”, Electronic Journal of Statistics, 2019. [4] X. Song, H. Choi, Y. Shi, “Permuted linear model for header-free communication via symmetric polynomials”, ISIT , 2018.

Ax = b

<latexit sha1_base64="+a9H+R06Yv8T41qcam0W1mUv46g=">AB7XicbVBNSwMxEJ34WetX1aOXYBE8lV0R9CJUvHiSCvYD2qVk02wbm02WJCuWpf/BiwdFvPp/vPlvTNs9aOuDgcd7M8zMCxPBjfW8b7S0vLK6tl7YKG5ube/slvb2G0almrI6VULpVkgME1yuVWsFaiGYlDwZrh8HriNx+ZNlzJeztKWBCTvuQRp8Q6qXH1hC9x2C2VvYo3BV4kfk7KkKPWLX1eoqmMZOWCmJM2/cSG2REW04FGxc7qWEJoUPSZ21HJYmZCbLptWN87JQejpR2JS2eqr8nMhIbM4pD1xkTOzDz3kT8z2unNroIMi6T1DJZ4uiVGCr8OR13OaUStGjhCqubsV0wHRhFoXUNGF4M+/vEgapxXfq/h3Z+XqbR5HAQ7hCE7Ah3Oowg3UoA4UHuAZXuENKfSC3tHrHUJ5TMH8Afo8wdbKY5a</latexit><latexit sha1_base64="+a9H+R06Yv8T41qcam0W1mUv46g=">AB7XicbVBNSwMxEJ34WetX1aOXYBE8lV0R9CJUvHiSCvYD2qVk02wbm02WJCuWpf/BiwdFvPp/vPlvTNs9aOuDgcd7M8zMCxPBjfW8b7S0vLK6tl7YKG5ube/slvb2G0almrI6VULpVkgME1yuVWsFaiGYlDwZrh8HriNx+ZNlzJeztKWBCTvuQRp8Q6qXH1hC9x2C2VvYo3BV4kfk7KkKPWLX1eoqmMZOWCmJM2/cSG2REW04FGxc7qWEJoUPSZ21HJYmZCbLptWN87JQejpR2JS2eqr8nMhIbM4pD1xkTOzDz3kT8z2unNroIMi6T1DJZ4uiVGCr8OR13OaUStGjhCqubsV0wHRhFoXUNGF4M+/vEgapxXfq/h3Z+XqbR5HAQ7hCE7Ah3Oowg3UoA4UHuAZXuENKfSC3tHrHUJ5TMH8Afo8wdbKY5a</latexit><latexit sha1_base64="+a9H+R06Yv8T41qcam0W1mUv46g=">AB7XicbVBNSwMxEJ34WetX1aOXYBE8lV0R9CJUvHiSCvYD2qVk02wbm02WJCuWpf/BiwdFvPp/vPlvTNs9aOuDgcd7M8zMCxPBjfW8b7S0vLK6tl7YKG5ube/slvb2G0almrI6VULpVkgME1yuVWsFaiGYlDwZrh8HriNx+ZNlzJeztKWBCTvuQRp8Q6qXH1hC9x2C2VvYo3BV4kfk7KkKPWLX1eoqmMZOWCmJM2/cSG2REW04FGxc7qWEJoUPSZ21HJYmZCbLptWN87JQejpR2JS2eqr8nMhIbM4pD1xkTOzDz3kT8z2unNroIMi6T1DJZ4uiVGCr8OR13OaUStGjhCqubsV0wHRhFoXUNGF4M+/vEgapxXfq/h3Z+XqbR5HAQ7hCE7Ah3Oowg3UoA4UHuAZXuENKfSC3tHrHUJ5TMH8Afo8wdbKY5a</latexit><latexit sha1_base64="+a9H+R06Yv8T41qcam0W1mUv46g=">AB7XicbVBNSwMxEJ34WetX1aOXYBE8lV0R9CJUvHiSCvYD2qVk02wbm02WJCuWpf/BiwdFvPp/vPlvTNs9aOuDgcd7M8zMCxPBjfW8b7S0vLK6tl7YKG5ube/slvb2G0almrI6VULpVkgME1yuVWsFaiGYlDwZrh8HriNx+ZNlzJeztKWBCTvuQRp8Q6qXH1hC9x2C2VvYo3BV4kfk7KkKPWLX1eoqmMZOWCmJM2/cSG2REW04FGxc7qWEJoUPSZ21HJYmZCbLptWN87JQejpR2JS2eqr8nMhIbM4pD1xkTOzDz3kT8z2unNroIMi6T1DJZ4uiVGCr8OR13OaUStGjhCqubsV0wHRhFoXUNGF4M+/vEgapxXfq/h3Z+XqbR5HAQ7hCE7Ah3Oowg3UoA4UHuAZXuENKfSC3tHrHUJ5TMH8Afo8wdbKY5a</latexit>

known only up to a permutation

slide-3
SLIDE 3
  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>

Shuffled Linear Regression

slide-4
SLIDE 4
  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>

Shuffled Linear Regression

slide-5
SLIDE 5

unknown known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

π1(v1)

<latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>

known

Shuffled Linear Regression

unknown permutation

π1

<latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit>
slide-6
SLIDE 6

Reconstruct the original signal from its shuffled measurements.

unknown known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

π1(v1)

<latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>

reconstruction?

Shuffled Linear Regression

unknown permutation

π1

<latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit>

known

slide-7
SLIDE 7
  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

Reconstruct the original signal from its shuffled measurements.

unknown known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

π1(v1)

<latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>

unique reconstruction?

Shuffled Linear Regression

reconstruction? unknown permutation

π1

<latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit>

known

slide-8
SLIDE 8

Reconstruct the original signal from its shuffled measurements.

unknown known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

π1(v1)

<latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>

unique reconstruction:

v1 6= v2 2 V ) π1(v1) 6= π2(v2)

<latexit sha1_base64="9D9C9GpnDE/orIvK13ui6Fs/JvM=">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</latexit><latexit sha1_base64="9D9C9GpnDE/orIvK13ui6Fs/JvM=">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</latexit><latexit sha1_base64="9D9C9GpnDE/orIvK13ui6Fs/JvM=">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</latexit><latexit sha1_base64="9D9C9GpnDE/orIvK13ui6Fs/JvM=">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</latexit>

permutations

Shuffled Linear Regression

reconstruction? unknown permutation

π1

<latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit>

known

slide-9
SLIDE 9

Reconstruct the original signal from its shuffled and downsampled measurements.

known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>
  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

ρ1π1(v1)

<latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit>

downsampled values

Unlabeled Sensing

reconstruction? unknown

ρ1π1

<latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit>

unknown known

slide-10
SLIDE 10

known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>
  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

ρ1π1(v1)

<latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit>

downsampled values

v1 6= v2 2 V ) ρ1π1(v1) 6= ρ2π2(v2)

<latexit sha1_base64="vTXEj4dZiqrCrK4LA0objqObY=">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</latexit><latexit sha1_base64="vTXEj4dZiqrCrK4LA0objqObY=">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</latexit><latexit sha1_base64="vTXEj4dZiqrCrK4LA0objqObY=">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</latexit><latexit sha1_base64="W5HERqf396Qm+xfIqzu4Edc1mYw=">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</latexit>

Unlabeled Sensing

reconstruction? unknown

ρ1π1

<latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit>

Reconstruct the original signal from its shuffled and downsampled measurements.

unknown known

slide-11
SLIDE 11

The Problem

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

T : a finite set of linear transformations τ : Rm → Rm

<latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit>
slide-12
SLIDE 12

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

T : a finite set of linear transformations τ : Rm → Rm

<latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">ACTHicbVA9bxNBFNwzEIL5MlDSPGEjUVl3aUCpIqUJXUBxEslnrHfrd/Yq+3HafQeyTv6BNBR0/AoaChBCYu/igiSMtNJo5s3u2ykqrQKn6bekd+v2nZ27u/f69x8fPR48OTpaXC1lzSRTjt/XmAgrSxNWLGm8oTmkLTWXFx2PpnH8kH5ewJryuaGVxaVSqJHKX5QI5yg7ySqJuTzWgfEplFRMEYnAltPeiB/ZoQ+m86WIBRjljvQ9dtia95sPBnKvlitG792nK8ZoPhim47QD3CTZlgzFsfzwd84WRtyLUGMI0SyueNehZSU2bfl4HqlBe4JKmkVo0FGZNV8YGXkZlAXHXeCxDp/6baNCEsDZFnGyXDNe9VvyfN625fDNrlK1qJisvHyprDeygbRYWypNkvY4EpVdxV5Ar9Cg59t+PJWTXv3yTnO6Ns3ScvdsbHrzd1rErnosX4pXIxGtxI7EsZgIKT6L7+Kn+JV8SX4kv5M/l6O9ZJt5Jq6gt/MXgt20RA=</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit>

“Under what conditions on dim(V) can we uniquely reconstruct (sense) points in V from their (homomorphic) images under the transformations in T ?”

<latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit>

The Problem

slide-13
SLIDE 13

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

T : a finite set of linear transformations τ : Rm → Rm

<latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit>

homomorphic sensing property v1 6= v2 2 V ) τ1(v1) 6= τ2(v2), 8τ1, τ2 2 T

<latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit><latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit><latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit><latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit>

The Problem

HSP

“Under what conditions on dim(V) can we uniquely reconstruct (sense) points in V from their (homomorphic) images under the transformations in T ?”

<latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit>
slide-14
SLIDE 14

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

HSP

  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-15
SLIDE 15

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">ACQHicbZBJSwMxFMcz7tat6tFLsAgKUmaKoMeCF72p2AU6ZXiTpm0wkxmTN5Uy9KN58SN48+zFgyJePZkuQl0eCfz5vSV5/zCRwqDrPjkzs3PzC4tLy7mV1bX1jfzmVtXEqWa8wmIZ63oIhkuheAUFSl5PNIcolLwW3pwO87Ue10bE6hr7CW9G0FGiLRigRUG+1gs86it+S3tBifpCUT8C7DKQWXVA/cPvcyU6XQSt47spiJAG3r6dcDAeMQIlC0oHQb7gFt1R0L/Cm4gCmcRFkH/0WzFLI6QSTCm4bkJNjPQKJjkg5yfGp4Au4EOb1ipIOKmY0MGNA9S1q0HWt7FdIRne7IDKmH4W2crid+Z0bwv9yjRTbJ81MqCRFrtj4oXYqKcZ06CZtCc0Zyr4VwLSwf6WsCxoYWs9z1gTv98p/RbVU9Nyid3lUKJ9P7FgiO2SX7BOPHJMyOSMXpEIYuSfP5JW8OQ/Oi/PufIxLZ5xJzb5Ec7nF73+q60=</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

ρ: any projection onto im(τ2)

<latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit>

HSP

  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-16
SLIDE 16

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

ρ: any projection onto im(τ2)

<latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit>

Y =

  • w ∈ Cm : ρτ1(w), τ2(w) are linearly dependent
<latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit>

HSP

  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-17
SLIDE 17

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">ACQHicbZBJSwMxFMcz7tat6tFLsAgKUmaKoMeCF72p2AU6ZXiTpm0wkxmTN5Uy9KN58SN48+zFgyJePZkuQl0eCfz5vSV5/zCRwqDrPjkzs3PzC4tLy7mV1bX1jfzmVtXEqWa8wmIZ63oIhkuheAUFSl5PNIcolLwW3pwO87Ue10bE6hr7CW9G0FGiLRigRUG+1gs86it+S3tBifpCUT8C7DKQWXVA/cPvcyU6XQSt47spiJAG3r6dcDAeMQIlC0oHQb7gFt1R0L/Cm4gCmcRFkH/0WzFLI6QSTCm4bkJNjPQKJjkg5yfGp4Au4EOb1ipIOKmY0MGNA9S1q0HWt7FdIRne7IDKmH4W2crid+Z0bwv9yjRTbJ81MqCRFrtj4oXYqKcZ06CZtCc0Zyr4VwLSwf6WsCxoYWs9z1gTv98p/RbVU9Nyid3lUKJ9P7FgiO2SX7BOPHJMyOSMXpEIYuSfP5JW8OQ/Oi/PufIxLZ5xJzb5Ec7nF73+q60=</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

ρ: any projection onto im(τ2)

<latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit>

Y =

  • w ∈ Cm : ρτ1(w), τ2(w) are linearly dependent
<latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit>

(determinantal variety)

HSP

ρτ1(w) = λτ2(w)

<latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit>

τ2(w) = λρτ1(w)

<latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit>
  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-18
SLIDE 18

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

ρ: any projection onto im(τ2)

<latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit>

Z = ker(ρτ1) ∪ ker(τ2) ∪ ker(ρτ1 − τ2)

<latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit>

HSP

Y =

  • w ∈ Cm : ρτ1(w), τ2(w) are linearly dependent
<latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">ACX3icbVFRS9xAEN6ktdrTamyfSl+WnoKCHMm9KIg+GLfLPTUcjmPyWbubnGzCbsT9Qj5k74JvSfdJO7QqsduHj+2Z2Zr5NCiUtheGT5795u/Jude19Z3jw+ZWsP3x0ualETgQucrNdQIWldQ4IEkKrwuDkCUKr5Lbs0a/ukNjZa5/0LzAUQZTLSdSADlqHNztxBnQTICqftYnPE7kNK7ueSw1b4Ukqc7qm+yYxwftMbOcxwTlONq73z9YwL6Df/SGwgeqwCBvZgKj5jzFAnWKmupFg3pnHTDXtgGfw2iJeiyZVyMg8c4zUWZuUeEAmuHUVjQyLUhKRTWnbi0WIC4hSkOHdSQoR1VrT813VMyie5cVcTb9m/KyrIrJ1nictsdrYvtYb8nzYsaXI0qQuSkItFo0mpeKU8ZsnkqDghoHJAgj3axczMCAIPclHWdC9HLl1+Cy34vCXvS93z39trRjX1hX9kei9ghO2Xn7INmGDPnu+texveL3/V3/SDRarvLWs+sX/C/wbHkCynQ=</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit>

(determinantal variety)

ρτ1(w) = λτ2(w)

<latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit>

τ2(w) = λρτ1(w)

<latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit>

λ = 0

<latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit>

λ = 1

<latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit>
  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-19
SLIDE 19

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">ACQHicbZBJSwMxFMcz7tat6tFLsAgKUmaKoMeCF72p2AU6ZXiTpm0wkxmTN5Uy9KN58SN48+zFgyJePZkuQl0eCfz5vSV5/zCRwqDrPjkzs3PzC4tLy7mV1bX1jfzmVtXEqWa8wmIZ63oIhkuheAUFSl5PNIcolLwW3pwO87Ue10bE6hr7CW9G0FGiLRigRUG+1gs86it+S3tBifpCUT8C7DKQWXVA/cPvcyU6XQSt47spiJAG3r6dcDAeMQIlC0oHQb7gFt1R0L/Cm4gCmcRFkH/0WzFLI6QSTCm4bkJNjPQKJjkg5yfGp4Au4EOb1ipIOKmY0MGNA9S1q0HWt7FdIRne7IDKmH4W2crid+Z0bwv9yjRTbJ81MqCRFrtj4oXYqKcZ06CZtCc0Zyr4VwLSwf6WsCxoYWs9z1gTv98p/RbVU9Nyid3lUKJ9P7FgiO2SX7BOPHJMyOSMXpEIYuSfP5JW8OQ/Oi/PufIxLZ5xJzb5Ec7nF73+q60=</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

ρ: any projection onto im(τ2)

<latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit>

Z = ker(ρτ1) ∪ ker(τ2) ∪ ker(ρτ1 − τ2)

<latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit>

U = Y \ Z

<latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit><latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit><latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit><latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit>

(quasi-variety)

HSP

Y =

  • w ∈ Cm : ρτ1(w), τ2(w) are linearly dependent
<latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit>

(determinantal variety)

ρτ1(w) = λτ2(w)

<latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit>

τ2(w) = λρτ1(w)

<latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit>

λ = 0

<latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit>

λ = 1

<latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit>
  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-20
SLIDE 20

Main Result

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

homomorphic sensing property

v1 6= v2 2 V ) τ1(v1) 6= τ2(v2)

<latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit><latexit sha1_base64="9p6YEQYJFyahSr30PlQvHvq3MCs=">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</latexit>

τ1, τ2: linear transformations Rm → Rm

<latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit><latexit sha1_base64="l/EvGVNJ7TAW0enOvjZh3mJf/0=">ACOXicbVA9T8MwEHX4LOWrwMhi0SIxoCrpAmKqxAJbQRQqNaW6uG5r4TiRfQFVUf8WC/+CDYmFAYRY+QM4oQNQTrL89N6d/e4FsRQGXfJmZmdm19YLCwVl1dW19ZLG5uXJko040WyUi3AjBcCsWbKFDyVqw5hIHkV8HNcaZf3XJtRKQucBTzTgDJfqCAVqW2pUfISk6+3T/K5Vjmj2FGiKGpTpRzrMOw2t+BYNgyA9H1+H1NdiMETQOrqjP4VKt1R2q25edBp4E1Amk2p0S49+L2JyBUyCca0PTfGTgoaBZN8XPQTw2NgNzDgbQsVhNx0nzMd21TI9al/YopDn7cyKF0JhRGNjOzKT5q2Xkf1o7wf5hJxUqTpAr9v1RP5EUI5rFSHtCc4ZyZAEwLaxXyoagaENu2hD8P6uPA0ua1XPrXpntXL9dBJHgWyTHbJHPHJA6uSENEiTMHJPnskreXMenBfn3fn4bp1xJjNb5Fc5n18m8q07</latexit>

ρ: any projection onto im(τ2)

<latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit><latexit sha1_base64="HpG1iPYh43aCGIaLTYt2Bb9/XB4=">ACGHicbVDLTgIxFO3gC/E16tJNI5jgBmfYaFyRuNEdJvJIGEI6pUC107aO0Yy4TPc+CtuXGiMW3b+jeWxUPAkTU7OuTe354Sx4AY879vJrKyurW9kN3Nb2zu7e+7+Qd2oRFNWo0o3QyJYJLVgMOgjVjzUgUCtYIH64mfuORacOVvINhzNoR6Uve45SAlTruWSHQA1W4xEQOcazVPaMTAysJChcCYE+Q8mhUDIAknfJpoePmvZI3BV4m/pzk0RzVjsOuomEZNABTGm5XsxtFOigVPBRrkgMSwm9IH0WctSJm2uk02AifWKWLe0rbJwFP1d8bKYmMGUahnYwIDMyiNxH/81oJ9C7aKZdxAkzS2aFeIrANPWkJd7m2RYihJYRqbv+K6YBoQsF2mbMl+IuRl0m9XPK9kn9bzldu5nVk0RE6RkXko3NUQdeoimqIomf0it7Rh/PivDmfztdsNOPMdw7RHzjHxoYnzE=</latexit>

Z = ker(ρτ1) ∪ ker(τ2) ∪ ker(ρτ1 − τ2)

<latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit><latexit sha1_base64="RkWLmromQTs9ylOHsfijYoqKBU=">ACTHicbVDLSgMxFM3UV62vqks3wSLowjJTBN0IBTe6q2Af2Cklk97a0MyD5I5Yhn6gGxfu/Ao3LhQRzLQDWuBwMk595Dc40VSaLTtFyu3sLi0vJfLaytb2xuFbd3GjqMFYc6D2WoWh7TIEUAdRQoRUpYL4noekNL1K/eQ9KizC4wVEHZ/dBaIvOEMjdYvc9RkOJPJ7ZieUxfhAZMhqPGhqwahubO46xRl8fRjJnqlX/0nxA9ptlUt1iy/YEdJ4GSmRDLVu8dnthTz2IUAumdZtx46wkzCFgksYF9xYQ8T4kN1B29CA+aA7yaSMT0wSo/2Q2VOgHSi/k4kzNd65HtmMl1d/VS8T+vHWP/rJOIoRAj59qB9LiFNm6U9oYCjHBnCuBLmr5QPmGIcTf8FU4Lzd+V50qiUHbvsXJ+UqldZHXmyR/bJIXHIKamS1IjdcLJI3kl7+TDerLerE/razqas7LMLplBbvkb/hCzMw=</latexit>

U = Y \ Z

<latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit><latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit><latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit><latexit sha1_base64="zA+TCjwaxulFU1aNp0jv1mZ6jYo=">ACGHicbVDLSgMxFM3UV62vUZdugkVwVWdE0I1QcKO7Ck6rdoaSdM2NMkMSUYow3yG3/FjQtF3Hbn35hpBx+tBwIn59zLvfeEMaNKO86nVpYXFpeKa9W1tY3Nrfs7Z2mihKJiYcjFsnbECnCqCepqR21gSxENGWuHwIvdbD0QqGokbPYpJwFf0B7FSBupYx/5HOkBRiz1MngOv393GfQV0ZyKRP2o91nHrjo1ZwI4T9yCVEGBRsce+90IJ5wIjRlSqu06sQ5SJDXFjGQVP1EkRniI+qRtqECcqCdHJbBA6N0YS+S5gkNJ+rvjhRxpUY8NJX5imrWy8X/vHaie2dBSkWcaCLwdFAvYVBHME8JdqkWLORIQhLanaFeIAkwtpkWTEhuLMnz5Pmc1au71SbV+VcRBntgHxwCF5yCOrgEDeABDB7BM3gFb9aT9WK9Wx/T0pJV9OyCP7DGX2/4oKs=</latexit>

(quasi-variety)

dim(V) ≤ min n1 2rank(τ2), codim(U)

  • <latexit sha1_base64="pReEch5dyu2oUx5Nn6OAwehT5U=">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</latexit><latexit sha1_base64="pReEch5dyu2oUx5Nn6OAwehT5U=">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</latexit><latexit sha1_base64="pReEch5dyu2oUx5Nn6OAwehT5U=">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</latexit><latexit sha1_base64="pReEch5dyu2oUx5Nn6OAwehT5U=">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</latexit>

Theorem HSP HSP

Y =

  • w ∈ Cm : ρτ1(w), τ2(w) are linearly dependent
<latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit><latexit sha1_base64="x/ZQtFxROFOiI3YfqDBEzDfahc=">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</latexit>

(determinantal variety)

ρτ1(w) = λτ2(w)

<latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit><latexit sha1_base64="Ev0LYKeDGZPX6endIpH+l3GSo=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR4i6GiS9Hp9hieQUeY9IDMtYrR+nYRlRFCGOYEVw9RYbU67UKrkGcWQZFsECzb787g4gmAQs1FUSpHkaxdlMiNaeCTfNOolhM6JgMWc/QkARMuenslik8MsoA+pE0L9Rwpn5PpCRQahJ4ZjIgeqR+e5n4l9dLtF9zUx7GiWYhnS/yEwF1BLNi4IBLRrWYGEKo5OavkI6IJFSb+vKmhK9L4f+kXSljVMZXJ8XG5aKOHDgEBVACGFRBA1yAJmgBCu7A3gCz9a9Wi9WK/z0SVrkTkAP2C9fQJST5lY</latexit>

τ2(w) = λρτ1(w)

<latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit><latexit sha1_base64="g3woX2QfuHpcHqYvRURfsMLSlQ=">AC3icdVDLSgMxFM34rPU16tJNaBHqpiRFbLsQCm50V8E+oDOUTJpQzMPkoxShu7d+CtuXCji1h9w59+YaSuo6IHA4Zx7uLnHiwVXGqEPa2l5ZXVtPbeR39za3tm19/bKkokZS0aiUh2PaKY4CFra4F68aSkcATrONzO/c8Ok4lF4rScxcwMyDLnPKdFG6tsFR5OkXyndHsMz6AgTHBDoyFEZwY2Rt8uojJCGM4Krp8iQer1WwTWIM8ugCBZo9u13ZxDRJGChpoIo1cMo1m5KpOZUsGneSRSLCR2TIesZGpKAKTed3TKFR0YZQD+S5oUaztTviZQESk0Cz0wGRI/Uby8T/J6ifZrbsrDONEspPNFfiKgjmBWDBxwyagWE0MIldz8FdIRkYRqU1/elPB1KfyftCtljMr46qTYuFzUkQOHoABKAIMqaIAL0AQtQMEdeABP4Nm6tx6tF+t1PrpkLTIH4Aest09OvJlY</latexit>

λ = 0

<latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit><latexit sha1_base64="bHKn5/bSIRwRBsTpNwleG9Olu7A=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA0KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEMeKQjw=</latexit>

λ = 1

<latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit><latexit sha1_base64="i4OlgVMgHweJAy5eRv960/aMg=">AB8nicdVDNSgMxGMzWv1r/qh69BIvgqSRFbHsQCl70VsHawnYp2Wy2Dc0mS5IVSuljePGgiFefxptvY7atoKIDgWFmPvJ9E6aCG4vQh1dYWV1b3yhulra2d3b3yvsHd0ZlmrIOVULpXkgME1yjuVWsF6qGUlCwbrh+DL3u/dMG67krZ2kLEjIUPKYU2Kd5PeFi0YEXkA8KFdQFSGEMY5wfVz5Eiz2ajhBsS5VABS7QH5fd+pGiWMGmpIMb4GKU2mBJtORVsVupnhqWEjsmQ+Y5KkjATOcrz+CJUyIYK+2etHCufp+YksSYSRK6ZELsyPz2cvEvz89s3AimXKaZIuPozAa2C+f0w4pRKyaOEKq52xXSEdGEWtdSyZXwdSn8n9zVqhV8c1ZpXW9rKMIjsAxOAUY1ELXIE26AKFHgAT+DZs96j9+K9LqIFbzlzCH7Ae/sEM2aQkA=</latexit>
  • rank(τ2) ≥ rank(τ1)
  • <latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit><latexit sha1_base64="7krNuNETJg3Lm3YQMQHjtcBr1Qw=">ACIHicdVBNaxsxFNSmSZs4Ses0x1xETcG5GMmE2rkFemluDsQf4F2MVn52hLXaRXpbYhb/lF76V3rpISGkt+bXRGu70CbpgGCYecPTmzjTyiFjv4ONV5tbr9s71R29/bfvqsevO+5NLcSujLVqR3EwoFWBrqoUMgsyCSWEM/n0u/f5XsE6l5hLnGUSJmBo1UVKgl0bVhiraZ2GCNdYWGFmi3qIh81j2k4hec697pPHI+qNdZgjHOaUl46xPz5PS03eRtykvLo0bW6Iyqv8JxKvMEDEotnBtylmFUCItKalhUwtxBJuRMTGHoqREJuKhYHrigH70ypPU+meQLtW/E4VInJsnsZ9MBF65p14pvuQNc5y0o0KZLEcwcrVokmuKS3bomNlQaKeyKkVf6vVF4JKyT6Tiu+hD+X0v+TXrPBWYNfnNTOztd1bJMj8oHUCSctcka+kA7pEkm+kR/khtwG34OfwV1wvxrdCNaZQ/IPgodHw8uivA=</latexit>
slide-21
SLIDE 21

Reconstruct the original signal from its shuffled measurements.

unknown known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

π1(v1)

<latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit><latexit sha1_base64="ptijceLtN2rjES1nATsgTcN/O0=">AB83icdVDLSgMxFM34rPVdekmWIS6KUkR2+4KbnRXwT6gMwyZNOGZmZCkimUob/hxoUibv0Zd/6NmbaCih64cDjnXu69J5Ca4PQh7O2vrG5tV3YKe7u7R8clo6OuzpJFWUdmohE9QOimeAx6xhuBOtLxUgUCNYLJte535sypXkS35uZF5ERjEPOSXGSq4ruY8rUx9fQNcvlVEVIYQxhjnB9StkSbPZqOEGxLlUQYrtP3SuztMaBqx2FBtB5gJI2XEWU4FWxedFPNJKETMmIDS2MSMe1li5vn8NwqQxgmylZs4EL9PpGRSOtZFNjOiJix/u3l4l/eIDVhw8t4LFPDYrpcFKYCmgTmAcAhV4waMbOEUMXtrZCOiSLU2JiKNoSvT+H/pFurYlTFd5fl1u0qjgI4BWegAjCogxa4AW3QARI8ACewLOTOo/Oi/O6bF1zVjMn4Aect0+egJDH</latexit>

Shuffled Linear Regression

Corollary HSP

dim(V) ≤ m 2

<latexit sha1_base64="rHalJ6uQVpu1Fi7s4yipjVshxE4=">ACnicdVDLSsNAFJ34rPUVdelmtAh1U5IY2roruNFdBfuApTJdNIOnUnCzEQoIWs3/obF4q49Qvc+TdO2goqeuDC4Zx7ufceP2ZUKsv6MJaWV1bX1gsbxc2t7Z1dc2+/LaNEYNLCEYtE10eSMBqSlqKkW4sCOI+Ix1/cpH7nVsiJI3CGzWNSZ+jUgDipHS0sA8oaUlz2O1BgjlrazU+gxAr1AIJzyLHWygVmyKuf1quNWoVWxrJrt2Dlxau6ZC2t5CiBZoD890bRjhJFSYISl7thWrfoqEopiRrOglksQIT9CI9DQNESeyn85eyeCJVoYwiISuUMGZ+n0iRVzKfd1Z36z/O3l4l9eL1FBvZ/SME4UCfF8UZAwqCKY5wKHVBCs2FQThAXVt0I8RjoFpdMr6hC+PoX/k7ZTsa2Kfe2WGleLOArgEByDMrBDTAJWiCFsDgDjyAJ/Bs3BuPxovxOm9dMhYzB+AHjLdPMTWanQ=</latexit><latexit sha1_base64="rHalJ6uQVpu1Fi7s4yipjVshxE4=">ACnicdVDLSsNAFJ34rPUVdelmtAh1U5IY2roruNFdBfuApTJdNIOnUnCzEQoIWs3/obF4q49Qvc+TdO2goqeuDC4Zx7ufceP2ZUKsv6MJaWV1bX1gsbxc2t7Z1dc2+/LaNEYNLCEYtE10eSMBqSlqKkW4sCOI+Ix1/cpH7nVsiJI3CGzWNSZ+jUgDipHS0sA8oaUlz2O1BgjlrazU+gxAr1AIJzyLHWygVmyKuf1quNWoVWxrJrt2Dlxau6ZC2t5CiBZoD890bRjhJFSYISl7thWrfoqEopiRrOglksQIT9CI9DQNESeyn85eyeCJVoYwiISuUMGZ+n0iRVzKfd1Z36z/O3l4l9eL1FBvZ/SME4UCfF8UZAwqCKY5wKHVBCs2FQThAXVt0I8RjoFpdMr6hC+PoX/k7ZTsa2Kfe2WGleLOArgEByDMrBDTAJWiCFsDgDjyAJ/Bs3BuPxovxOm9dMhYzB+AHjLdPMTWanQ=</latexit><latexit sha1_base64="rHalJ6uQVpu1Fi7s4yipjVshxE4=">ACnicdVDLSsNAFJ34rPUVdelmtAh1U5IY2roruNFdBfuApTJdNIOnUnCzEQoIWs3/obF4q49Qvc+TdO2goqeuDC4Zx7ufceP2ZUKsv6MJaWV1bX1gsbxc2t7Z1dc2+/LaNEYNLCEYtE10eSMBqSlqKkW4sCOI+Ix1/cpH7nVsiJI3CGzWNSZ+jUgDipHS0sA8oaUlz2O1BgjlrazU+gxAr1AIJzyLHWygVmyKuf1quNWoVWxrJrt2Dlxau6ZC2t5CiBZoD890bRjhJFSYISl7thWrfoqEopiRrOglksQIT9CI9DQNESeyn85eyeCJVoYwiISuUMGZ+n0iRVzKfd1Z36z/O3l4l9eL1FBvZ/SME4UCfF8UZAwqCKY5wKHVBCs2FQThAXVt0I8RjoFpdMr6hC+PoX/k7ZTsa2Kfe2WGleLOArgEByDMrBDTAJWiCFsDgDjyAJ/Bs3BuPxovxOm9dMhYzB+AHjLdPMTWanQ=</latexit><latexit sha1_base64="rHalJ6uQVpu1Fi7s4yipjVshxE4=">ACnicdVDLSsNAFJ34rPUVdelmtAh1U5IY2roruNFdBfuApTJdNIOnUnCzEQoIWs3/obF4q49Qvc+TdO2goqeuDC4Zx7ufceP2ZUKsv6MJaWV1bX1gsbxc2t7Z1dc2+/LaNEYNLCEYtE10eSMBqSlqKkW4sCOI+Ix1/cpH7nVsiJI3CGzWNSZ+jUgDipHS0sA8oaUlz2O1BgjlrazU+gxAr1AIJzyLHWygVmyKuf1quNWoVWxrJrt2Dlxau6ZC2t5CiBZoD890bRjhJFSYISl7thWrfoqEopiRrOglksQIT9CI9DQNESeyn85eyeCJVoYwiISuUMGZ+n0iRVzKfd1Z36z/O3l4l9eL1FBvZ/SME4UCfF8UZAwqCKY5wKHVBCs2FQThAXVt0I8RjoFpdMr6hC+PoX/k7ZTsa2Kfe2WGleLOArgEByDMrBDTAJWiCFsDgDjyAJ/Bs3BuPxovxOm9dMhYzB+AHjLdPMTWanQ=</latexit>
  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

reconstruction? unknown permutation

π1

<latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit><latexit sha1_base64="UCxoicMNtitC/qIYwyzn0edbM=">AB7HicdVBNS8NAEN3Ur1q/qh69LBbBU8jG0NZbwYveKpi20Iay2W7apZtN2N0IJfQ3ePGgiFd/kDf/jZu2go+GHi8N8PMvDlTGnH+bBKa+sbm1vl7crO7t7+QfXwqKOSTBLqk4QnshdiRTkT1NdMc9pLJcVxyGk3nF4VfveSsUScadnKQ1iPBYsYgRrI/mDlA3RsFpz7Mtm3fXq0LEdp4FcVBC34V14EBmlQA2s0B5W3wejhGQxFZpwrFQfOakOciw1I5zOK4NM0RSTKR7TvqECx1QF+eLYOTwzyghGiTQlNFyo3ydyHCs1i0PTGWM9Ub+9QvzL62c6agY5E2mqSDLRVHGoU5g8TkcMUmJ5jNDMJHM3ArJBEtMtMmnYkL4+hT+TzqujRwb3Xq1s0qjI4AafgHCDQAC1wDdrABwQw8ACewLMlrEfrxXpdtpas1cwx+AHr7RPY0Y6</latexit>

4-dimensional signal subspace V ⊂ Rm

<latexit sha1_base64="IT2/TZjr4d7iWM8aipH9CZ7UCc=">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</latexit><latexit sha1_base64="IT2/TZjr4d7iWM8aipH9CZ7UCc=">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</latexit><latexit sha1_base64="IT2/TZjr4d7iWM8aipH9CZ7UCc=">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</latexit><latexit sha1_base64="IT2/TZjr4d7iWM8aipH9CZ7UCc=">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</latexit>

known

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>
slide-22
SLIDE 22

unknown known

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

v1 ∈ V

<latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit><latexit sha1_base64="J0l8cFqPvc3biR8NuFdbWLFrcKA=">AB/HicdVDLSsNAFJ34rPUV7dLNYBFclSGtu4KbnRXwT6gCWUynbRDJ5MwMymEUH/FjQtF3Poh7vwbJ20FT0wcDjnXu6ZEySMSmVZH8ba+sbm1nZp7y7t39waB4d2WcCkw6OGax6AdIEkY56SiqGOkngqAoYKQXTK8KvzcjQtKY36ksIX6ExpyGFCOlpaFZmQ1t6FEOvQipCUYs786HZtWqXTbrjluHVs2yGrZjF8RpuBcutLVSoApWaA/Nd28U4zQiXGpBzYVqL8HAlFMSPzspdKkiA8RWMy0JSjiEg/X4SfwzOtjGAYC/24gv1+0aOIimzKNCTRUT52yvEv7xBqsKmn1OepIpwvDwUpgyqGBZNwBEVBCuWaYKwoDorxBMkEFa6r7Iu4eun8H/SdWq2VbNv3WrZlVHCZyAU3AObNALXAN2qADMjA3gCz8a98Wi8GK/L0TVjtVMBP2C8fQJ2uJSv</latexit>

4-dimensional signal subspace V

<latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit><latexit sha1_base64="Gou/+BdLalt1pM2a+Bo4i0QplWQ=">ACFHicdVDLSgMxFM3UV62vUZdugq0giCUpYtdwY3uKtgHtEPJZDJtaOZBkhHK0I9w46+4caGIWxfu/BszbQUVvRByOc+jxsLrjRCH1ZuaXldS2/XtjY3NresXf32ipKJGUtGolIdl2imOAha2muBevGkpHAFazji8yvXPLpOJReKMnMXMCMgy5zynRhrYJ2enHg9YmCUQARUfzr7EVTGhDJb6AdEjSkTanpYGdhGVEUIY5gBXD1HBtTrtQquQZxJopgEc2B/d73IpqY9poKolQPo1g7KZGaU8GmhX6imBkzJkPWMzAkAVNOjtqCo8M40E/kuaFGs7Y7xUpCZSaBK7JzHZUv7WM/EvrJdqvOSkP40SzkM4H+YmAOoKZQ9DjklEtJgYQKrnZFdIRkYRq42PBmPB1KfwftCtljMr4ulJsXC3syIMDcAiOAQZV0ACXoAlagI78ACewLN1bz1aL9brPDVnLWr2wY+w3j4BjCKeiA=</latexit>
  • 1
  • 0.5

0.5 1 1.5 2 4 6 8 10 12 14 16 18 20

ρ1π1(v1)

<latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit><latexit sha1_base64="iK8c63OZ+sKpA/2ePHqOIA6uKg=">AB+nicdVDLSgMxFM3UV62vqS7dBItQNyUpYtdwY3uKtgHdIYhk2ba0MyDJFMptZ/ixoUibv0Sd/6NmbaCih64cDjnXu69x08EVxqhDyu3tr6xuZXfLuzs7u0f2MXDjopTSVmbxiKWPZ8oJnjE2prwXqJZCT0Bev648vM706YVDyObvU0YW5IhEPOCXaSJ5dOQo9jB0Eu7h8sTDZ5dQhWEMYZgTXLpAhjUa9iusQZ5ZBCazQ8ux3ZxDTNGSRpoIo1co0e6MSM2pYPOCkyqWEDomQ9Y3NCIhU+5scfocnhplAINYmo0XKjfJ2YkVGoa+qYzJHqkfnuZ+JfXT3VQd2c8SlLNIrpcFKQC6hmOcABl4xqMTWEUMnNrZCOiCRUm7QKJoSvT+H/pFOtYFTBN+el5vUqjw4BiegDCogSa4Ai3QBhTcgQfwBJ6te+vRerFel605azVzBH7AevsEtcGTAw=</latexit>

Unlabeled Sensing

Corollary HSP

dim(V) ≤ rank(ρ1) 2

<latexit sha1_base64="Dw56kH60nCszXwfITbzgD+8QaKs=">ACHicdVBNa9tAFwlTeo6X056zGWpCTgXI8nGTm6BXNqbA/EHWMas1k/x4tVK7D6VGKEfkv+Si45tJRcij03SVONCGZGBhmHmPtzNhKoVB1/3jrK1/2Nj8WPlU3dre2d2r7R8MTJpDn2eyESPQmZACgV9FChlGpgcShGC7OS3/4HbQRibrEZQqTmF0pEQnO0ErTWiuYibgRxAznMl8UBzTQAINIs14HiBcY6ZWhSNQM+TqXdc5H4xrdXd5ulJx293qNt03a7neyXxu+1Wm3pWKVEnK/SmtcdglvAsBoVcMmPGnpviJGcaBZdQVIPMQMr4gl3B2FLFYjCT/ClcQY+sMqNRou1TSJ/UfzdyFhuzjEM7WaYwr71SfMsbZxidTHKh0gxB8edDUSYpJrRsis6EBo5yaQnjWti/Uj5nthe0fVZtCS9J6ftk4Dc9t+ldtOtn31Z1VMgh+UIaxCNdcka+kh7pE05uyB35QX46t8698t5eB5dc1Y7n8l/cH7/BVYQoi8=</latexit><latexit sha1_base64="Dw56kH60nCszXwfITbzgD+8QaKs=">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</latexit><latexit sha1_base64="Dw56kH60nCszXwfITbzgD+8QaKs=">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</latexit><latexit sha1_base64="Dw56kH60nCszXwfITbzgD+8QaKs=">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</latexit>

reconstruction? unknown

ρ1π1

<latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit><latexit sha1_base64="X6Z61pfk0eUwdlUf978PtplBgl8=">AB8nicdVDLSsNAFJ3UV62vqks3g0VwVTIxtHVXcKO7CvYBSQiT6aQdOsmEmYlQSj/DjQtF3Po17vwbJ20FT1w4XDOvdx7T5RxprRtf1iltfWNza3ydmVnd2/oHp41FMil4R2ieBCDiKsKGcp7WqmOR1kuIk4rQfTa4Kv39PpWIivdPTjAYJHqUsZgRrI3m+HIsQ+RkLUVit2fXLVsNxG9Cu23YTOagTtO9cCEySoEaWKETVt/9oSB5QlNOFbKQ3amgxmWmhFO5xU/VzTDZIJH1DM0xQlVwWx8hyeGWUIYyFNpRou1O8TM5woNU0i05lgPVa/vUL8y/NyHbeCGUuzXNOULBfFOYdawOJ/OGSEs2nhmAimbkVkjGWmGiTUsWE8PUp/J/0nDqy6+jWrbVvVnGUwQk4BecAgSZog2vQAV1AgAP4Ak8W9p6tF6s12VryVrNHIMfsN4+ASAokSs=</latexit>

Reconstruct the original signal from its shuffled and downsampled measurements.

known

slide-23
SLIDE 23

V: generic linear subspace of Rm

<latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit><latexit sha1_base64="7JK7SQ2OWOY/ftCbkeOMwYw+8wM=">ACIXicbVC7TsMwFHV4lvIKMLJYtEhMVdKFiqkSC2wF0YfUhspxb1qrjhPZDlIV9VdY+BUWBhDqhvgZnDYDtBzJ0tE59/re/yYM6Ud58taW9/Y3Nou7BR39/YPDu2j45aKEkmhSMeyY5PFHAmoKmZ5tCJZDQ59D2x9eZ34CqVgkHvQkBi8kQ8ECRok2Ut+ulXsh0SNKeNqalq/wEARIRnH2H5FYJb6KCQUcBXhR6fvp/fQxLPftklNx5sCrxM1JCeVo9O1ZbxDRJAShKSdKdV0n1l5KpGaUw7TYSxSYUWMyhK6hgoSgvHR+4RSfG2WAg0iaJzSeq787UhIqNQl9U5ktqZa9TPzP6yY6qHkpE3GiQdDFoCDhWEc4iwsPmASq+cQiUzu2I6IpJQbUItmhDc5ZNXSatacZ2Ke1ct1W/zOAroFJ2hC+SiS1RHN6iBmoiZ/SK3tGH9WK9WZ/WbFG6ZuU9J+gPrO8foIqjMA=</latexit>

T : a finite set of linear transformations τ : Rm → Rm

<latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit><latexit sha1_base64="t/fn4mLxRczszU2Sbzx0JH6DlzE=">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</latexit>

homomorphic sensing property v1 6= v2 2 V ) τ1(v1) 6= τ2(v2), 8τ1, τ2 2 T

<latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit><latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit><latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit><latexit sha1_base64="U6ayeUH5gE1NLibQr+/B+BjcAbU=">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</latexit>

Homomorphic Sensing

HSP

“Under what conditions on dim(V) can we uniquely reconstruct (sense) points in V from their (homomorphic) images under the transformations in T ?”

<latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit><latexit sha1_base64="1EqtWTgqC+NZwscmJR8m+AxQ1wQ=">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</latexit>

Theorem HSP

dim(V) ≤ min n1 2rank(τ), codim(U)

  • <latexit sha1_base64="59ofaMlsqjDvHPo/Sis3IXYsOXI=">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</latexit><latexit sha1_base64="59ofaMlsqjDvHPo/Sis3IXYsOXI=">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</latexit><latexit sha1_base64="59ofaMlsqjDvHPo/Sis3IXYsOXI=">ACRnicdVBNbxMxEJ0NHy3hK4UjF4sIKZVQtLtETXur4AK3IpG0UryKvI43tWJ7V/Zs1cjaX9cLZ278BC4cQIgr3iaID8GTLD29mXkzfnmlpM4/h1bty8dXtn90737r37Dx729h5NXVlbLia8VKU9y5kTShoxQYlKnFVWMJ0rcZqvXrX10wthnSzNO1xXItNsaWQhOcMgzXsZXUg9oJrhOWfKT5t9QpUgVEtDX8ol9bSwjPuk8WlDUVyit8ysmgFVu8/3yi8DB7NL5NJa9ION/NePx4eHR6kowMSD+N4nKRJS9Lx6MWIJEFp0YctTua9D3R8loLg1wx52ZJXGHmUXJlWi6tHaiYnzFlmIWqGFauMxfx9CQZ0FZkK04Rk1+rvE5p59Y6D53tqe7vWiv+qzarsTjMvDRVjcLwzaKiVgRL0mZKFtIKjmodCONWhlsJP2chNwzJd0MIP39K/k+m6TCJh8nbUf/4zTaOXgCT2EACYzhGF7DCUyAwxV8gi/wNXofY6+Rd83rZ1oO/MY/kAHfgBQt7Ln</latexit><latexit sha1_base64="59ofaMlsqjDvHPo/Sis3IXYsOXI=">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</latexit>