Generative models as data-driven priors: how to learn them e ffi - - PowerPoint PPT Presentation

generative models as data driven priors how to learn them
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Generative models as data-driven priors: how to learn them e ffi - - PowerPoint PPT Presentation

<latexit


slide-1
SLIDE 1

Generative models as data-driven priors: how to learn them efficiently?

Vincent Schellekens & Laurent Jacques

1

...

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xi

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P∗

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b PX

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X

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θ

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b Pθ

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b PZ

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Pz

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A

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A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

'

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zX

A( b Pθ)

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UCLouvain

slide-2
SLIDE 2

2

Motivation: inverse problems

y = Φ(x∗) + w

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x∗

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Signal of interest

Measurements

y

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Forward model

slide-3
SLIDE 3

3

Motivation: inverse problems

y = Φ(x∗) + w

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b x = arg min

x

l(x; y) + λr(x)

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Variational formulation

x∗

<latexit sha1_base64="BPHaAhzJsUHTQ2CuHwCs2Ub0LJE=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRIiKFKChKMFSyMRaIPqQ2V4zitVceJbAdRon4JCwMIsfIpbPwNbpsBWo5k+eice+Xj4yecKe0431ZhZXVtfaO4Wdra3tkt23v7LRWnktAmiXksOz5WlDNBm5pTjuJpDjyOW37o+up36gUrFY3OlxQr0IDwQLGcHaSH273PNjHqhxZC70eH/atytO1ZkBLRM3JxXI0ejbX70gJmlEhSYcK9V1nUR7GZaEU4npV6qaILJCA9o1CBI6q8bBZ8go6NEqAwluYIjWbq740MR2qazUxGWA/VojcV/O6qQ4vYyJNVUkPlDYcqRjtG0BRQwSYnmY0MwkcxkRWSIJSbadFUyJbiLX14mrVrVPavWbs8r9au8jiIcwhGcgAsXUIcbaEATCKTwDK/wZj1ZL9a79TEfLVj5zgH8gfX5A3zgkvg=</latexit>

Signal of interest

Measurements

y

<latexit sha1_base64="MJOf/whPwuBDVu9ufkLIT2hyj1o=">AB9HicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LoxmUF+4B2KJlMpg3NJGOSKQyl3+HGhSJu/Rh3/o2ZdhbaeiDkcM695OQECWfauO63s7a+sbm1Xdop7+7tHxWjo7bWqaK0BaRXKpugDXlTNCWYbTbqIojgNO8H4Lvc7E6o0k+LRZAn1YzwULGIEGyv5/UDyUGexvVA2qFTdmjsHWiVeQapQoDmofPVDSdKYCkM41rnuYnxp1gZRjidlfupgkmYzykPUsFjqn2p/PQM3RulRBFUtkjDJqrvzemONZ5MjsZYzPSy14u/uf1UhPd+FMmktRQRYPRSlHRqK8ARQyRYnhmSWYKGazIjLChNjeyrbErzlL6+Sdr3mXdbqD1fVxm1RwlO4QwuwINraMA9NKEFBJ7gGV7hzZk4L86787EYXOKnRP4A+fzB+PDkiw=</latexit>

Forward model

slide-4
SLIDE 4

4

Motivation: inverse problems

y = Φ(x∗) + w

<latexit sha1_base64="kxdayRVYLlbhqHCJZbwEvlOdPE4=">ACInicbVDLSgMxFM3UV62vqks3wSJUhTJTBXUhFN24rGAf0BlLJpNpQzOZIcmoZhvceOvuHGhqCvBjzF9LPrwQMjhnHu59x43YlQq0/wxMguLS8sr2dXc2vrG5lZ+e6cuw1hgUsMhC0XTRZIwyklNUcVIMxIEBS4jDbd3PfAbD0RIGvI71Y+IE6AOpz7FSGmpnb+w3ZB5sh/oL+mn8BLa1S4twkn5Kb0/OoTHU9pj2s4XzJI5BJwn1pgUwBjVdv7L9kIcB4QrzJCULcuMlJMgoShmJM3ZsSQRwj3UIS1NOQqIdJLhiSk80IoH/VDoxUcqpMdCQrkYDVdGSDVlbPeQPzPa8XKP3cSyqNYEY5Hg/yYQRXCQV7Qo4JgxfqaICyo3hXiLhIK51qTodgzZ48T+rlknVSKt+eFipX4ziyYA/sgyKwBmogBtQBTWAwTN4Be/gw3gx3oxP43tUmjHGPbtgCsbvH5LypFA=</latexit>

b x = arg min

x

l(x; y) + λr(x)

<latexit sha1_base64="zW2stFTOLhB5WS9OiDvOnQGfKjQ=">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</latexit>

Variational formulation

l(x; y) = ky Φ(x)k2

2

<latexit sha1_base64="LEjX/1hoUaJ3m+FkNDT+3joVtcs=">ACPXicbVDLSgMxFM3UV62vqks3F4vQLiwzVAQoejGZYW+oFNLJk3b0MyDJCOWsT/mxn9w586NC0XcujXTzsK2Xg5Ofcu9xAs6kMs1XI7W0vLK6l7PbGxube9kd/fq0g8FoTXic180HSwpZx6tKaY4bQaCYtfhtOEMr+N+454KyXyvqkYBbu47EeI1hpqpOt8jzYjs+7cuTqCx7gYuY9AigAXIL9OEcfg10ZsPyMt6BlndJdqZPNmUVzUrAIrATkUFKVTvbF7vokdKmnCMdStiwzUO0IC8UIp+OMHUoaYDLEfdrS0Mule1osv0YjThZ4v9PEUTNi/jgi7Mp5QK12sBnK+F5P/9Vqh6p23I+YFoaIemX7UCzkoH+IocsEJYqPNMBEMD0rkAEWmCgdeEaHYM2vAjqpaJ1UizdnubKV0kcaXSADlEeWegMldENqAaIugJvaEP9Gk8G+/Gl/E9laMxLOPZsr4+QUVBawm</latexit>

x∗

<latexit sha1_base64="BPHaAhzJsUHTQ2CuHwCs2Ub0LJE=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRIiKFKChKMFSyMRaIPqQ2V4zitVceJbAdRon4JCwMIsfIpbPwNbpsBWo5k+eice+Xj4yecKe0431ZhZXVtfaO4Wdra3tkt23v7LRWnktAmiXksOz5WlDNBm5pTjuJpDjyOW37o+up36gUrFY3OlxQr0IDwQLGcHaSH273PNjHqhxZC70eH/atytO1ZkBLRM3JxXI0ejbX70gJmlEhSYcK9V1nUR7GZaEU4npV6qaILJCA9o1CBI6q8bBZ8go6NEqAwluYIjWbq740MR2qazUxGWA/VojcV/O6qQ4vYyJNVUkPlDYcqRjtG0BRQwSYnmY0MwkcxkRWSIJSbadFUyJbiLX14mrVrVPavWbs8r9au8jiIcwhGcgAsXUIcbaEATCKTwDK/wZj1ZL9a79TEfLVj5zgH8gfX5A3zgkvg=</latexit>

Signal of interest

Measurements

y

<latexit sha1_base64="MJOf/whPwuBDVu9ufkLIT2hyj1o=">AB9HicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LoxmUF+4B2KJlMpg3NJGOSKQyl3+HGhSJu/Rh3/o2ZdhbaeiDkcM695OQECWfauO63s7a+sbm1Xdop7+7tHxWjo7bWqaK0BaRXKpugDXlTNCWYbTbqIojgNO8H4Lvc7E6o0k+LRZAn1YzwULGIEGyv5/UDyUGexvVA2qFTdmjsHWiVeQapQoDmofPVDSdKYCkM41rnuYnxp1gZRjidlfupgkmYzykPUsFjqn2p/PQM3RulRBFUtkjDJqrvzemONZ5MjsZYzPSy14u/uf1UhPd+FMmktRQRYPRSlHRqK8ARQyRYnhmSWYKGazIjLChNjeyrbErzlL6+Sdr3mXdbqD1fVxm1RwlO4QwuwINraMA9NKEFBJ7gGV7hzZk4L86787EYXOKnRP4A+fzB+PDkiw=</latexit>

Data-fidelity loss

  • Euclidean

Forward model

slide-5
SLIDE 5

5

Motivation: inverse problems

Regularization (= prior)

y = Φ(x∗) + w

<latexit sha1_base64="kxdayRVYLlbhqHCJZbwEvlOdPE4=">ACInicbVDLSgMxFM3UV62vqks3wSJUhTJTBXUhFN24rGAf0BlLJpNpQzOZIcmoZhvceOvuHGhqCvBjzF9LPrwQMjhnHu59x43YlQq0/wxMguLS8sr2dXc2vrG5lZ+e6cuw1hgUsMhC0XTRZIwyklNUcVIMxIEBS4jDbd3PfAbD0RIGvI71Y+IE6AOpz7FSGmpnb+w3ZB5sh/oL+mn8BLa1S4twkn5Kb0/OoTHU9pj2s4XzJI5BJwn1pgUwBjVdv7L9kIcB4QrzJCULcuMlJMgoShmJM3ZsSQRwj3UIS1NOQqIdJLhiSk80IoH/VDoxUcqpMdCQrkYDVdGSDVlbPeQPzPa8XKP3cSyqNYEY5Hg/yYQRXCQV7Qo4JgxfqaICyo3hXiLhIK51qTodgzZ48T+rlknVSKt+eFipX4ziyYA/sgyKwBmogBtQBTWAwTN4Be/gw3gx3oxP43tUmjHGPbtgCsbvH5LypFA=</latexit>

b x = arg min

x

l(x; y) + λr(x)

<latexit sha1_base64="zW2stFTOLhB5WS9OiDvOnQGfKjQ=">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</latexit>
  • Tikhonov
  • Sparsity (in wavelets)
  • TV-norm
  • Sparsity (in dictionary)
  • Generative priors

Variational formulation

l(x; y) = ky Φ(x)k2

2

<latexit sha1_base64="LEjX/1hoUaJ3m+FkNDT+3joVtcs=">ACPXicbVDLSgMxFM3UV62vqks3F4vQLiwzVAQoejGZYW+oFNLJk3b0MyDJCOWsT/mxn9w586NC0XcujXTzsK2Xg5Ofcu9xAs6kMs1XI7W0vLK6l7PbGxube9kd/fq0g8FoTXic180HSwpZx6tKaY4bQaCYtfhtOEMr+N+454KyXyvqkYBbu47EeI1hpqpOt8jzYjs+7cuTqCx7gYuY9AigAXIL9OEcfg10ZsPyMt6BlndJdqZPNmUVzUrAIrATkUFKVTvbF7vokdKmnCMdStiwzUO0IC8UIp+OMHUoaYDLEfdrS0Mule1osv0YjThZ4v9PEUTNi/jgi7Mp5QK12sBnK+F5P/9Vqh6p23I+YFoaIemX7UCzkoH+IocsEJYqPNMBEMD0rkAEWmCgdeEaHYM2vAjqpaJ1UizdnubKV0kcaXSADlEeWegMldENqAaIugJvaEP9Gk8G+/Gl/E9laMxLOPZsr4+QUVBawm</latexit>

r(x) = kxk2

2

<latexit sha1_base64="CnHzqRWZFcA7at25lNLhi4dY464=">ACEXicbVC7TsMwFHV4lvIKMLJYVEhlqZKABAtSBQtjkehDakLkOE5r1XnIdhBV2l9g4VdYGECIlY2Nv8FpM9CWI1k+Oude3XuPlzAqpGH8aEvLK6tr6WN8ubW9s6uvrfEnHKMWnimMW84yFBGI1IU1LJSCfhBIUeI21vcJ37QfCBY2jOzlMiBOiXkQDipFUkqtXedX2YuaLYag+HgCL6E9mpHskWvdW65eMWrGBHCRmAWpgAINV/+2/RinIYkZkiIrmk0skQlxQzMi7bqSAJwgPUI1FIxQS4WSTi8bwWCk+DGKuXiThRP3bkaFQ5PupyhDJvpj3cvE/r5vK4MLJaJSkR4OihIGZQxzOBPuUESzZUBGFO1a4Q9xFHWKoQyoEc/7kRdKyauZpzbo9q9SvijhK4BAcgSowTmogxvQAE2AwRN4AW/gXvWXrUP7XNauqQVPQdgBtrXL0IynKY=</latexit>

r(x) = kxkT V

<latexit sha1_base64="vlC4U5myq8BhqnzdOl4FEZ7oWM=">ACEnicbVDLSsNAFJ3UV62vqEs3g0VoNyWpgm6EohuXFfqCJoTJZNIOnTyYmYgl7Te48VfcuFDErSt3/o2TNgvbemCYwzn3cu89bsyokIbxoxXW1jc2t4rbpZ3dvf0D/fCoI6KEY9LGEYt4z0WCMBqStqSkV7MCQpcRru6Dbzuw+ECxqFLTmOiR2gQUh9ipFUkqNXecVyI+aJcaA+FiF19CaLEjWxElbnamjl42aMQNcJWZOyiBH09G/LS/CSUBCiRkSom8asbRTxCXFjExLViJIjPAIDUhf0RAFRNjp7KQpPFOKB/2IqxdKOFP/dqQoENmCqjJAciWvUz8z+sn0r+yUxrGiSQhng/yEwZlBLN8oEc5wZKNFUGYU7UrxEPEZYqxZIKwVw+eZV06jXzvFa/vyg3bvI4iuAEnIKMElaIA70ARtgMETeAFv4F171l61D+1zXlrQ8p5jsADt6xfKZJ2Q</latexit>

r(x) = kΨxk1

<latexit sha1_base64="eHaOW+zfVxRyb1uzHuCMHKJuL8=">ACFHicbVDLSsNAFJ34rPUVdelmsAgVoSRV0I1QdOygn1AE8JkMmHTiZhZiKWtB/hxl9x40IRty7c+TdO2i5s64FhDufcy73+AmjUlnWj7G0vLK6tl7YKG5ube/smnv7TRmnApMGjlks2j6ShFOGoqRtqJICjyGWn5/Zvcbz0QIWnM79UgIW6EupyGFCOlJc8FWXHj1kgB5H+4OMJvILO0KlLCmd0Z+jZnlmyKtYcJHYU1ICU9Q989sJYpxGhCvMkJQd20qUmyGhKGZkVHRSRKE+6hLOpyFBHpZuOjRvBYKwEMY6EfV3Cs/u3IUCTz7XRlhFRPznu5+J/XSV46WaUJ6kiHE8GhSmDKoZ5QjCgmDFBpogLKjeFeIeEgrnWNRh2DPn7xImtWKfVap3p2XatfTOArgEByBMrDBaiBW1AHDYDBE3gBb+DdeDZejQ/jc1K6ZEx7DsAMjK9fgT+d2w=</latexit>

r(x) = kDxk1

<latexit sha1_base64="RdijtjQI3hx0UyftyGnfi06FBTg=">ACEXicbVC7TsMwFHV4lvIKMLJYVEhlqZKCBAtSBQyMRaIPqYkix3Fbq4T2Q6iSvsLPwKCwMIsbKx8Tc4bQbaciTLR+fcq3v8WNGpbKsH2NpeWV1b2wUdzc2t7ZNf2mzJKBCYNHLFItH0kCaOcNBRVjLRjQVDoM9LyB9eZ3ogQtKI36thTNwQ9TjtUoyUljyzLMqOH7FADkP9wcTeAmd0Q2cEZ2RZ3tmyapYE8BFYuekBHLUPfPbCSKchIQrzJCUHduKlZsioShmZFx0EklihAeoRzqachQS6aTi8bwWCsB7EZCP67gRP3bkaJQZtvpyhCpvpz3MvE/r5Oo7oWbUh4ninA8HdRNGFQRzOKBARUEKzbUBGFB9a4Q95FAWOkQizoEe/7kRdKsVuzTSvXurFS7yuMogENwBMrABuegBm5BHTQABk/gBbyBd+PZeDU+jM9p6ZKR9xyAGRhfv/dRnHk=</latexit>

x∗

<latexit sha1_base64="BPHaAhzJsUHTQ2CuHwCs2Ub0LJE=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRIiKFKChKMFSyMRaIPqQ2V4zitVceJbAdRon4JCwMIsfIpbPwNbpsBWo5k+eice+Xj4yecKe0431ZhZXVtfaO4Wdra3tkt23v7LRWnktAmiXksOz5WlDNBm5pTjuJpDjyOW37o+up36gUrFY3OlxQr0IDwQLGcHaSH273PNjHqhxZC70eH/atytO1ZkBLRM3JxXI0ejbX70gJmlEhSYcK9V1nUR7GZaEU4npV6qaILJCA9o1CBI6q8bBZ8go6NEqAwluYIjWbq740MR2qazUxGWA/VojcV/O6qQ4vYyJNVUkPlDYcqRjtG0BRQwSYnmY0MwkcxkRWSIJSbadFUyJbiLX14mrVrVPavWbs8r9au8jiIcwhGcgAsXUIcbaEATCKTwDK/wZj1ZL9a79TEfLVj5zgH8gfX5A3zgkvg=</latexit>

Signal of interest

Measurements

y

<latexit sha1_base64="MJOf/whPwuBDVu9ufkLIT2hyj1o=">AB9HicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LoxmUF+4B2KJlMpg3NJGOSKQyl3+HGhSJu/Rh3/o2ZdhbaeiDkcM695OQECWfauO63s7a+sbm1Xdop7+7tHxWjo7bWqaK0BaRXKpugDXlTNCWYbTbqIojgNO8H4Lvc7E6o0k+LRZAn1YzwULGIEGyv5/UDyUGexvVA2qFTdmjsHWiVeQapQoDmofPVDSdKYCkM41rnuYnxp1gZRjidlfupgkmYzykPUsFjqn2p/PQM3RulRBFUtkjDJqrvzemONZ5MjsZYzPSy14u/uf1UhPd+FMmktRQRYPRSlHRqK8ARQyRYnhmSWYKGazIjLChNjeyrbErzlL6+Sdr3mXdbqD1fVxm1RwlO4QwuwINraMA9NKEFBJ7gGV7hzZk4L86787EYXOKnRP4A+fzB+PDkiw=</latexit>

Data-fidelity loss

  • Euclidean

Forward model

slide-6
SLIDE 6

6

Motivation: inverse problems

Regularization (= prior)

y = Φ(x∗) + w

<latexit sha1_base64="kxdayRVYLlbhqHCJZbwEvlOdPE4=">ACInicbVDLSgMxFM3UV62vqks3wSJUhTJTBXUhFN24rGAf0BlLJpNpQzOZIcmoZhvceOvuHGhqCvBjzF9LPrwQMjhnHu59x43YlQq0/wxMguLS8sr2dXc2vrG5lZ+e6cuw1hgUsMhC0XTRZIwyklNUcVIMxIEBS4jDbd3PfAbD0RIGvI71Y+IE6AOpz7FSGmpnb+w3ZB5sh/oL+mn8BLa1S4twkn5Kb0/OoTHU9pj2s4XzJI5BJwn1pgUwBjVdv7L9kIcB4QrzJCULcuMlJMgoShmJM3ZsSQRwj3UIS1NOQqIdJLhiSk80IoH/VDoxUcqpMdCQrkYDVdGSDVlbPeQPzPa8XKP3cSyqNYEY5Hg/yYQRXCQV7Qo4JgxfqaICyo3hXiLhIK51qTodgzZ48T+rlknVSKt+eFipX4ziyYA/sgyKwBmogBtQBTWAwTN4Be/gw3gx3oxP43tUmjHGPbtgCsbvH5LypFA=</latexit>

b x = arg min

x

l(x; y) + λr(x)

<latexit sha1_base64="zW2stFTOLhB5WS9OiDvOnQGfKjQ=">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</latexit>
  • Tikhonov
  • Sparsity (in wavelets)
  • TV-norm
  • Sparsity (in dictionary)
  • Generative priors

From model-based to data-driven priors!

Variational formulation

l(x; y) = ky Φ(x)k2

2

<latexit sha1_base64="LEjX/1hoUaJ3m+FkNDT+3joVtcs=">ACPXicbVDLSgMxFM3UV62vqks3F4vQLiwzVAQoejGZYW+oFNLJk3b0MyDJCOWsT/mxn9w586NC0XcujXTzsK2Xg5Ofcu9xAs6kMs1XI7W0vLK6l7PbGxube9kd/fq0g8FoTXic180HSwpZx6tKaY4bQaCYtfhtOEMr+N+454KyXyvqkYBbu47EeI1hpqpOt8jzYjs+7cuTqCx7gYuY9AigAXIL9OEcfg10ZsPyMt6BlndJdqZPNmUVzUrAIrATkUFKVTvbF7vokdKmnCMdStiwzUO0IC8UIp+OMHUoaYDLEfdrS0Mule1osv0YjThZ4v9PEUTNi/jgi7Mp5QK12sBnK+F5P/9Vqh6p23I+YFoaIemX7UCzkoH+IocsEJYqPNMBEMD0rkAEWmCgdeEaHYM2vAjqpaJ1UizdnubKV0kcaXSADlEeWegMldENqAaIugJvaEP9Gk8G+/Gl/E9laMxLOPZsr4+QUVBawm</latexit>

r(x) = kxk2

2

<latexit sha1_base64="CnHzqRWZFcA7at25lNLhi4dY464=">ACEXicbVC7TsMwFHV4lvIKMLJYVEhlqZKABAtSBQtjkehDakLkOE5r1XnIdhBV2l9g4VdYGECIlY2Nv8FpM9CWI1k+Oude3XuPlzAqpGH8aEvLK6tr6WN8ubW9s6uvrfEnHKMWnimMW84yFBGI1IU1LJSCfhBIUeI21vcJ37QfCBY2jOzlMiBOiXkQDipFUkqtXedX2YuaLYag+HgCL6E9mpHskWvdW65eMWrGBHCRmAWpgAINV/+2/RinIYkZkiIrmk0skQlxQzMi7bqSAJwgPUI1FIxQS4WSTi8bwWCk+DGKuXiThRP3bkaFQ5PupyhDJvpj3cvE/r5vK4MLJaJSkR4OihIGZQxzOBPuUESzZUBGFO1a4Q9xFHWKoQyoEc/7kRdKyauZpzbo9q9SvijhK4BAcgSowTmogxvQAE2AwRN4AW/gXvWXrUP7XNauqQVPQdgBtrXL0IynKY=</latexit>

r(x) = kxkT V

<latexit sha1_base64="vlC4U5myq8BhqnzdOl4FEZ7oWM=">ACEnicbVDLSsNAFJ3UV62vqEs3g0VoNyWpgm6EohuXFfqCJoTJZNIOnTyYmYgl7Te48VfcuFDErSt3/o2TNgvbemCYwzn3cu89bsyokIbxoxXW1jc2t4rbpZ3dvf0D/fCoI6KEY9LGEYt4z0WCMBqStqSkV7MCQpcRru6Dbzuw+ECxqFLTmOiR2gQUh9ipFUkqNXecVyI+aJcaA+FiF19CaLEjWxElbnamjl42aMQNcJWZOyiBH09G/LS/CSUBCiRkSom8asbRTxCXFjExLViJIjPAIDUhf0RAFRNjp7KQpPFOKB/2IqxdKOFP/dqQoENmCqjJAciWvUz8z+sn0r+yUxrGiSQhng/yEwZlBLN8oEc5wZKNFUGYU7UrxEPEZYqxZIKwVw+eZV06jXzvFa/vyg3bvI4iuAEnIKMElaIA70ARtgMETeAFv4F171l61D+1zXlrQ8p5jsADt6xfKZJ2Q</latexit>

r(x) = kΨxk1

<latexit sha1_base64="eHaOW+zfVxRyb1uzHuCMHKJuL8=">ACFHicbVDLSsNAFJ34rPUVdelmsAgVoSRV0I1QdOygn1AE8JkMmHTiZhZiKWtB/hxl9x40IRty7c+TdO2i5s64FhDufcy73+AmjUlnWj7G0vLK6tl7YKG5ube/smnv7TRmnApMGjlks2j6ShFOGoqRtqJICjyGWn5/Zvcbz0QIWnM79UgIW6EupyGFCOlJc8FWXHj1kgB5H+4OMJvILO0KlLCmd0Z+jZnlmyKtYcJHYU1ICU9Q989sJYpxGhCvMkJQd20qUmyGhKGZkVHRSRKE+6hLOpyFBHpZuOjRvBYKwEMY6EfV3Cs/u3IUCTz7XRlhFRPznu5+J/XSV46WaUJ6kiHE8GhSmDKoZ5QjCgmDFBpogLKjeFeIeEgrnWNRh2DPn7xImtWKfVap3p2XatfTOArgEByBMrDBaiBW1AHDYDBE3gBb+DdeDZejQ/jc1K6ZEx7DsAMjK9fgT+d2w=</latexit>

r(x) = kDxk1

<latexit sha1_base64="RdijtjQI3hx0UyftyGnfi06FBTg=">ACEXicbVC7TsMwFHV4lvIKMLJYVEhlqZKCBAtSBQyMRaIPqYkix3Fbq4T2Q6iSvsLPwKCwMIsbKx8Tc4bQbaciTLR+fcq3v8WNGpbKsH2NpeWV1b2wUdzc2t7ZNf2mzJKBCYNHLFItH0kCaOcNBRVjLRjQVDoM9LyB9eZ3ogQtKI36thTNwQ9TjtUoyUljyzLMqOH7FADkP9wcTeAmd0Q2cEZ2RZ3tmyapYE8BFYuekBHLUPfPbCSKchIQrzJCUHduKlZsioShmZFx0EklihAeoRzqachQS6aTi8bwWCsB7EZCP67gRP3bkaJQZtvpyhCpvpz3MvE/r5Oo7oWbUh4ninA8HdRNGFQRzOKBARUEKzbUBGFB9a4Q95FAWOkQizoEe/7kRdKsVuzTSvXurFS7yuMogENwBMrABuegBm5BHTQABk/gBbyBd+PZeDU+jM9p6ZKR9xyAGRhfv/dRnHk=</latexit>

x∗

<latexit sha1_base64="BPHaAhzJsUHTQ2CuHwCs2Ub0LJE=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRIiKFKChKMFSyMRaIPqQ2V4zitVceJbAdRon4JCwMIsfIpbPwNbpsBWo5k+eice+Xj4yecKe0431ZhZXVtfaO4Wdra3tkt23v7LRWnktAmiXksOz5WlDNBm5pTjuJpDjyOW37o+up36gUrFY3OlxQr0IDwQLGcHaSH273PNjHqhxZC70eH/atytO1ZkBLRM3JxXI0ejbX70gJmlEhSYcK9V1nUR7GZaEU4npV6qaILJCA9o1CBI6q8bBZ8go6NEqAwluYIjWbq740MR2qazUxGWA/VojcV/O6qQ4vYyJNVUkPlDYcqRjtG0BRQwSYnmY0MwkcxkRWSIJSbadFUyJbiLX14mrVrVPavWbs8r9au8jiIcwhGcgAsXUIcbaEATCKTwDK/wZj1ZL9a79TEfLVj5zgH8gfX5A3zgkvg=</latexit>

Signal of interest

Measurements

y

<latexit sha1_base64="MJOf/whPwuBDVu9ufkLIT2hyj1o=">AB9HicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LoxmUF+4B2KJlMpg3NJGOSKQyl3+HGhSJu/Rh3/o2ZdhbaeiDkcM695OQECWfauO63s7a+sbm1Xdop7+7tHxWjo7bWqaK0BaRXKpugDXlTNCWYbTbqIojgNO8H4Lvc7E6o0k+LRZAn1YzwULGIEGyv5/UDyUGexvVA2qFTdmjsHWiVeQapQoDmofPVDSdKYCkM41rnuYnxp1gZRjidlfupgkmYzykPUsFjqn2p/PQM3RulRBFUtkjDJqrvzemONZ5MjsZYzPSy14u/uf1UhPd+FMmktRQRYPRSlHRqK8ARQyRYnhmSWYKGazIjLChNjeyrbErzlL6+Sdr3mXdbqD1fVxm1RwlO4QwuwINraMA9NKEFBJ7gGV7hzZk4L86787EYXOKnRP4A+fzB+PDkiw=</latexit>

Data-fidelity loss

  • Euclidean

Forward model

slide-7
SLIDE 7

7

Motivation: inverse problems

Regularization (= prior)

y = Φ(x∗) + w

<latexit sha1_base64="kxdayRVYLlbhqHCJZbwEvlOdPE4=">ACInicbVDLSgMxFM3UV62vqks3wSJUhTJTBXUhFN24rGAf0BlLJpNpQzOZIcmoZhvceOvuHGhqCvBjzF9LPrwQMjhnHu59x43YlQq0/wxMguLS8sr2dXc2vrG5lZ+e6cuw1hgUsMhC0XTRZIwyklNUcVIMxIEBS4jDbd3PfAbD0RIGvI71Y+IE6AOpz7FSGmpnb+w3ZB5sh/oL+mn8BLa1S4twkn5Kb0/OoTHU9pj2s4XzJI5BJwn1pgUwBjVdv7L9kIcB4QrzJCULcuMlJMgoShmJM3ZsSQRwj3UIS1NOQqIdJLhiSk80IoH/VDoxUcqpMdCQrkYDVdGSDVlbPeQPzPa8XKP3cSyqNYEY5Hg/yYQRXCQV7Qo4JgxfqaICyo3hXiLhIK51qTodgzZ48T+rlknVSKt+eFipX4ziyYA/sgyKwBmogBtQBTWAwTN4Be/gw3gx3oxP43tUmjHGPbtgCsbvH5LypFA=</latexit>

b x = arg min

x

l(x; y) + λr(x)

<latexit sha1_base64="zW2stFTOLhB5WS9OiDvOnQGfKjQ=">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</latexit>
  • Tikhonov
  • Sparsity (in wavelets)
  • TV-norm
  • Sparsity (in dictionary)
  • Generative priors

From model-based to data-driven priors!

Variational formulation

l(x; y) = ky Φ(x)k2

2

<latexit sha1_base64="LEjX/1hoUaJ3m+FkNDT+3joVtcs=">ACPXicbVDLSgMxFM3UV62vqks3F4vQLiwzVAQoejGZYW+oFNLJk3b0MyDJCOWsT/mxn9w586NC0XcujXTzsK2Xg5Ofcu9xAs6kMs1XI7W0vLK6l7PbGxube9kd/fq0g8FoTXic180HSwpZx6tKaY4bQaCYtfhtOEMr+N+454KyXyvqkYBbu47EeI1hpqpOt8jzYjs+7cuTqCx7gYuY9AigAXIL9OEcfg10ZsPyMt6BlndJdqZPNmUVzUrAIrATkUFKVTvbF7vokdKmnCMdStiwzUO0IC8UIp+OMHUoaYDLEfdrS0Mule1osv0YjThZ4v9PEUTNi/jgi7Mp5QK12sBnK+F5P/9Vqh6p23I+YFoaIemX7UCzkoH+IocsEJYqPNMBEMD0rkAEWmCgdeEaHYM2vAjqpaJ1UizdnubKV0kcaXSADlEeWegMldENqAaIugJvaEP9Gk8G+/Gl/E9laMxLOPZsr4+QUVBawm</latexit>

r(x) = kxk2

2

<latexit sha1_base64="CnHzqRWZFcA7at25lNLhi4dY464=">ACEXicbVC7TsMwFHV4lvIKMLJYVEhlqZKABAtSBQtjkehDakLkOE5r1XnIdhBV2l9g4VdYGECIlY2Nv8FpM9CWI1k+Oude3XuPlzAqpGH8aEvLK6tr6WN8ubW9s6uvrfEnHKMWnimMW84yFBGI1IU1LJSCfhBIUeI21vcJ37QfCBY2jOzlMiBOiXkQDipFUkqtXedX2YuaLYag+HgCL6E9mpHskWvdW65eMWrGBHCRmAWpgAINV/+2/RinIYkZkiIrmk0skQlxQzMi7bqSAJwgPUI1FIxQS4WSTi8bwWCk+DGKuXiThRP3bkaFQ5PupyhDJvpj3cvE/r5vK4MLJaJSkR4OihIGZQxzOBPuUESzZUBGFO1a4Q9xFHWKoQyoEc/7kRdKyauZpzbo9q9SvijhK4BAcgSowTmogxvQAE2AwRN4AW/gXvWXrUP7XNauqQVPQdgBtrXL0IynKY=</latexit>

r(x) = kxkT V

<latexit sha1_base64="vlC4U5myq8BhqnzdOl4FEZ7oWM=">ACEnicbVDLSsNAFJ3UV62vqEs3g0VoNyWpgm6EohuXFfqCJoTJZNIOnTyYmYgl7Te48VfcuFDErSt3/o2TNgvbemCYwzn3cu89bsyokIbxoxXW1jc2t4rbpZ3dvf0D/fCoI6KEY9LGEYt4z0WCMBqStqSkV7MCQpcRru6Dbzuw+ECxqFLTmOiR2gQUh9ipFUkqNXecVyI+aJcaA+FiF19CaLEjWxElbnamjl42aMQNcJWZOyiBH09G/LS/CSUBCiRkSom8asbRTxCXFjExLViJIjPAIDUhf0RAFRNjp7KQpPFOKB/2IqxdKOFP/dqQoENmCqjJAciWvUz8z+sn0r+yUxrGiSQhng/yEwZlBLN8oEc5wZKNFUGYU7UrxEPEZYqxZIKwVw+eZV06jXzvFa/vyg3bvI4iuAEnIKMElaIA70ARtgMETeAFv4F171l61D+1zXlrQ8p5jsADt6xfKZJ2Q</latexit>

r(x) = kΨxk1

<latexit sha1_base64="eHaOW+zfVxRyb1uzHuCMHKJuL8=">ACFHicbVDLSsNAFJ34rPUVdelmsAgVoSRV0I1QdOygn1AE8JkMmHTiZhZiKWtB/hxl9x40IRty7c+TdO2i5s64FhDufcy73+AmjUlnWj7G0vLK6tl7YKG5ube/smnv7TRmnApMGjlks2j6ShFOGoqRtqJICjyGWn5/Zvcbz0QIWnM79UgIW6EupyGFCOlJc8FWXHj1kgB5H+4OMJvILO0KlLCmd0Z+jZnlmyKtYcJHYU1ICU9Q989sJYpxGhCvMkJQd20qUmyGhKGZkVHRSRKE+6hLOpyFBHpZuOjRvBYKwEMY6EfV3Cs/u3IUCTz7XRlhFRPznu5+J/XSV46WaUJ6kiHE8GhSmDKoZ5QjCgmDFBpogLKjeFeIeEgrnWNRh2DPn7xImtWKfVap3p2XatfTOArgEByBMrDBaiBW1AHDYDBE3gBb+DdeDZejQ/jc1K6ZEx7DsAMjK9fgT+d2w=</latexit>

r(x) = kDxk1

<latexit sha1_base64="RdijtjQI3hx0UyftyGnfi06FBTg=">ACEXicbVC7TsMwFHV4lvIKMLJYVEhlqZKCBAtSBQyMRaIPqYkix3Fbq4T2Q6iSvsLPwKCwMIsbKx8Tc4bQbaciTLR+fcq3v8WNGpbKsH2NpeWV1b2wUdzc2t7ZNf2mzJKBCYNHLFItH0kCaOcNBRVjLRjQVDoM9LyB9eZ3ogQtKI36thTNwQ9TjtUoyUljyzLMqOH7FADkP9wcTeAmd0Q2cEZ2RZ3tmyapYE8BFYuekBHLUPfPbCSKchIQrzJCUHduKlZsioShmZFx0EklihAeoRzqachQS6aTi8bwWCsB7EZCP67gRP3bkaJQZtvpyhCpvpz3MvE/r5Oo7oWbUh4ninA8HdRNGFQRzOKBARUEKzbUBGFB9a4Q95FAWOkQizoEe/7kRdKsVuzTSvXurFS7yuMogENwBMrABuegBm5BHTQABk/gBbyBd+PZeDU+jM9p6ZKR9xyAGRhfv/dRnHk=</latexit>

x∗

<latexit sha1_base64="BPHaAhzJsUHTQ2CuHwCs2Ub0LJE=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRIiKFKChKMFSyMRaIPqQ2V4zitVceJbAdRon4JCwMIsfIpbPwNbpsBWo5k+eice+Xj4yecKe0431ZhZXVtfaO4Wdra3tkt23v7LRWnktAmiXksOz5WlDNBm5pTjuJpDjyOW37o+up36gUrFY3OlxQr0IDwQLGcHaSH273PNjHqhxZC70eH/atytO1ZkBLRM3JxXI0ejbX70gJmlEhSYcK9V1nUR7GZaEU4npV6qaILJCA9o1CBI6q8bBZ8go6NEqAwluYIjWbq740MR2qazUxGWA/VojcV/O6qQ4vYyJNVUkPlDYcqRjtG0BRQwSYnmY0MwkcxkRWSIJSbadFUyJbiLX14mrVrVPavWbs8r9au8jiIcwhGcgAsXUIcbaEATCKTwDK/wZj1ZL9a79TEfLVj5zgH8gfX5A3zgkvg=</latexit>

Signal of interest

Measurements

y

<latexit sha1_base64="MJOf/whPwuBDVu9ufkLIT2hyj1o=">AB9HicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LoxmUF+4B2KJlMpg3NJGOSKQyl3+HGhSJu/Rh3/o2ZdhbaeiDkcM695OQECWfauO63s7a+sbm1Xdop7+7tHxWjo7bWqaK0BaRXKpugDXlTNCWYbTbqIojgNO8H4Lvc7E6o0k+LRZAn1YzwULGIEGyv5/UDyUGexvVA2qFTdmjsHWiVeQapQoDmofPVDSdKYCkM41rnuYnxp1gZRjidlfupgkmYzykPUsFjqn2p/PQM3RulRBFUtkjDJqrvzemONZ5MjsZYzPSy14u/uf1UhPd+FMmktRQRYPRSlHRqK8ARQyRYnhmSWYKGazIjLChNjeyrbErzlL6+Sdr3mXdbqD1fVxm1RwlO4QwuwINraMA9NKEFBJ7gGV7hzZk4L86787EYXOKnRP4A+fzB+PDkiw=</latexit>

Data-fidelity loss

  • Euclidean

Forward model

Idea: approximate true prior distribution from learning examples

slide-8
SLIDE 8

8

Generative priors in inverse problems (1)

1) Learn a generative network

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

Dataset

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

We have samples (signals)

Idea: approximate true prior distribution from learning examples

b PX = 1 N X

xi∈X

δxi

<latexit sha1_base64="J8vzyx4sK2/+tiaeHBbDkf83bQE=">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</latexit>

Empirical distribution

slide-9
SLIDE 9

9

Generative priors in inverse problems (1)

1) Learn a generative network

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

Dataset

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

We have samples (signals) from a high-dimensional distribution…

Idea: approximate true prior distribution from learning examples

b PX = 1 N X

xi∈X

δxi

<latexit sha1_base64="J8vzyx4sK2/+tiaeHBbDkf83bQE=">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</latexit>

Empirical distribution

slide-10
SLIDE 10

10

Generative priors in inverse problems (1)

1) Learn a generative network

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

Dataset

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

… and a way to generate “artificial” samples…

Gθ(z) = ρ(ΘL · ρ(ΘL−1 · · · ρ(Θ2 · ρ(Θ1 · z))))

<latexit sha1_base64="imPpbKexev6yCcD8c8N7WzdnIU=">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</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

b PX = 1 N X

xi∈X

δxi

<latexit sha1_base64="J8vzyx4sK2/+tiaeHBbDkf83bQE=">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</latexit>

Empirical distribution

Idea: approximate true prior distribution from learning examples

We have samples (signals) from a high-dimensional distribution…

Random “noise”

slide-11
SLIDE 11

11

Generative priors in inverse problems (1)

1) Learn a generative network

θ

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b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

Dataset

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

… and a way to generate “artificial” samples…

Gθ(z) = ρ(ΘL · ρ(ΘL−1 · · · ρ(Θ2 · ρ(Θ1 · z))))

<latexit sha1_base64="imPpbKexev6yCcD8c8N7WzdnIU=">ACenicbVFNSwMxEM2u3/Wr6lGEYFbxLJbBb0Iogc9FDBqtAtSzabtsHsZklmhbrsj/CvefOXePFgtq1g1Qkhj/dmMmbIBFcg+O8W/bM7Nz8wuJSaXldW29vLF5r2WqKGtTKaR6DIhmgsesDRwEe0wUI1Eg2EPwdFXoD89MaS7jOxgmrBuRfsx7nBIwlF9+9SICA0oEvYzDwYMSF71AilCPYzMk73kNXyOPTWQVe+ukP0m9mgoYrLmkduPub1lND4J9n95qba4Nr4+OWKU3dGgf8CdwIqaBItv/zmhZKmEYuBCqJ1x3US6GZEAaeC5SUv1Swh9In0WcfAmERMd7ORdTneM0yIe1KZGwMesT8rMhLpYkSTWRilf2sF+Z/WSaF31s14nKTAYjpu1EsFBomLPeCQK0ZBDA0gVHEzK6YDogFs62SMcH9/eW/4L5Rd4/rjduTysXlxI5FtI12URW56BRdoBvUQm1E0Ye1Y+1bB9anvWvX7MNxqm1NarbQVNgnX2GTvpY=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

b PX = 1 N X

xi∈X

δxi

<latexit sha1_base64="J8vzyx4sK2/+tiaeHBbDkf83bQE=">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</latexit>

Empirical distribution

Idea: approximate true prior distribution from learning examples

We have samples (signals) from a high-dimensional distribution…

Point-wise nonlinearity

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

(leaky ReLU)

Random “noise”

slide-12
SLIDE 12

12

Generative priors in inverse problems (1)

1) Learn a generative network

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

Dataset Generated samples

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

b PX = 1 N X

xi∈X

δxi

<latexit sha1_base64="J8vzyx4sK2/+tiaeHBbDkf83bQE=">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</latexit>

Empirical distribution

b Pθ = 1 Nz X

zi∈Z

δGθ(zi)

<latexit sha1_base64="qCYeWJfhliG/ZdnevZ+TsiVj1g=">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</latexit>

Idea: approximate true prior distribution from learning examples

… and a way to generate “artificial” samples…

Gθ(z) = ρ(ΘL · ρ(ΘL−1 · · · ρ(Θ2 · ρ(Θ1 · z))))

<latexit sha1_base64="imPpbKexev6yCcD8c8N7WzdnIU=">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</latexit>

We have samples (signals) from a high-dimensional distribution…

Random “noise”

slide-13
SLIDE 13

13

Generative priors in inverse problems (1)

1) Learn a generative network

...

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xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

Dataset Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Generated samples

Goal: mimic sampling from the data-generating distribution

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

(implicit manifold learning) ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

b PX = 1 N X

xi∈X

δxi

<latexit sha1_base64="J8vzyx4sK2/+tiaeHBbDkf83bQE=">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</latexit>

Empirical distribution

b Pθ = 1 Nz X

zi∈Z

δGθ(zi)

<latexit sha1_base64="qCYeWJfhliG/ZdnevZ+TsiVj1g=">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</latexit>

Idea: approximate true prior distribution from learning examples

Random “noise”

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>
slide-14
SLIDE 14

14

Generative priors in inverse problems (1)

1) Learn a generative network

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

Dataset Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Generated samples

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

Goal: mimic sampling from the data-generating distribution

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

(implicit manifold learning) ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

b PX = 1 N X

xi∈X

δxi

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Empirical distribution

b Pθ = 1 Nz X

zi∈Z

δGθ(zi)

<latexit sha1_base64="qCYeWJfhliG/ZdnevZ+TsiVj1g=">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</latexit>

Idea: approximate true prior distribution from learning examples

Random “noise”

…proxy for… '

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>
slide-15
SLIDE 15

15

Generative priors in inverse problems (2)

1) Learn a generative network 2) Use the generative network in the inverse problem

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>
slide-16
SLIDE 16

16

Generative priors in inverse problems (2)

1) Learn a generative network 2) Use the generative network in the inverse problem

e.g. reconstructed signal in the range of the network

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

Generative network

Latent space (“coefficients”)

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Reconstructed signal

b x

<latexit sha1_base64="ofrym1LSBd6VytHtmHs4S4w/mKE=">ACAHicdVDLSsNAFJ3UV62vqAsXbgaL4CokaWtcFt24rGAf0IQymUzboZMHMxO1hGz8FTcuFHrZ7jzb5y0FVT0wDCHc+7l3nv8hFEhTfNDKy0tr6yuldcrG5tb2zv67l5HxCnHpI1jFvOejwRhNCJtSUjvYQTFPqMdP3JReF3bwgXNI6u5TQhXohGER1SjKSBvqBe0sDMkYyc/2YBWIaqg/e5QO9ahpOo2Y7NjQN67RmNqyC2HWn3oCWYc5QBQu0Bvq7G8Q4DUkMUNC9C0zkV6GuKSYkbzipoIkCE/QiPQVjVBIhJfNDsjhsVICOIy5epGEM/V7R4ZCUaymKkMkx+K3V4h/ef1UDs+8jEZJKkmE54OGKYMyhkUaMKCcYMmiDMqdoV4jHiCEuVWUWF8HUp/J90bMOqGfZVvdo8X8RBofgCJwACzigCS5BC7QBjl4AE/gWbvXHrUX7XVeWtIWPfvgB7S3T9vHlzs=</latexit>

b z

<latexit sha1_base64="m6iLdZjbnXJ8MJmbgh3rau8UBNg=">ACAHicdVDLSsNAFJ3UV62vqAsXbgaL4CokaWtcFt24rGAf0IQymUzboZMHMxOlhmz8FTcuFHrZ7jzb5y0FVT0wDCHc+7l3nv8hFEhTfNDKy0tr6yuldcrG5tb2zv67l5HxCnHpI1jFvOejwRhNCJtSUjvYQTFPqMdP3JReF3bwgXNI6u5TQhXohGER1SjKSBvqBe0sDMkYyc/2YBWIaqg/e5QO9ahpOo2Y7NjQN67RmNqyC2HWn3oCWYc5QBQu0Bvq7G8Q4DUkMUNC9C0zkV6GuKSYkbzipoIkCE/QiPQVjVBIhJfNDsjhsVICOIy5epGEM/V7R4ZCUaymKkMkx+K3V4h/ef1UDs+8jEZJKkmE54OGKYMyhkUaMKCcYMmiDMqdoV4jHiCEuVWUWF8HUp/J90bMOqGfZVvdo8X8RBofgCJwACzigCS5BC7QBjl4AE/gWbvXHrUX7XVeWtIWPfvgB7S3T97Rlz0=</latexit>

b x = Gθ(b z) with b z = arg min

z

ky Φ(Gθ(z))k2

2

<latexit sha1_base64="KAfVPSMHn/hK2DVQH5tg10q7g=">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</latexit>
slide-17
SLIDE 17

17

Generative priors in inverse problems

Some examples…

slide-18
SLIDE 18

18

Generative priors in inverse problems

Some examples…

slide-19
SLIDE 19

19

Generative priors in inverse problems

Some examples…

slide-20
SLIDE 20

20

Generative priors in inverse problems

Some examples…

slide-21
SLIDE 21

21

Generative priors in inverse problems

Some examples…

slide-22
SLIDE 22

22

It’s nice! …where’s the catch?

How to learn the generative network Gθ

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

Distance??

slide-23
SLIDE 23

23

It’s nice! …where’s the catch?

How to learn the generative network Gθ

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

1) Golden standard: Generative Adversarial Networks

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

<latexit sha1_base64="BiZWCF/MYtFKY8FAVLaO+kitF4=">AB+nicbVDLSsNAFL3xWesr1aWbwSK4KkVdFnUhcsK9gFNCJPptB06mYSZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnTDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHshlhRzgRtaY57SaS4ijktBOr3O/80ClYrG415OE+hEeCjZgBGsjBXbFi7AeEczRTZB5yYhNA7vq1JwZ0DJxC1KFAs3A/vL6MUkjKjThWKme6yTaz7DUjHA6LXupogkmYzykPUMFjqjys1n0KToxSh8NYme0Gim/t7IcKTUJArNZB5ULXq5+J/XS/Xg0s+YSFJNBZkfGqQc6RjlPaA+k5RoPjE8lMVkRGWGKiTVtlU4K7+OVl0q7X3LNa/e682rgq6ijBERzDKbhwAQ24hSa0gMAjPMrvFlP1ov1bn3MR1esYucQ/sD6/AEcR5Pn</latexit>

X

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Z

<latexit sha1_base64="FO5NRjT0DapzdaQZYFw0PtqJG0U=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkUeEDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJb3ZpygH9GB5CFn1Fip/tArltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+ALoPjOI=</latexit>

Real data

L

<latexit sha1_base64="n2ZkyHRQfuxyMWApcvw1zS/q5hw=">AB8XicbVC7SgNBFL3rM8ZX1NJmMAhWYTcKWgZtLCwimAcmS5id3E2GzM4uM7NCPkLGwtFbP0bO/G2WQLTwcDjnXubcEySCa+O6387K6tr6xmZhq7i9s7u3Xzo4bOo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdJP5rSdUmsfywYwT9CM6kDzkjBorPXYjaoaMCnLXK5XdijsDWSZeTsqQo94rfX7MUsjlIYJqnXHcxPjT6gynAmcFrupxoSyER1gx1JI9T+ZJZ4Sk6t0idhrOyThszU3xsTGmk9jgI7mSXUi14m/ud1UhNe+RMuk9SgZPOPwlQE5PsfNLnCpkRY0soU9xmJWxIFWXGlS0JXiLJy+TZrXinVeq9xfl2nVeRwGO4QTOwINLqMEt1KEBDCQ8wyu8Odp5cd6dj/noipPvHMEfOJ8/EZaQhA=</latexit>

Generator

Discriminator

Distance??

slide-24
SLIDE 24

24

It’s nice! …where’s the catch?

How to learn the generative network Gθ

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

1) Golden standard: Generative Adversarial Networks

On twitter…

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

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'

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Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

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Very difficult to train (due to balancing of training discriminator/generator)

X

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Z

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Real data

L

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Generator

Discriminator

Distance??

min

θ

max

φ

L(θ, φ)

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slide-25
SLIDE 25

25

It’s nice! …where’s the catch?

How to learn the generative network Gθ

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1) Golden standard: Generative Adversarial Networks

P∗

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b PX

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<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

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'

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Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

<latexit sha1_base64="BiZWCF/MYtFKY8FAVLaO+kitF4=">AB+nicbVDLSsNAFL3xWesr1aWbwSK4KkVdFnUhcsK9gFNCJPptB06mYSZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnTDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHshlhRzgRtaY57SaS4ijktBOr3O/80ClYrG415OE+hEeCjZgBGsjBXbFi7AeEczRTZB5yYhNA7vq1JwZ0DJxC1KFAs3A/vL6MUkjKjThWKme6yTaz7DUjHA6LXupogkmYzykPUMFjqjys1n0KToxSh8NYme0Gim/t7IcKTUJArNZB5ULXq5+J/XS/Xg0s+YSFJNBZkfGqQc6RjlPaA+k5RoPjE8lMVkRGWGKiTVtlU4K7+OVl0q7X3LNa/e682rgq6ijBERzDKbhwAQ24hSa0gMAjPMrvFlP1ov1bn3MR1esYucQ/sD6/AEcR5Pn</latexit>

Very difficult to train (due to balancing of training discriminator/generator)

min

θ

max

φ

L(θ, φ)

<latexit sha1_base64="6fnrI4Xbh6A57YzD7kaGYj8dTPc=">ACGXicbVDLSgMxFM3UV62vqks3wSJUkDJTBV0W3bhwUcE+oDOUTJq2oZnMkNwRyzC/4cZfceNCEZe68m/MtF1o64HAyTn3cu89fiS4Btv+tnJLyura/n1wsbm1vZOcXevqcNYUdagoQhV2yeaCS5ZAzgI1o4UI4EvWMsfXWV+654pzUN5B+OIeQEZSN7nlICRukXbDbjsJi4MGZDUDciD+URDnmLDYUiJwDflqXuS6cfdYsmu2BPgReLMSAnNUO8WP91eSOASaCaN1x7Ai8hCjgVLC04MaRYSOyIB1DJUkYNpLJpel+MgoPdwPlXkS8ET93ZGQOtx4JvKbF0972Xif14nhv6Fl3AZxcAknQ7qxwJDiLOYcI8rRkGMDSFUcbMrpkOiCAUTZsGE4MyfvEia1YpzWqnenpVql7M48ugAHaIyctA5qFrVEcNRNEjekav6M16sl6sd+tjWpqzZj376A+srx/7SqDc</latexit>

X

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Z

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Real data

L

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Generator

Discriminator

2) Maximum Mean Discrepancy

Easier to train (no balancing)…

Distance??

min

θ

MMDκ( b PX, b Pθ)

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slide-26
SLIDE 26

26

It’s nice! …where’s the catch?

How to learn the generative network Gθ

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1) Golden standard: Generative Adversarial Networks

P∗

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b PX

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<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

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<latexit sha1_base64="BiZWCF/MYtFKY8FAVLaO+kitF4=">AB+nicbVDLSsNAFL3xWesr1aWbwSK4KkVdFnUhcsK9gFNCJPptB06mYSZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnTDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHshlhRzgRtaY57SaS4ijktBOr3O/80ClYrG415OE+hEeCjZgBGsjBXbFi7AeEczRTZB5yYhNA7vq1JwZ0DJxC1KFAs3A/vL6MUkjKjThWKme6yTaz7DUjHA6LXupogkmYzykPUMFjqjys1n0KToxSh8NYme0Gim/t7IcKTUJArNZB5ULXq5+J/XS/Xg0s+YSFJNBZkfGqQc6RjlPaA+k5RoPjE8lMVkRGWGKiTVtlU4K7+OVl0q7X3LNa/e682rgq6ijBERzDKbhwAQ24hSa0gMAjPMrvFlP1ov1bn3MR1esYucQ/sD6/AEcR5Pn</latexit>

Very difficult to train (due to balancing of training discriminator/generator)

min

θ

max

φ

L(θ, φ)

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X

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Z

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Real data

L

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Generator

Discriminator

2) Maximum Mean Discrepancy

Easier to train (no balancing)…

Distance??

X

xi∈X xj∈X

κ(xi, xj) − 2 X

xi∈X zj∈Z

κ(xi, Gθ(zj)) + X

zi∈Z zj∈Z

κ(Gθ(zi), Gθ(zj))

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min

θ

MMDκ( b PX, b Pθ)

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slide-27
SLIDE 27

27

It’s nice! …where’s the catch?

How to learn the generative network Gθ

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1) Golden standard: Generative Adversarial Networks

P∗

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b PX

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<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

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'

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Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

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Very difficult to train (due to balancing of training discriminator/generator)

min

θ

max

φ

L(θ, φ)

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X

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Z

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Real data

L

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Generator

Discriminator

2) Maximum Mean Discrepancy

Easier to train (no balancing)…

Distance??

X

xi∈X xj∈X

κ(xi, xj) − 2 X

xi∈X zj∈Z

κ(xi, Gθ(zj)) + X

zi∈Z zj∈Z

κ(Gθ(zi), Gθ(zj))

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min

θ

MMDκ( b PX, b Pθ)

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Similarity between samples

slide-28
SLIDE 28

28

It’s nice! …where’s the catch?

How to learn the generative network Gθ

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1) Golden standard: Generative Adversarial Networks

P∗

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b PX

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<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

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'

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Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

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Very difficult to train (due to balancing of training discriminator/generator)

min

θ

max

φ

L(θ, φ)

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X

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Z

<latexit sha1_base64="FO5NRjT0DapzdaQZYFw0PtqJG0U=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkUeEDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJb3ZpygH9GB5CFn1Fip/tArltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+ALoPjOI=</latexit>

Real data

L

<latexit sha1_base64="n2ZkyHRQfuxyMWApcvw1zS/q5hw=">AB8XicbVC7SgNBFL3rM8ZX1NJmMAhWYTcKWgZtLCwimAcmS5id3E2GzM4uM7NCPkLGwtFbP0bO/G2WQLTwcDjnXubcEySCa+O6387K6tr6xmZhq7i9s7u3Xzo4bOo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdJP5rSdUmsfywYwT9CM6kDzkjBorPXYjaoaMCnLXK5XdijsDWSZeTsqQo94rfX7MUsjlIYJqnXHcxPjT6gynAmcFrupxoSyER1gx1JI9T+ZJZ4Sk6t0idhrOyThszU3xsTGmk9jgI7mSXUi14m/ud1UhNe+RMuk9SgZPOPwlQE5PsfNLnCpkRY0soU9xmJWxIFWXGlS0JXiLJy+TZrXinVeq9xfl2nVeRwGO4QTOwINLqMEt1KEBDCQ8wyu8Odp5cd6dj/noipPvHMEfOJ8/EZaQhA=</latexit>

Generator

Discriminator

2) Maximum Mean Discrepancy

Easier to train (no balancing)…

Eω∼Λ

  • X

xi∈X

eiωT xi − X

zi∈Z

eiωT Gθ(zi)

  • 2
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Distance??

X

xi∈X xj∈X

κ(xi, xj) − 2 X

xi∈X zj∈Z

κ(xi, Gθ(zj)) + X

zi∈Z zj∈Z

κ(Gθ(zi), Gθ(zj))

<latexit sha1_base64="XTroVO1M/1xaVH6QUthopzlnrmA=">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</latexit>

min

θ

MMDκ( b PX, b Pθ)

<latexit sha1_base64="U+5y4O6U3b6SxZQKgwTybchUaE=">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</latexit>

Similarity between samples

With Λ = Fκ

<latexit sha1_base64="WNr8NYe90zLKRHOsKmY6pNZ8NsM=">ACBHicbVDLSsNAFJ34rPUVdnNYBFclaQKuhGKgrhwUcE+oAnlZjJph04ezEyErpw46+4caGIWz/CnX/jpM1CWw8MHM45l7n3eAlnUlnWt7G0vLK6tl7aKG9ube/smnv7bRmngtAWiXksuh5IylEW4opTruJoB6nHa80VXudx6okCyO7tU4oW4Ig4gFjIDSUt+sOLc67AO+wE4IakiA42vsjCBJoG9WrZo1BV4kdkGqECzb345fkzSkEaKcJCyZ1uJcjMQihFOJ2UnlTQBMoIB7WkaQUilm02PmOAjrfg4iIV+kcJT9fdEBqGU49DTyXxROe/l4n9eL1XBuZuxKEkVjcjsoyDlWMU4bwT7TFCi+FgTILpXTEZgCidG9lXYI9f/Iiadr9kmtfndabVwWdZRQBR2iY2SjM9RAN6iJWoigR/SMXtGb8WS8GO/Gxy6ZBQzB+gPjM8f51KW/A=</latexit>

equivalent to (for later)

slide-29
SLIDE 29

29

It’s nice! …where’s the catch?

How to learn the generative network Gθ

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

1) Golden standard: Generative Adversarial Networks

P∗

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b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

<latexit sha1_base64="BiZWCF/MYtFKY8FAVLaO+kitF4=">AB+nicbVDLSsNAFL3xWesr1aWbwSK4KkVdFnUhcsK9gFNCJPptB06mYSZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnTDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHshlhRzgRtaY57SaS4ijktBOr3O/80ClYrG415OE+hEeCjZgBGsjBXbFi7AeEczRTZB5yYhNA7vq1JwZ0DJxC1KFAs3A/vL6MUkjKjThWKme6yTaz7DUjHA6LXupogkmYzykPUMFjqjys1n0KToxSh8NYme0Gim/t7IcKTUJArNZB5ULXq5+J/XS/Xg0s+YSFJNBZkfGqQc6RjlPaA+k5RoPjE8lMVkRGWGKiTVtlU4K7+OVl0q7X3LNa/e682rgq6ijBERzDKbhwAQ24hSa0gMAjPMrvFlP1ov1bn3MR1esYucQ/sD6/AEcR5Pn</latexit>

Very difficult to train (due to balancing of training discriminator/generator)

min

θ

max

φ

L(θ, φ)

<latexit sha1_base64="6fnrI4Xbh6A57YzD7kaGYj8dTPc=">ACGXicbVDLSgMxFM3UV62vqks3wSJUkDJTBV0W3bhwUcE+oDOUTJq2oZnMkNwRyzC/4cZfceNCEZe68m/MtF1o64HAyTn3cu89fiS4Btv+tnJLyura/n1wsbm1vZOcXevqcNYUdagoQhV2yeaCS5ZAzgI1o4UI4EvWMsfXWV+654pzUN5B+OIeQEZSN7nlICRukXbDbjsJi4MGZDUDciD+URDnmLDYUiJwDflqXuS6cfdYsmu2BPgReLMSAnNUO8WP91eSOASaCaN1x7Ai8hCjgVLC04MaRYSOyIB1DJUkYNpLJpel+MgoPdwPlXkS8ET93ZGQOtx4JvKbF0972Xif14nhv6Fl3AZxcAknQ7qxwJDiLOYcI8rRkGMDSFUcbMrpkOiCAUTZsGE4MyfvEia1YpzWqnenpVql7M48ugAHaIyctA5qFrVEcNRNEjekav6M16sl6sd+tjWpqzZj376A+srx/7SqDc</latexit>

X

<latexit sha1_base64="kpPTAtGMnO2krFRSnNra2xDivU=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZqdfrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU67nJsbPqDKcCZyWeqnGhLIxHWLXUkj1H42P3RKzqwyIGsbElD5urviYxGWk+iwHZG1Iz0sjcT/O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaijJjsynZELzl1dJu1b1Lq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8ftweM4A=</latexit>

Z

<latexit sha1_base64="FO5NRjT0DapzdaQZYFw0PtqJG0U=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkUeEDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJb3ZpygH9GB5CFn1Fip/tArltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+ALoPjOI=</latexit>

Real data

L

<latexit sha1_base64="n2ZkyHRQfuxyMWApcvw1zS/q5hw=">AB8XicbVC7SgNBFL3rM8ZX1NJmMAhWYTcKWgZtLCwimAcmS5id3E2GzM4uM7NCPkLGwtFbP0bO/G2WQLTwcDjnXubcEySCa+O6387K6tr6xmZhq7i9s7u3Xzo4bOo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdJP5rSdUmsfywYwT9CM6kDzkjBorPXYjaoaMCnLXK5XdijsDWSZeTsqQo94rfX7MUsjlIYJqnXHcxPjT6gynAmcFrupxoSyER1gx1JI9T+ZJZ4Sk6t0idhrOyThszU3xsTGmk9jgI7mSXUi14m/ud1UhNe+RMuk9SgZPOPwlQE5PsfNLnCpkRY0soU9xmJWxIFWXGlS0JXiLJy+TZrXinVeq9xfl2nVeRwGO4QTOwINLqMEt1KEBDCQ8wyu8Odp5cd6dj/noipPvHMEfOJ8/EZaQhA=</latexit>

Generator

Discriminator

2) Maximum Mean Discrepancy

Easier to train (no balancing) but quadratic complexity…

Eω∼Λ

  • X

xi∈X

eiωT xi − X

zi∈Z

eiωT Gθ(zi)

  • 2
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Distance??

X

xi∈X xj∈X

κ(xi, xj) − 2 X

xi∈X zj∈Z

κ(xi, Gθ(zj)) + X

zi∈Z zj∈Z

κ(Gθ(zi), Gθ(zj))

<latexit sha1_base64="XTroVO1M/1xaVH6QUthopzlnrmA=">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</latexit>

min

θ

MMDκ( b PX, b Pθ)

<latexit sha1_base64="U+5y4O6U3b6SxZQKgwTybchUaE=">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</latexit>

Similarity between samples

With Λ = Fκ

<latexit sha1_base64="WNr8NYe90zLKRHOsKmY6pNZ8NsM=">ACBHicbVDLSsNAFJ34rPUVdnNYBFclaQKuhGKgrhwUcE+oAnlZjJph04ezEyErpw46+4caGIWz/CnX/jpM1CWw8MHM45l7n3eAlnUlnWt7G0vLK6tl7aKG9ube/smnv7bRmngtAWiXksuh5IylEW4opTruJoB6nHa80VXudx6okCyO7tU4oW4Ig4gFjIDSUt+sOLc67AO+wE4IakiA42vsjCBJoG9WrZo1BV4kdkGqECzb345fkzSkEaKcJCyZ1uJcjMQihFOJ2UnlTQBMoIB7WkaQUilm02PmOAjrfg4iIV+kcJT9fdEBqGU49DTyXxROe/l4n9eL1XBuZuxKEkVjcjsoyDlWMU4bwT7TFCi+FgTILpXTEZgCidG9lXYI9f/Iiadr9kmtfndabVwWdZRQBR2iY2SjM9RAN6iJWoigR/SMXtGb8WS8GO/Gxy6ZBQzB+gPjM8f51KW/A=</latexit>

equivalent to (for later)

slide-30
SLIDE 30

30

It’s nice! …where’s the catch?

How to learn the generative network Gθ

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1) Golden standard: Generative Adversarial Networks

P∗

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b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

'

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Learn a second “discriminator” network that classifies real/ fake at the same time as the generator

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

<latexit sha1_base64="BiZWCF/MYtFKY8FAVLaO+kitF4=">AB+nicbVDLSsNAFL3xWesr1aWbwSK4KkVdFnUhcsK9gFNCJPptB06mYSZiVJiP8WNC0Xc+iXu/BsnbRbaemDgcM693DMnTDhT2nG+rZXVtfWNzdJWeXtnd2/frhy0VZxKQlsk5rHshlhRzgRtaY57SaS4ijktBOr3O/80ClYrG415OE+hEeCjZgBGsjBXbFi7AeEczRTZB5yYhNA7vq1JwZ0DJxC1KFAs3A/vL6MUkjKjThWKme6yTaz7DUjHA6LXupogkmYzykPUMFjqjys1n0KToxSh8NYme0Gim/t7IcKTUJArNZB5ULXq5+J/XS/Xg0s+YSFJNBZkfGqQc6RjlPaA+k5RoPjE8lMVkRGWGKiTVtlU4K7+OVl0q7X3LNa/e682rgq6ijBERzDKbhwAQ24hSa0gMAjPMrvFlP1ov1bn3MR1esYucQ/sD6/AEcR5Pn</latexit>

Very difficult to train (due to balancing of training discriminator/generator)

min

θ

max

φ

L(θ, φ)

<latexit sha1_base64="6fnrI4Xbh6A57YzD7kaGYj8dTPc=">ACGXicbVDLSgMxFM3UV62vqks3wSJUkDJTBV0W3bhwUcE+oDOUTJq2oZnMkNwRyzC/4cZfceNCEZe68m/MtF1o64HAyTn3cu89fiS4Btv+tnJLyura/n1wsbm1vZOcXevqcNYUdagoQhV2yeaCS5ZAzgI1o4UI4EvWMsfXWV+654pzUN5B+OIeQEZSN7nlICRukXbDbjsJi4MGZDUDciD+URDnmLDYUiJwDflqXuS6cfdYsmu2BPgReLMSAnNUO8WP91eSOASaCaN1x7Ai8hCjgVLC04MaRYSOyIB1DJUkYNpLJpel+MgoPdwPlXkS8ET93ZGQOtx4JvKbF0972Xif14nhv6Fl3AZxcAknQ7qxwJDiLOYcI8rRkGMDSFUcbMrpkOiCAUTZsGE4MyfvEia1YpzWqnenpVql7M48ugAHaIyctA5qFrVEcNRNEjekav6M16sl6sd+tjWpqzZj376A+srx/7SqDc</latexit>

X

<latexit sha1_base64="kpPTAtGMnO2krFRSnNra2xDivU=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZqdfrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU67nJsbPqDKcCZyWeqnGhLIxHWLXUkj1H42P3RKzqwyIGsbElD5urviYxGWk+iwHZG1Iz0sjcT/O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaijJjsynZELzl1dJu1b1Lq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8ftweM4A=</latexit>

Z

<latexit sha1_base64="FO5NRjT0DapzdaQZYFw0PtqJG0U=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkUeEDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJb3ZpygH9GB5CFn1Fip/tArltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+ALoPjOI=</latexit>

Real data

L

<latexit sha1_base64="n2ZkyHRQfuxyMWApcvw1zS/q5hw=">AB8XicbVC7SgNBFL3rM8ZX1NJmMAhWYTcKWgZtLCwimAcmS5id3E2GzM4uM7NCPkLGwtFbP0bO/G2WQLTwcDjnXubcEySCa+O6387K6tr6xmZhq7i9s7u3Xzo4bOo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdJP5rSdUmsfywYwT9CM6kDzkjBorPXYjaoaMCnLXK5XdijsDWSZeTsqQo94rfX7MUsjlIYJqnXHcxPjT6gynAmcFrupxoSyER1gx1JI9T+ZJZ4Sk6t0idhrOyThszU3xsTGmk9jgI7mSXUi14m/ud1UhNe+RMuk9SgZPOPwlQE5PsfNLnCpkRY0soU9xmJWxIFWXGlS0JXiLJy+TZrXinVeq9xfl2nVeRwGO4QTOwINLqMEt1KEBDCQ8wyu8Odp5cd6dj/noipPvHMEfOJ8/EZaQhA=</latexit>

Generator

Discriminator

2) Maximum Mean Discrepancy

Easier to train (no balancing) but quadratic complexity…

Eω∼Λ

  • X

xi∈X

eiωT xi − X

zi∈Z

eiωT Gθ(zi)

  • 2
<latexit sha1_base64="Upa/Vc+j9E+3G7EQvF9CMrA6GOw=">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</latexit>

Distance??

X

xi∈X xj∈X

κ(xi, xj) − 2 X

xi∈X zj∈Z

κ(xi, Gθ(zj)) + X

zi∈Z zj∈Z

κ(Gθ(zi), Gθ(zj))

<latexit sha1_base64="XTroVO1M/1xaVH6QUthopzlnrmA=">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</latexit>

min

θ

MMDκ( b PX, b Pθ)

<latexit sha1_base64="U+5y4O6U3b6SxZQKgwTybchUaE=">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</latexit>

Similarity between samples

equivalent to (for later)

With Λ = Fκ

<latexit sha1_base64="WNr8NYe90zLKRHOsKmY6pNZ8NsM=">ACBHicbVDLSsNAFJ34rPUVdnNYBFclaQKuhGKgrhwUcE+oAnlZjJph04ezEyErpw46+4caGIWz/CnX/jpM1CWw8MHM45l7n3eAlnUlnWt7G0vLK6tl7aKG9ube/smnv7bRmngtAWiXksuh5IylEW4opTruJoB6nHa80VXudx6okCyO7tU4oW4Ig4gFjIDSUt+sOLc67AO+wE4IakiA42vsjCBJoG9WrZo1BV4kdkGqECzb345fkzSkEaKcJCyZ1uJcjMQihFOJ2UnlTQBMoIB7WkaQUilm02PmOAjrfg4iIV+kcJT9fdEBqGU49DTyXxROe/l4n9eL1XBuZuxKEkVjcjsoyDlWMU4bwT7TFCi+FgTILpXTEZgCidG9lXYI9f/Iiadr9kmtfndabVwWdZRQBR2iY2SjM9RAN6iJWoigR/SMXtGb8WS8GO/Gxy6ZBQzB+gPjM8f51KW/A=</latexit>

Training generative networks typically requires massive amounts of data!

slide-31
SLIDE 31

31

Compressive Learning to the rescue!

slide-32
SLIDE 32

32

Compressive Learning: principle

Usual machine learning

Learning

θ

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… large memory & training time!

X

<latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit><latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit><latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit><latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit>

… … Large means…

N

<latexit sha1_base64="1geheZ8OEVc4P/xyjJq8gn3vuMI=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OJWrAf0Iay2U7atZtN2N0IJfQXePGgiFd/kjf/jds2B219YeHhnRl25g0SwbVx3W+nsLa+sblV3C7t7O7tH5QPj1o6ThXDJotFrDoB1Si4xKbhRmAnUijQGA7GN/O6u0nVJrH8sFMEvQjOpQ85IwazXu+WKW3XnIqvg5VCBXPV+as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXYuSRqj9bL7olJxZ0DCWNknDZm7vycyGmk9iQLbGVEz0su1mflfrZua8NrPuExSg5ItPgpTQUxMZleTAVfIjJhYoExuythI6oMzabkg3BWz5FVoXVc9y47JSu8njKMIJnMI5eHAFNbiDOjSBAcIzvMKb8+i8O/Ox6K14OQzx/BHzucPpOM0g=</latexit><latexit sha1_base64="1geheZ8OEVc4P/xyjJq8gn3vuMI=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OJWrAf0Iay2U7atZtN2N0IJfQXePGgiFd/kjf/jds2B219YeHhnRl25g0SwbVx3W+nsLa+sblV3C7t7O7tH5QPj1o6ThXDJotFrDoB1Si4xKbhRmAnUijQGA7GN/O6u0nVJrH8sFMEvQjOpQ85IwazXu+WKW3XnIqvg5VCBXPV+as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXYuSRqj9bL7olJxZ0DCWNknDZm7vycyGmk9iQLbGVEz0su1mflfrZua8NrPuExSg5ItPgpTQUxMZleTAVfIjJhYoExuythI6oMzabkg3BWz5FVoXVc9y47JSu8njKMIJnMI5eHAFNbiDOjSBAcIzvMKb8+i8O/Ox6K14OQzx/BHzucPpOM0g=</latexit><latexit sha1_base64="1geheZ8OEVc4P/xyjJq8gn3vuMI=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OJWrAf0Iay2U7atZtN2N0IJfQXePGgiFd/kjf/jds2B219YeHhnRl25g0SwbVx3W+nsLa+sblV3C7t7O7tH5QPj1o6ThXDJotFrDoB1Si4xKbhRmAnUijQGA7GN/O6u0nVJrH8sFMEvQjOpQ85IwazXu+WKW3XnIqvg5VCBXPV+as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXYuSRqj9bL7olJxZ0DCWNknDZm7vycyGmk9iQLbGVEz0su1mflfrZua8NrPuExSg5ItPgpTQUxMZleTAVfIjJhYoExuythI6oMzabkg3BWz5FVoXVc9y47JSu8njKMIJnMI5eHAFNbiDOjSBAcIzvMKb8+i8O/Ox6K14OQzx/BHzucPpOM0g=</latexit><latexit sha1_base64="1geheZ8OEVc4P/xyjJq8gn3vuMI=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRY9OJWrAf0Iay2U7atZtN2N0IJfQXePGgiFd/kjf/jds2B219YeHhnRl25g0SwbVx3W+nsLa+sblV3C7t7O7tH5QPj1o6ThXDJotFrDoB1Si4xKbhRmAnUijQGA7GN/O6u0nVJrH8sFMEvQjOpQ85IwazXu+WKW3XnIqvg5VCBXPV+as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXYuSRqj9bL7olJxZ0DCWNknDZm7vycyGmk9iQLbGVEz0su1mflfrZua8NrPuExSg5ItPgpTQUxMZleTAVfIjJhYoExuythI6oMzabkg3BWz5FVoXVc9y47JSu8njKMIJnMI5eHAFNbiDOjSBAcIzvMKb8+i8O/Ox6K14OQzx/BHzucPpOM0g=</latexit>

Multiple passes over X

<latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit><latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit><latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit><latexit sha1_base64="O1GCNFOLEfXrH+80JBkvTQl62o4=">AB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA20m7drMJuxuhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEiuDau+0UNja3tneKu6W9/YPDo/LxSVvHqWLYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFjrWZ3UK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm/VeqkJb/yMyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2poyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz0oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AtbuM3A=</latexit>
slide-33
SLIDE 33

33

θ

<latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit><latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit><latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit><latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit>

… constant memory & training time!

X

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… Large means…

N

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zX

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Dataset sketch (summary)

m

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“observations”

m ' size(θ) ⌧ Nn

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Single pass over X

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Usual machine learning

Learning

θ

<latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit><latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit><latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit><latexit sha1_base64="9nw3eiH0vxpghXegSZAQaK2DTls=">AB7nicbZDLSgNBEVrfMb4irp0xgEV2FGBF0G3biMYB6QDKGnU0ma9DzorhHCkI9w40IRt36PO/GnmQWmnih4XCriq6QaKkIdf9dtbWNza3tks75d29/YPDytFxy8SpFtgUsYp1J+AGlYywSZIUdhKNPAwUtoPJXV5vP6E2Mo4eaZqgH/JRJIdScLJWu0djJF7uV6puzZ2LrYJXQBUKNfqVr94gFmIEQnFjel6bkJ+xjVJoXBW7qUGEy4mfIRdixEP0fjZfN0ZO7fOgA1jbV9EbO7+nsh4aMw0DGxnyGlslmu5+V+tm9Lwxs9klKSEkVh8NEwVo5jlt7OB1ChITS1woaXdlYkx1yQTSgPwVs+eRValzXP8sNVtX5bxFGCUziDC/DgGupwDw1ogoAJPMrvDmJ8+K8Ox+L1jWnmDmBP3I+fwDagY8</latexit>

… large memory & training time!

X

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… … Large means…

N

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Multiple passes over X

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“ S k e t c h i n g ” “Learning”

  • R. Gribonval et al., “Compressive Statistical Learning

with Random Feature Moments,” 2017

Compressive Learning

Compressive Learning: principle

slide-34
SLIDE 34

34

X = , , , , · · · , z }| { N examples xi ∈ Rn n-dimensional

Sketching a dataset

slide-35
SLIDE 35

35

  • Compressed representation
  • Preserves relevant information

, , , , · · · , z }| { N examples yi ∈ Rp Rn → Rp

  • Dim. reduction

X = , , , , · · · , z }| { N examples xi ∈ Rn

Sketching a dataset

slide-36
SLIDE 36

36

  • Compressed representation
  • Preserves relevant information
  • Constant number of examples

Rn → Rp N examples

N can be VERY large (“big data”)!

X = , , , , · · · , z }| { N examples xi ∈ Rn , , , · · · , z }| { yi ∈ Rp ,

  • Dim. reduction

Sketching a dataset

slide-37
SLIDE 37

37

  • Compressed representation
  • Preserves relevant information
  • Dataset summary = single vector

zX = ∈ Cm

[Gribonval17]

X = , , , , · · · , z }| { N examples xi ∈ Rn

Sketching

Sketching a dataset

slide-38
SLIDE 38

38

  • Compressed representation
  • Preserves relevant information
  • Dataset summary = single vector

zX = ∈ Cm

[Gribonval17]

X = , , , , · · · , z }| { N examples xi ∈ Rn

Sketching

?

Sketching a dataset

slide-39
SLIDE 39

39

xi = ΩT ·

  • 1. Project on m (random) vectors
  • 2. Nonlinear periodic signature function

zX

ωj ∼ Λ

e.g.

Λ

Controls the cluster scale

Sketching a dataset

slide-40
SLIDE 40

40

xi = ΩT ·

  • 1. Project on m (random) vectors
  • 2. Nonlinear periodic signature function

zX

ωj ∼ Λ

exp i !

e.g.

Λ

Sketching a dataset

slide-41
SLIDE 41

41

xi = ΩT ·

  • 1. Project on m (random) vectors
  • 2. Nonlinear periodic signature function

zxi xi

7!

Random Fourier Features

zX

ωj ∼ Λ

exp i !

[A. Rahimi, B. Recht, “Random Features for Large-scale Kernel Machines”,NIPS, 2007]

e.g.

Λ

Sketching a dataset

slide-42
SLIDE 42

42

xi = ΩT exp i ! ·

  • 1. Project on m (random) vectors
  • 2. Nonlinear periodic signature function
  • 3. Pooling (average)

zxi xi

7!

Random Fourier Features

zX

ωj ∼ Λ

1 N X

xi∈X

e.g.

Λ

[A. Rahimi, B. Recht, “Random Features for Large-scale Kernel Machines”,NIPS, 2007]

Sketching a dataset

slide-43
SLIDE 43

43

xi = ΩT exp i ! ·

  • 1. Project on m (random) vectors
  • 2. Nonlinear periodic signature function
  • 3. Pooling (average)

1 N X

xi∈X

zxi xi

7!

Random Fourier Features

zX

ωj ∼ Λ

zX = " 1 N X

xi∈X

eiωT

j xi

#m

j=1

e.g.

Λ

∈ Cm

[A. Rahimi, B. Recht, “Random Features for Large-scale Kernel Machines”,NIPS, 2007]

Sketching a dataset

slide-44
SLIDE 44

44

Compressive Learning: SoA in one slide

“Sketching”

X

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zX

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Generalized moments Linear sensing

θ

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“Learning” Inverse problem Moment matching Dataset Model Sketch

Advantages:

  • One single pass on dataset (parallelizable)
  • Harsh compression (ideal for large-scale

datasets)

  • Handy for privacy preservation

Existing (unsupervised) tasks:

  • clustering: k-means, subspace

clustering

  • mixture model estimation: GMM,

alpha-stable distributions

  • Principal Component Analysis
  • Independent Component Analysis
  • Generative networks (this work)
slide-45
SLIDE 45

45

Main idea: move to the compressed domain!

...

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xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

Dataset Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Empirical distribution Generated samples

Compressively learning generative networks

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>
slide-46
SLIDE 46

46

Main idea: move to the compressed domain!

Sketching

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

∈ Cm

<latexit sha1_base64="3qTgmJnSdMLSaUeyYhAJ3DKtmjE=">AB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GWxG5cV7AOaWCbTSTt0MgkzE6GfokbF4q49VPc+TdO2iy09cDA4Zx7uWdOkHCmtON8W2vrG5tb26Wd8u7e/kHFPjzqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeYNHO/+0ilYrG419OE+hEeCRYygrWRBnbFYwJ5EdbjIEDNh2hgV52aMwdaJW5BqlCgNbC/vGFM0ogKThWqu86ifYzLDUjnM7KXqpogskEj2jfUIEjqvxsHnyGzowyRGEszRMazdXfGxmOlJpGgZnMI6plLxf/8/qpDq/9jIk1VSQxaEw5UjHKG8BDZmkRPOpIZhIZrIiMsYSE26KpsS3OUvr5JOveZe1Op3l9XGTVFHCU7gFM7BhStowC20oA0EUniGV3iznqwX6936WIyuWcXOMfyB9fkD2niSkA=</latexit>

∈ Cm

<latexit sha1_base64="3qTgmJnSdMLSaUeyYhAJ3DKtmjE=">AB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GWxG5cV7AOaWCbTSTt0MgkzE6GfokbF4q49VPc+TdO2iy09cDA4Zx7uWdOkHCmtON8W2vrG5tb26Wd8u7e/kHFPjzqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeYNHO/+0ilYrG419OE+hEeCRYygrWRBnbFYwJ5EdbjIEDNh2hgV52aMwdaJW5BqlCgNbC/vGFM0ogKThWqu86ifYzLDUjnM7KXqpogskEj2jfUIEjqvxsHnyGzowyRGEszRMazdXfGxmOlJpGgZnMI6plLxf/8/qpDq/9jIk1VSQxaEw5UjHKG8BDZmkRPOpIZhIZrIiMsYSE26KpsS3OUvr5JOveZe1Op3l9XGTVFHCU7gFM7BhStowC20oA0EUniGV3iznqwX6936WIyuWcXOMfyB9fkD2niSkA=</latexit>

Dataset Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Empirical distribution Generated samples Sketching

A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

Compressively learning generative networks

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

zX

A( b Pθ)

<latexit sha1_base64="lw0JmqYGH7IdfQzb21w5RexgEko=">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</latexit>

Sketch of generated samples

A( b Pθ) = 1 N 0 X

zi2Z

eiΩT Gθ(zi)

<latexit sha1_base64="ZVrBu/F9WYZbR2CrDGlhc4Vvo=">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</latexit>
slide-47
SLIDE 47

47

Main idea: move to the compressed domain!

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

∈ Cm

<latexit sha1_base64="3qTgmJnSdMLSaUeyYhAJ3DKtmjE=">AB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GWxG5cV7AOaWCbTSTt0MgkzE6GfokbF4q49VPc+TdO2iy09cDA4Zx7uWdOkHCmtON8W2vrG5tb26Wd8u7e/kHFPjzqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeYNHO/+0ilYrG419OE+hEeCRYygrWRBnbFYwJ5EdbjIEDNh2hgV52aMwdaJW5BqlCgNbC/vGFM0ogKThWqu86ifYzLDUjnM7KXqpogskEj2jfUIEjqvxsHnyGzowyRGEszRMazdXfGxmOlJpGgZnMI6plLxf/8/qpDq/9jIk1VSQxaEw5UjHKG8BDZmkRPOpIZhIZrIiMsYSE26KpsS3OUvr5JOveZe1Op3l9XGTVFHCU7gFM7BhStowC20oA0EUniGV3iznqwX6936WIyuWcXOMfyB9fkD2niSkA=</latexit>

∈ Cm

<latexit sha1_base64="3qTgmJnSdMLSaUeyYhAJ3DKtmjE=">AB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GWxG5cV7AOaWCbTSTt0MgkzE6GfokbF4q49VPc+TdO2iy09cDA4Zx7uWdOkHCmtON8W2vrG5tb26Wd8u7e/kHFPjzqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeYNHO/+0ilYrG419OE+hEeCRYygrWRBnbFYwJ5EdbjIEDNh2hgV52aMwdaJW5BqlCgNbC/vGFM0ogKThWqu86ifYzLDUjnM7KXqpogskEj2jfUIEjqvxsHnyGzowyRGEszRMazdXfGxmOlJpGgZnMI6plLxf/8/qpDq/9jIk1VSQxaEw5UjHKG8BDZmkRPOpIZhIZrIiMsYSE26KpsS3OUvr5JOveZe1Op3l9XGTVFHCU7gFM7BhStowC20oA0EUniGV3iznqwX6936WIyuWcXOMfyB9fkD2niSkA=</latexit>

Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Generated samples Sketching

...

<latexit sha1_base64="1D1M+XvlZpNUCS4SsUNcbJjqZsU=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujed9O4W19Y3NreJ2aWd3b/+gfHjU1EmGDZYIhLVDqlGwSU2DcC26lCGocCW+Hodua3nlBpnshHM04xiOlA8ogzaqz04Lpur1zxXG8Oskr8nFQgR71X/ur2E5bFKA0TVOuO76UmFBlOBM4LXUzjSlIzrAjqWSxqiDyfzUKTmzSp9EibIlDZmrvycmNZ6HIe2M6ZmqJe9mfif18lMdB1MuEwzg5ItFkWZICYhs79JnytkRowtoUxeythQ6oMzadkg3BX35lTSrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c0Rzovz7nwsWgtOPnMf+B8/gBNMo0m</latexit>

xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

P∗

<latexit sha1_base64="3+qbKlUGe3cvT9KyZluRGM1HT1E=">AB83icdVDLSgMxFM3UV62vqks3wSKIiyHTlk67K7pxWcE+oDOWTJpQzMPkoxQhv6GxeKuPVn3Pk3ZqYVPRA4HDOvdyT48WcSYXQh1FYW9/Y3Cpul3Z29/YPyodHPRklgtAuiXgkBh6WlLOQdhVTnA5iQXHgcdr3ZleZ37+nQrIovFXzmLoBnoTMZwQrLTlOgNWUYA47dxejcgWZNRvVrRZEZqNp2zlpWU3UsKBlohwVsEJnVH53xhFJAhoqwrGUQwvFyk2xUIxwuig5iaQxJjM8oUNQxQ6aZ5gU808oY+pHQL1QwV79vpDiQch54ejLKH97mfiXN0yU3RTFsaJoiFZHvITDlUEswLgmAlKFJ9rgolgOiskUywUbqmki7h6fwf9KrmlbNrN7UK+3LVR1FcAJOwTmwgA3a4Bp0QBcQEIMH8ASejcR4NF6M1+VowVjtHIMfMN4+AbVZkXs=</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

Dataset

Empirical distribution Sketching

A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

Compressively learning generative networks

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

Sketch of generated samples

A( b Pθ) = 1 N 0 X

zi2Z

eiΩT Gθ(zi)

<latexit sha1_base64="ZVrBu/F9WYZbR2CrDGlhc4Vvo=">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</latexit>

A( b Pθ)

<latexit sha1_base64="lw0JmqYGH7IdfQzb21w5RexgEko=">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</latexit>

zX

slide-48
SLIDE 48

48

Compressively learning generative networks

Proposed approach: match the sketches of real and generated data

min

θ

  • zX − 1

N 0 X

zi2Z

eiΩT Gθ(zi)

  • 2

2

<latexit sha1_base64="qJO+l0X/jADmxCf3CL6h+IxLeXw=">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</latexit>
slide-49
SLIDE 49

49

Compressively learning generative networks

Proposed approach: match the sketches of real and generated data

Sampled (Monte Carlo) estimation to the MMD!

min

θ

  • zX − 1

N 0 X

zi2Z

eiΩT Gθ(zi)

  • 2

2

<latexit sha1_base64="qJO+l0X/jADmxCf3CL6h+IxLeXw=">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</latexit>

…but where dataset is accessed only once

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

ωj ∼ Λ

min

θ

Eω⇠Λ

  • 1

N X

xi2X

eiωT xi − 1 N 0 X

zi2Z

eiωT Gθ(zi)

  • 2
<latexit sha1_base64="IZrAj1TbLclhyJFuxm4wH+6AwUA=">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</latexit>
slide-50
SLIDE 50

50

Compressively learning generative networks

Proposed approach: match the sketches of real and generated data

Sampled (Monte Carlo) estimation to the MMD!

min

θ

  • zX − 1

N 0 X

zi2Z

eiΩT Gθ(zi)

  • 2

2

<latexit sha1_base64="qJO+l0X/jADmxCf3CL6h+IxLeXw=">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</latexit>

…but where dataset is accessed only once

'

<latexit sha1_base64="gnQ2L8m5VJr4MIZSVUK1PJqA8vE=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2WTMPNaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlLy34SGAg8kixnB1knNrmGCPvZKZb/iz4CWSZCTMuSo90pf3b4iqaDSEo6N6QR+YsMa8sIp5NiNzU0wWSEB7TjqMSCmjCbXTtBp07po1hpV9Kimfp7IsPCmLGIXKfAdmgWvan4n9dJbXwVZkwmqaWSzBfFKUdWoenrqM80JZaPHcFEM3crIkOsMbEuoKILIVh8eZk0q5XgvFK9uyjXrvM4CnAMJ3AGAVxCDW6hDg0g8ADP8ApvnvJevHfvY964uUzR/AH3ucPsumPNQ=</latexit>

ωj ∼ Λ

min

θ

Eω⇠Λ

  • 1

N X

xi2X

eiωT xi − 1 N 0 X

zi2Z

eiωT Gθ(zi)

  • 2
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Practical learning algorithm?

Differentiable by chain rule (feat. backprop)

slide-51
SLIDE 51

51

Preliminary results

Simple 2d signals… …it works (in principle)!

slide-52
SLIDE 52

To conclude: open challenges

Batch size?

min

θ

  • zX − 1

N 0 X

zi2Z

eiΩT Gθ(zi)

  • 2

2

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Sketch size? Kernel?

Sketching

...

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xi

<latexit sha1_base64="VawXLeIOkcuiVKEJ7hk/7w6cu4Q=">AB+HicbVC7TsMwFL0pr1IeDTCyWFRITFVSkGCsYGEsEn1IbRQ5jtNadZzIdhCl6pewMIAQK5/Cxt/gtBmg5UiWj865Vz4+QcqZ0o7zbZXW1jc2t8rblZ3dvf2qfXDYUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wvsn97gOViXiXk9S6sV4KFjECNZG8u3qIEh4qCaxudCjz3y75tSdOdAqcQtSgwIt3/4ahAnJYio04Vipvuk2ptiqRnhdFYZIqmIzxkPYNFTimypvOg8/QqVFCFCXSHKHRXP29McWxyrOZyRjrkVr2cvE/r5/p6MqbMpFmgqyeCjKONIJyltAIZOUaD4xBPJTFZERlhiok1XFVOCu/zlVdJp1N3zeuPuota8LuowzGcwBm4cAlNuIUWtIFABs/wCm/Wk/VivVsfi9GSVewcwR9Ynz/d4ZM4</latexit>

b PX

<latexit sha1_base64="DJDJqJ4pL/y4UrKbW91eGw1q+g=">AB/3icdVDLSsNAFJ3UV62vquDGzWARXIWkqabdFd24rGAf0IQwmUzaoZMHMxOlxC78FTcuFHrb7jzb5w+BU9MHA4517umeOnjApGB9aYWl5ZXWtuF7a2Nza3inv7nVEknFM2jhCe/5SBGY9KWVDLSzlBkc9I1x9dTP3uDeGCJvG1HKfEjdAgpiHFSCrJKx84tzQgQyRzJ0JyiBGDrYnX8oVQzerhm2Z0NBrZsOqW4qcNU7tug1N3ZihAhZoeV3J0hwFpFYoaE6JtGKt0cUkxI5OSkwmSIjxCA9JXNEYREW4+yz+Bx0oJYJhw9WIJZ+r3jRxFQowjX01OQ4rf3lT8y+tnMqy7OY3TJIYzw+FGYMygdMyYEA5wZKNFUGYU5UV4iHiCEtVWUmV8PVT+D/pVHXT0qtXtUrzfFHERyCI3ACTGCDJrgELdAGNyB/AEnrV7VF70V7nowVtsbMPfkB7+wSOFZ2</latexit>

X

<latexit sha1_base64="+SX5ghEw4dhNqJNmctBJ1YeqMs=">AB6HicdVDJSgNBEO2JW4xb1KOXxiB4GmYy0UluQS8eEzALJEPo6dQkbXoWunuEMOQLvHhQxKuf5M2/sbMIKvqg4PFeFVX1/IQzqSzrw8itrW9sbuW3Czu7e/sHxcOjtoxTQaFYx6Lrk8kcBZBSzHFoZsIKHPoeNPrud+5x6EZHF0q6YJeCEZRSxglCgtNbuDYsky7bLlOja2zIpdc6qOJpe1C7fqYtu0FihFRqD4nt/GNM0hEhRTqTs2VaivIwIxSiHWaGfSkgInZAR9DSNSAjSyxaHzvCZVoY4iIWuSOGF+n0iI6GU09DXnSFRY/nbm4t/eb1UBVUvY1GSKojoclGQcqxiP8aD5kAqvhUE0IF07diOiaCUKWzKegQvj7F/5N2bQds9yslOpXqzjy6ASdonNkIxfV0Q1qoBaiCNADekLPxp3xaLwYr8vWnLGaOUY/YLx9Ai+gjTQ=</latexit>

A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

∈ Cm

<latexit sha1_base64="3qTgmJnSdMLSaUeyYhAJ3DKtmjE=">AB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GWxG5cV7AOaWCbTSTt0MgkzE6GfokbF4q49VPc+TdO2iy09cDA4Zx7uWdOkHCmtON8W2vrG5tb26Wd8u7e/kHFPjzqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeYNHO/+0ilYrG419OE+hEeCRYygrWRBnbFYwJ5EdbjIEDNh2hgV52aMwdaJW5BqlCgNbC/vGFM0ogKThWqu86ifYzLDUjnM7KXqpogskEj2jfUIEjqvxsHnyGzowyRGEszRMazdXfGxmOlJpGgZnMI6plLxf/8/qpDq/9jIk1VSQxaEw5UjHKG8BDZmkRPOpIZhIZrIiMsYSE26KpsS3OUvr5JOveZe1Op3l9XGTVFHCU7gFM7BhStowC20oA0EUniGV3iznqwX6936WIyuWcXOMfyB9fkD2niSkA=</latexit>

Dataset

Empirical distribution

zX

θ

<latexit sha1_base64="QS5zpnJMtg1Afj50jAeQlCFYz0=">AB7XicdVDLSsNAFJ3UV62vqks3g0VwFZK0IS6LblxWsA9oQ5lMJ+3YySTM3Ail9B/cuFDErf/jzr9x+hBU9MCFwzn3cu89USa4Bsf5sApr6xubW8Xt0s7u3v5B+fCopdNcUdakqUhVJyKaCS5ZEzgI1skUI0kWDsaX839j1TmqfyFiYZCxMylDzmlICRWj0YMSD9csWx3Wrg+x527GoQuEHNENfz/ZqPXdtZoIJWaPTL71BSvOESaCaN1nQzCKVHAqWCzUi/XLCN0TIasa6gkCdPhdHtDJ8ZYDjVJmSgBfq94kpSbSeJHpTAiM9G9vLv7ldXOIL8Ipl1kOTNLlojgXGFI8fx0PuGIUxMQhU3t2I6IopQMAGVTAhfn+L/ScszQdneTa1Sv1zFUQn6BSdIxcFqI6uUQM1EUV36AE9oWcrtR6tF+t12VqwVjPH6Aest08QZY92</latexit>

<latexit sha1_base64="YhL1xTRGsbaPEOSVWRYfTORm0=">AB/HicdVDLSsNAFJ3UV62vaJduBovgKiRtXVXdOygrWFJoTJdNoOnTyYuRFKqL/ixoUibv0Qd/6Nk7aCih4YOJxzL/fMCRLBFdj2h1FYWV1b3yhulra2d3b3zP2DWxWnkrIOjUsewFRTPCIdYCDYL1EMhIGgnWDyWXud+YVDyObmCaMC8ko4gPOSWgJd8suyGBMSUCt/3MhTEDMvPNim01685Zo4Ftq+5Uq82aJrXTmn3exI5lz1FBS7R9890dxDQNWQRUEKX6jp2AlxEJnAo2K7mpYgmhEzJifU0jEjLlZfPwM3yslQEexlK/CPBc/b6RkVCpaRjoyTyq+u3l4l9eP4Vh08t4lKTAIro4NEwFhjnTeABl4yCmGpCqOQ6K6ZjIgkF3VdJl/D1U/w/ua1aTs2qXtcrYtlHUV0iI7QCXJQA7XQFWqjDqJoih7QE3o27o1H48V4XYwWjOVOGf2A8fYJPtqVLQ=</latexit>

b Pθ

<latexit sha1_base64="FyfHC/GQsu734QVf9+WxGrITCw4=">ACBnicdVBNSyNBEO1Rd9XsV9SjCM2GBU/DzCTZ7DHoxWMEkwiZEGo6FdPY80F3jUsYcvKyf2UvHhTx6m/w5r+xJ0ZQ2X1Q8Hiviqp6UakIc97dFZW1z58XN/YrHz6/OXrt+rWds+kuRbYFalK9WkEBpVMsEuSFJ5mGiGOFPaj8PS71+gNjJNTmiW4TCGs0ROpACy0qi6F/6WY5wCFWEMNBWgeGc+KkKaIsF8VK15bqtZD1oB91z/Z91r+iUJGq1Gk/ut0CNLdEZVR/CcSryGBMSCowZ+F5GwI0SaFwXglzgxmIczjDgaUJxGiGxeKNOf9hlTGfpNpWQnyhvp4oIDZmFke2s7zVvPdK8V/eIKfJr2EhkywnTMTzokmuOKW8zISPpUZBamYJC3trVxMQYMgm1zFhvDyKf8/6QWuX3eD40atfbCMY4Ptsu9sn/msxdrsiHVYlwl2yf6ya3bj/HGunFvn7rl1xVnO7LA3cO6fAGVYmbs=</latexit>

b PZ

<latexit sha1_base64="kcvi7lgZYVJ+mDVY6QjNLhlwaeU=">ACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5CkrXFZdOygn1gU8pkMmHTh7MTJQS4sZfceNCEbf+hTv/xklbQUPXDicy/3uMljApmh/awuLS8spqa28vrG5ta3v7LZFnHJMWjhmMe96SBGI9KSVDLSThBocdIxufF37nhnB4+hKThLSD9EwogHFSCpoO+7t9QnIyQzN0RyhBGDzXyQXecDvWIaTr1qOzY0Deukatatgtg1p1aHlmFOUQFzNAf6u+vHOA1JDFDQvQsM5H9DHFJMSN52U0FSRAeoyHpKRqhkIh+Nv0gh0dK8WEQc1WRhFP1+0SGQiEmoac6izPFb68Q/J6qQxO+xmNklSCM8WBSmDMoZFHNCnGDJogzKm6FeIR4ghLFVpZhfD1KfyftG3Dqhr2Za3SOJvHUQIH4BAcAws4oAEuQBO0AZ34AE8gWftXnvUXrTXWeuCNp/ZAz+gvX0CSueXcQ=</latexit>

Pz

<latexit sha1_base64="Xk6GVzuHyhClX+wybFEoso4Kd5s=">AB9XicdVDLSgMxFM3UV62vqks3wSK4GuZR7bgrunFZwT6gHUsmzbShmcyQZJQ69D/cuFDErf/izr8x01ZQ0QOBwzn3ck9OkDAqlWV9GIWl5ZXVteJ6aWNza3unvLvXknEqMGnimMWiEyBJGOWkqahipJMIgqKAkXYwvsj9i0Rksb8Wk0S4kdoyGlIMVJaulFSI0wYrDRz+6n/XLFMr2qfVqrQcus2o7juZq4J6515kHbtGaogAUa/fJ7bxDjNCJcYak7NpWovwMCUxI9NSL5UkQXiMhqSrKUcRkX42Sz2FR1oZwDAW+nEFZ+r3jQxFUk6iQE/mKeVvLxf/8rqpCj0/ozxJFeF4fihMGVQxzCuAyoIVmyiCcKC6qwQj5BAWOmiSrqEr5/C/0nLMW3XdK6qlfr5o4iOACH4BjYoAbq4BI0QBNgIMADeALPxp3xaLwYr/PRgrHY2Qc/YLx9AvLqktI=</latexit>

∈ Cm

<latexit sha1_base64="3qTgmJnSdMLSaUeyYhAJ3DKtmjE=">AB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GWxG5cV7AOaWCbTSTt0MgkzE6GfokbF4q49VPc+TdO2iy09cDA4Zx7uWdOkHCmtON8W2vrG5tb26Wd8u7e/kHFPjzqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeYNHO/+0ilYrG419OE+hEeCRYygrWRBnbFYwJ5EdbjIEDNh2hgV52aMwdaJW5BqlCgNbC/vGFM0ogKThWqu86ifYzLDUjnM7KXqpogskEj2jfUIEjqvxsHnyGzowyRGEszRMazdXfGxmOlJpGgZnMI6plLxf/8/qpDq/9jIk1VSQxaEw5UjHKG8BDZmkRPOpIZhIZrIiMsYSE26KpsS3OUvr5JOveZe1Op3l9XGTVFHCU7gFM7BhStowC20oA0EUniGV3iznqwX6936WIyuWcXOMfyB9fkD2niSkA=</latexit>

Generative network Latent space

<latexit sha1_base64="hMjPrjVhJCWHJIlFkq/WFtnYic=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GXRhS4r2Ac0IUymk3bo5MHMjRBC/RU3LhRx64e482+ctFlo64GBwzn3cs8cPxFcgWV9G5W19Y3Nrep2bWd3b/APDzqTiVlHVpLGI58IligkesCxwEGySkdAXrO9Pbwq/8ik4nH0AFnC3JCMIx5wSkBLnl3QgITSgS+9XIHJgzIzDMbVtOaA68SuyQNVKLjmV/OKZpyCKgig1tK0E3JxI4FSwWc1JFUsInZIxG2oakZApN5+Hn+FTrYxwEv9IsBz9fdGTkKlstDXk0VUtewV4n/eMIXgys15lKTAIro4FKQCQ4yLJvCIS0ZBZJoQKrnOiumESEJB91XTJdjLX14lvVbTPm+27i8a7euyjio6RifoDNnoErXRHeqgLqIoQ8/oFb0ZT8aL8W58LEYrRrlTR39gfP4Au4yU0w=</latexit>

Generated samples Sketching

A

<latexit sha1_base64="bwIXQVATSCxXpUQh/noUDMfH/o=">AB8XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6LqxmUF+8B2KJn0ThuayQxJRilf+HGhSJu/Rt3/o2ZdhbaeiBwOdecu4JEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6zfzWEyrNY/lgxgn6ER1IHnJGjZUeuxE1Q0YFue6Vym7FnYEsEy8nZchR75W+uv2YpRFKwTVuO5ifEnVBnOBE6L3VRjQtmIDrBjqaQRan8ySzwlp1bpkzBW9klDZurvjQmNtB5HgZ3MEupFLxP/8zqpCa/8CZdJalCy+UdhKoiJSXY+6XOFzIixJZQpbrMSNqSKMmNLKtoSvMWTl0mzWvHOK9X7i3LtJq+jAMdwAmfgwSXU4A7q0AGEp7hFd4c7bw4787HfHTFyXeO4A+czx8A6pB5</latexit>

A( b Pθ)

<latexit sha1_base64="lw0JmqYGH7IdfQzb21w5RexgEko=">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</latexit>
slide-53
SLIDE 53

53