Homomorphic Sensing
Manolis C. T
sakiris Liangzu Peng ShanghaiT
ech University
Homomorphic Sensing Manolis C. T sakiris Liangzu Peng - - PowerPoint PPT Presentation
Homomorphic Sensing Manolis C. T sakiris Liangzu Peng ShanghaiT ech University <latexit
Manolis C. T
ech University
[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.
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
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>0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>unknown known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>known
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>unknown known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>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
0.5 1 1.5 2 4 6 8 10 12 14 16 18 20
unknown known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>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
unknown known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>permutations
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
known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>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
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
known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>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
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
(determinantal variety)
(determinantal variety)
(quasi-variety)
(determinantal variety)
(quasi-variety)
(determinantal variety)
unknown known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 2 4 6 8 10 12 14 16 18 20
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=">ACKXicdVDLSgMxFM34rPVdekm2ApuLEkR2+4KbnRXxT6grSWT3rahmQdJRihDf8eNv+JGQVG3/oiZtoKXg5nHMv597jhlJoQ8ibs7C4tLymlpLr29sbm1ndnbrOogUhxoPZKCaLtMghQ81I4yEZqiAea6Ehjs6S/TGLSgtAv/ajEPoeGzgi7gzFiqm6mcHPeEB37SwCTWYjD9IleHjAPOtT1mhpzJuD7B7YQGg6ec68ZXkxsv181kSZ4QinFCaDFU2JBuVwq0BKmiWQri+ZV7Wae2r2AR9bTcMm0blESmk7MlBFcwiTdjRY7xEbQMtCn3mgO/H0gk+tEwP9wNln2/wlP0+ETNP67Hn2s5kSf1bS8i/tFZk+qVOLPwMuDzmVE/ktgEOIkN94QCbuTYAsaVsLtiPmSKcWPDTdsQvi7F/4N6IU9Jnl4WspWLeRwptI8O0BGiqIgq6BxVUQ1xdIce0DN6ce6dR+fVeZ+1LjzmT30o5yPT3pKp2s=</latexit><latexit sha1_base64="IT2/TZjr4d7iWM8aipH9CZ7UCc=">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</latexit><latexit sha1_base64="IT2/TZjr4d7iWM8aipH9CZ7UCc=">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</latexit>known
unknown known
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 2 4 6 8 10 12 14 16 18 20
0.5 1 1.5 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>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>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>known