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Scaling Optimal Transport to High Dimensional Learning

Gabriel Peyré
September 02, 2018

Scaling Optimal Transport to High Dimensional Learning

Talk at the Workshop "From Stochastic Geometric Mechanics to Mass Transportation Problems" in Lisboa. Updated on June 2nd.

Gabriel Peyré

September 02, 2018
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  1. Scaling Optimal Transport to High Dimensional Learning Gabriel Peyré É

    C O L E N O R M A L E S U P É R I E U R E RESEARCH UNIVERSITY PARIS Joint works with: Francis Bach François-Xavier Vialard Jean Feydy Thibault Séjourné Lénaic Chizat Aude Genevay Marco Cuturi Alain Trouvé Shun'ichi Amari
  2. Source image (X) Style image (Y) Sliced Wasserstein projection of

    X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization ! images, vision, graphics and machine learning, . . . Probability distributions and histograms <latexit 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Probability Distributions in Data Sciences
  3. Source image (X) Style image (Y) Sliced Wasserstein projection of

    X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization ! images, vision, graphics and machine learning, . . . Probability distributions and histograms <latexit 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<latexit 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<latexit sha1_base64="NHBrzuHY/DIiTVSeYz9st2EbBcI=">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</latexit> Parametric model: ✓ 7! ↵✓ <latexit 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<latexit 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<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Probability Distributions in Data Sciences <latexit 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<latexit 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<latexit 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↵✓ <latexit 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  4. Source image (X) Style image (Y) Sliced Wasserstein projection of

    X to style image color statistics Y Source image after color transfer J. Rabin Wasserstein Regularization ! images, vision, graphics and machine learning, . . . Probability distributions and histograms <latexit 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<latexit 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min ✓ D(↵✓, ) <latexit 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<latexit 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Density fitting: <latexit 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<latexit 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<latexit 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<latexit 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Parametric model: ✓ 7! ↵✓ <latexit 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<latexit 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<latexit sha1_base64="YYkL6JTK2uYxh8MNcanIViL0GK4=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Probability Distributions in Data Sciences <latexit 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<latexit 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<latexit 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↵✓ <latexit 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  5. 3. Sinkhorn Divergences <latexit 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<latexit 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<latexit 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<latexit

    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<latexit 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x <latexit 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<latexit 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d(x,y) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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<latexit 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  6. Kantorovitch’s Formulation Points (xi)i, (yj)j Input distributions ↵ = Pn

    i=1 ai xi = Pm j=1 bj yj Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 xi yj
  7. Kantorovitch’s Formulation Points (xi)i, (yj)j Input distributions ↵ = Pn

    i=1 ai xi = Pm j=1 bj yj Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 xi yj Couplings: <latexit sha1_base64="MCNqjwUgnSKqr3h3uG6OPk9oPyc=">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</latexit> <latexit 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<latexit 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<latexit 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U(a, b) def. = P 2 Rn⇥m + ; P1m = a, P>1n = b <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  8. Kantorovitch’s Formulation Points (xi)i, (yj)j Input distributions ↵ = Pn

    i=1 ai xi = Pm j=1 bj yj Weights ai > 0, bj > 0. Pn i=1 ai = Pm j=1 bj = 1 xi yj d(xi ,yj ) <latexit 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<latexit 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Couplings: <latexit 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<latexit 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U(a, b) def. = P 2 Rn⇥m + ; P1m = a, P>1n = b <latexit 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<latexit 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[Kantorovich 1942] min nP i,j d(xi, yj)pPi,j ; P 2 U(a, b) o <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  9. Optimal Transport Distances ↵ b <latexit 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<latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">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</latexit> <latexit

    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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="n2zdOYSE8oWxrpoEULZvL+5/PUM=">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</latexit> ⇡ = P i,j Pi,j xi,yj <latexit 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<latexit 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<latexit 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Wp(↵, )p def. = min ⇡2M1 + (X2) ⇢Z X2 d(x, y)pd⇡(x, y) ; ⇡1 = ↵, ⇡2 = <latexit 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<latexit 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<latexit 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  10. Optimal Transport Distances ↵ Theorem: <latexit sha1_base64="cHzNFJNn6IKb0rmopfqU8lH/agk=">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</latexit> <latexit sha1_base64="cHzNFJNn6IKb0rmopfqU8lH/agk=">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</latexit> <latexit

    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<latexit 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Wp is a distance and <latexit 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↵n * , Wp(↵n, ) ! 0 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="n2zdOYSE8oWxrpoEULZvL+5/PUM=">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</latexit> ⇡ = P i,j Pi,j xi,yj <latexit 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<latexit 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<latexit 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Wp(↵, )p def. = min ⇡2M1 + (X2) ⇢Z X2 d(x, y)pd⇡(x, y) ; ⇡1 = ↵, ⇡2 = <latexit 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<latexit 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<latexit sha1_base64="lJUzn7m52pXEhLImUg62Dqwze80=">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</latexit> Weak⇤ (aka in law) convergence: <latexit 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<latexit 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. . . x1 <latexit 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x2 <latexit 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<latexit 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<latexit 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x3 <latexit 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R X fd↵n ! R X fd <latexit 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<latexit 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<latexit 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|| xn x ||1 = 2 <latexit 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<latexit 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<latexit sha1_base64="Iaq3mbirAEh6zcC4R4ECZTExtSk=">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</latexit> Wp( xn , x) = d(xn, x) <latexit 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<latexit 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  11. 3. Sinkhorn Divergences <latexit 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<latexit 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<latexit 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<latexit

    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<latexit 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x <latexit 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<latexit 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d(x,y) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  12. Entropic Regularization Schr¨ odinger’s problem: [1931] Erwin Schrödinger ↵ "

    ↵ min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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⇡ = P i,j Pi,j xi,yj <latexit 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⇡" <latexit 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<latexit 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<latexit 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<latexit sha1_base64="GFDRy6M/TlDbNJvkPgJDOhXXs88=">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</latexit>
  13. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">AABB1nictVzbchu5EYU3l107N29SecrLJFqnvFuOV1a2KpetVK0sybLWtE2blC+7tF28jGjaQw7NIWXZXFWeUnnNL+Q1+Yh8R/4gecovpC/AAENipjGKY5QkDIjT3egBGt0N0L1pMsrmm5v/PPfet779ne++/8H5C9/7/g9++KOLH/74QZYuZv34sJ8m6exRr5vFyWgSH85H8yR+NJ3F3XEviR/2Xu7g5w+P41k2Sift+Ztp/GTcHU5GR6N+dw5Nzy7+tDOPTwC3bM7SaZqNsPX3p88ubmxe3aR/0Xrlmq5sKP2vmX4YHaqOGqhU9dVCjVWsJmoO9UR1VQbla3VNbaoptD1RS2ibQW1En8fqVF0A7AJ6xdCjC60v4fcQnr7WrRN4RpoZofvAJYGfGSAjdQkwKfSbQR25RfT5gihjaxntJdFE2d7A356mNYbWuXoOrRLO9AzF4Vjm6kj9lsYwgjFNqQVH19dUFqQVlDxyRjUHClNow/oAPp9BvU9Io+eIMBmNHXXbpc//RT2xFZ/7uu9C/ZukvAQlUi09+jSn0FXHRD+it7mAz1ieBDgPgUKsx4i116TrMY1+Av2X0H4HyinVjE56UJbUelqJ3IHiQ+6IyH0oPuS+iGxA8SEbIrIJxYdsaiRiZ6RzP74FxYdviZzvQfEh74nI+1B8yPsi8gEUH/KBiPwKig/5lYi8AcWHvCEib0HxIW+JyDYUH7ItIg+h+JCHInIPig+5p5HlK3UGJSU6I2FVbkO9yAMtRQIt26J818k6+rDXA9Z0vwQrr+pd+OvH7gboNC7B7gXMu6MSrDzz9sFG+rGyLbpJu4kPe1PEHsAM8GMPROyX6kUJ9suAlfayBCuvtQb082Nl63sbnvzY2yL2DtT8WHmPugstfuzdgB1jWoJtith76lUJNsTqz0qwst1vgV3xY+V9qg39/dgQa7oowcr29AF4MH6svFs9hFY/9qGIfaROSrCPROxjsO5+7OOAHfZtCdbssRdoBxmSPxLDiq2i1s1XJdamQK0r8E/yvSUh37gH7RJmmGOGhBmLiP0csR+IaOSIRrBcWW5HM/J3ZS6tHNEKRPTyvQlrc7H/IO+PtSQAsZsjdlcQVR4pvmszlmPyLkyLhJznOxfWQsaU5vYba7GeD9WW1yDuFhA8t5/TzL9C0RJGUKipKmrP8z2ekRE9VyFeU/RmRml4yLh5bhVc1ImI6nlQPRH1xoN6I6IWHtRCRB17UMciyq58F9cJmAFW//gulvTEM4B95PISgVewDbvOTVijEcyfJniB96nlLvxtUewtlSrJMJrHfRKzHE8KlngGtaXagHYbFe5SfJ3QCotBMu55V8f4+IS5jaVec2yFT/OdPMozJuF0RiTPMKeD3mJE66kenVvUckreHdfq4W/m697U6uH3SOOn5MVzrR5+rqWfn0H2tsa2z4BtwWqaau3bel0anH9hGqZ+gXZdtLj4Vsd6ziC9k5r0D/SbOTjDe9mhGuvH1uvRyJzxZYXx1aFh9Zw5eq5HBb0n9npNLao9komOe229rgwp7aITLYd9qvtmsM9AvxlTr0ejCR7XDsXcS6ded/ZO89HYej0aDxTnPU/Jkzf1ejSG9Mz6sPV6NDDb0tVxvq3XteyoAY6dbb2uVZ9QFhhzQDznucV6RTPykxaa2oj8g+psjevzr+9jmLN5mscI1ZSsb1tOp5fvZdUSGX8hBqs2rykH+hcLxwcr0liqLTG+Yhnmhf19nY7d41HzDdBiBKufzwCknHkCEpqcBFrvBCheE6Ou4sgMbkvE4Sw5WkF1dOtc9BYtX84aFdueUasUl9nRWj12yF5nNPem5BM2SLOSHhqlb7iMoqShRkFDMr06unur12tR+5sibrqCmOYzrU8nQnySVh2n+rTecnR8SZ/yzKHwmY+dv5htPtLWBmOelGwRylLF0+1n8khuG+6rV5TNcfNnEb1RtFfHZDVGdCKViVGoyRazN76kZ0v7kM7kkAfT6MN7jDSVqeJTM8yiYz49Iovq2luJN+rLZOi4npHVNfa4Gj100EMPun6MswM7xh2otSFmOISndkCUcyHXVUoan6lf5aejKb3B6og+KVhIQ4PtTVywkFVR9vMCldeAxtnAUXo4jVU6Bt9ZoyRH/T55bOxatPyX6OTWnG93aY6Xz+byTMyAuG4R14hWDZ/q8tMqB5Zg6f1ki/zX6lEivzoc0YZKXJ86nFkvEzrxjymCnZJnnNBqk1ZHsbebn1r9xHBqKnN2jqfZKVnIiOxfBPtTSnMyoh/37oA5QWeLkJCNDLE7o9y78fk6I3GOWT9upPhWg51vMdmyBfE3dN3VldFc5IiB94HTlbltdNIgXzAmrjNt3e3art59EGnvSbizhCnauXKZ+H9Mv82PmScbazMCNYxvINO2zvc+UopZUEdd2uWrbZDp60r5US7DUy213f+sTB8VJNuliAvlwd16AJz79My8cJbMSO5srQ/vo1XZXKQ8XdEjjvaIoni2+0O9A6PcV2iX3KA116FZMoRZMM+jCNNXyiKv8q3mVaQeRjv7v1C3ui5qDSlGymZwWUNSfj+maM2VMoFZzfP3Ja0mv9ZnK72q+UxoLo6dtfwNtP4cfhu5zXMYnV7BKlynOcAU7JPVCLdEaz3CeF0v8DIz09Cyz5afnZOml9tylviarZuNsY9rU2nSrDnRWQtTPwuNFw6NF4E6bNNZo9WiaTeW6JkYW7T1aWUovzrc2jUoL0TKskdmUKMAKd1YKozqQKQqx/gG9VaktSnS6sJqdU8D3DUfgvSv9dXV/U2+u0fqBvk2ffLAOH4Z0Codkc9lWqsjNaaAnD/T9tVd/R1qQe49sqBIme9x4orhU6c+ldNc0l/qnS0lO28tgrm39Fr3MTa2Q/VfryHHtCYyWpcG8Rn1iLX8rhzRikW66vgcEWX+u+RTsd9RHTO7ve07iQr+hI03eVVZXhwpTEj/UubtYC16PXDi14hiwoX2rntAq/4bRgqMMZkEv2eZ0RvCXY5PEtij7ZH9XLdTfIo3cSS6SlIv1R8CbAxHvXauu3PLjNiM7RPoiVq3b93XQ+aXBHOU+J3lRK9Lu9pY+6jLleez0erqXa74XKWHxQpfq48F9XEjCxvlFTEd9XkwF5aoHhfGhHCpN4o68teTvI7MfDoVStn0NpSLmQa2Mc8pXpLugSLC591d9npzHwvj6K3R6xHWpcYtEiXMxqU6P+BaWsxKnV/bh7j1fOVulDg7UdlOYai7u4W132whY7J+iZJyNtzblb1TiFLkLAxT6Cu+0VsWH7o0P4eCvyPliw4Nx5DcYQv82221o/bewW2IV7rOGc2IWtAWDFZi764eZ7FHtY5eOdRd+iEcwnmMQNeS9CPaSevKzpRlyV3q4fRfkxWYqViU3vasPwaXizySdU51xjMiyyaPZqTMd3HqjsVwCBlJkUs4Hz7XkEZxpMx3muqNwVCXR1DkUIeHuccQ9s5t7/q8XE7V+lrnEsqDdwFz4mJwePJXHqvYfiEWaua8kXfPAa3DUQV1s1v8r+MwfCyn+rxCuWX0XbMXAW+d+8U6I4v+cP01Y7mFzOZyjuE803x01lvy82O/L6r1plJnNO+ePvqjdg4YXkvFeVBZOsa7s8jKG0oFzwV8MqTqP+of5+RvI7zKaZTJUYeSOacop2Z6yNTMNy59ozOfhchk6ZTJVKRm44gW3YjdUQfqBvzs5B5g3duh/F1K/otY//dnB9B6RNbDZNE5c9ChtpiyH/YUbUDP9v5smcR4l5fv9rahBc/CG9SK93zvUH+869sujK38GyS81m+rVA0KEcnq6Z5dVz0YQfHkjXNA5nu+Ed2l5ywW3zwbB5wtmvtTqxIt6RP5ZkGvFN9zpOzTXJ3qs3o8OcAb9t08PxSpT6mtq+087rkS52Yp5+YK54y0U+Rw4nxWfTerjMuOw2WQ586Odb+U4mx7nledG90t5cJ30Kvxwwr80JGyRdp/SZHwTFVn8xYVNBdaJveEdaJMJpL1gHFmN3/f1ZHtcQWv44Dx3ypF33Ik3QdZepT/juiEbUb0Eq2bPZKebzpWZ1JvVkirv0f57OLGtdX/y2C9crh19XdXr93b2vjiuv5vDj5QP1O/UJdhif9GfQHEmuqQDr3/qv6m/r79ePuP23/a/jN3fe+cxvxEFf5t/+W/mrqpxQ==</latexit> <latexit 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<latexit

    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<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tM9ZoNF+/Iw0I2pZoWiOJ0AgbhI=">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</latexit>
  14. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit 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<latexit

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<latexit 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u (Kv) = a <latexit 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sha1_base64="L0a9yaq7SoDrCaDqUKFsvOMilSM=">AABELnictVxbbxu5FWa2t016y7YvBfoyW2+KbJGmdhqgBRYLbCI7jtdO4kSyk91VEugyUpSMNYpGUi5at3+mry36D/ovCvSh6Ov2qX+h50IOORJnDsdNQ9jmUPzOOTxDHp5zSKU7SUbZbHPzH+fe+9a3v/Pd771//sL3f/DDH/344gc/Oc7S+bQXH/XSJJ0+6nayOBmN46PZaJbEjybTuHPSTeKH3RcN/PzhIp5mo3Tcmr2ZxI9POsPxaDDqdWbQ9PTiz9rzRtRO++ksutzej9qLxsfRp1G703h6cWPz6ib9i9YrW7qyofS/w/SDD/+o2qqvUtVTc3WiYjVWM6gnqqMyKF+pLbWpJtD2WC2hbQq1EX0eq1N1AbBz6BVDjw60voDfQ3j6SreO4RlpZoTuAZcEfqaAjNQlwKTQbwp15BbR53OijK1ltJdEE2V7A3+7mtYJtM7UM2iVcKZnKA7HMlMD9XsawwjGNKEWHF1PU5mTVlDyyBnVDChMoA3rffh8CvUeIY2eI8JkNHbUbYc+/4Z6Yis+93Tfufo3SXkJSqSaevRpTqGjFkQ/orc5h89YngQ4D4FCrMeItVek6xMa/Rj6L6H9LpRTqhmddKEsqfW0EtmA4kM2ROQuFB9yV0QeQPEhD0TkIRQf8lAjETslnfvxTSg+fFPkfB+KD3lfRD6A4kM+EJHHUHzIYxH5JRQf8ksReQuKD3lLRO5D8SH3RWQLig/ZEpFHUHzIIxG5A8WH3NHI8pU6hZISnZGwKm9AvcgDLUUCLTdE+W6SdfRhbwas6V4JVl7V2/DXj90O0Glcgt0JmHeDEqw883bBRvqxsi26TbuJD3tbxO7BDPBj90Ts5+p5CfbzgJX2ogQrr7UD6OfHytb3Djz5sXdE7F2o+bHyHnUPWvzYewE7xqQEeyhi76uXJdgQqz8twcp2vwl2xY+V96kW9PdjQ6zpvAQr29Nj8GD8WHm3egitfuxDEftIvS7BPhKxX4B192O/CNhh35ZgzR57gXaQIfkjMazYKmqdfFVibQLUOgL/JN9bEvKNu9AuYYY5ZkiYExGxmyN2AxEHOeIgWK4st6MZ+bsyl2aOaAYiuvnehLWZ2L+f98daEoDYzhHbK4gqjxTftRnLgrwL0yIhZ/nOhbWQMaW5/cZarOdDteU1iHsFBM/tZzTzr1C0hBEUaqqK2rN8j2dkRM9ViFcUvZlRGh4ybpZbBRf1WkR1PaiuiHrjQb0RUXMPai6iFh7UQkTZle/i2gEzwOof38WSnngGsI9cXiLwCm7ArnMb1mgE8+cQvMAH1HIP/jYp9pZKlWQYzeM+iVmOxwVLPIXaUm1Au40Ktym+TmiFxSAZ97ynY3x8wtzGUq85tsKn+U4e5RmTcDojkmeY00FvMaL1VI/OPrWcknfHtXr42/m6N7V6+B3S+Cl58Vyrh59p6WdnkL2lsa0zYJuwmiZa+7ZelwbnX5iGqV+gXRctLr7VEz1nkN7rmvT39JvZO8N7aVCN9WPr9Whkzviywvjq0LB6zhw916OC3hN7vaYW1R7JWMe9tl5XhpR20bGWwz7VfTPYp6/fjKnXo3EIHleDYu6lU687eyf5aGy9Ho1jxXnPU/LkTb0ejSE9sz5svR4NzLZ0dJxv63UtO2qAY2dbr2vVx5QFxhwQz3lusV7RlPykuaY2Iv+gOlvj+vzr+xjmbJ7kMUI1JevbltPp5ntZtUTGX4jBqs1qyoH+xdzxwYo0luqaGF+xDLPC/r5Ox+7xqPkD0GIEq5/PAKSceQISmpwEWu8EKG6JUVdxZAZ3TcThLBmsoNq6dSZ6i5YvZ42KbU+pVYrL7GitHttkrzOaexPyCQ9Is5IeDkrfcBlFSUMHBQ3J9Oro7q1er0Xtb4q4yQpiks+0Hp0I8UladZzq03rT0fElfcozg8JnPnb+YrZ5oK0Nxjwp2SKUpYqn28/kkdw23FevKJvj5s8ieqNorxZkNUZ0IpWJUajJFrM3vqRnS/uIzuSQB9PowXuMNJWJ4lMzzKJjPj0ii+raW4k36stk6LiekdU19rgaPXTQQw+6fozTgB3jLtRaEDMcwVMrIMq5kOsqJY1P1a/z09GU3mB1RJ8ULKShwfYmLljIqij7WYHKK0DjbOAoPZzGKh2Db69RkqN+nzw2di1a/kt0cmvOtzs0x8tnc3kmpk9crxHXiFYNn+ry0yoHlmDp/eQa+a/Vo0R+dTiiDZW4PnE4s17GdOIfUwQ7Ic84odUmrY5ibzc/tfqJ4XSozNk5nmanZCEjsn8R7E8pzcmIfty7A+YEnS1CQjYyxO6Mcu/G5+uMxDlm/biR4lsNdr7FZMvmxN/QdVdXRnORIwbeB05X5rbRyQH5gjFxnWrrbtd29e6DSHtPwp0lTNHOlcvE/2P6bX7MPNlYmxGoYXwDmbZ1vveRUsyCOurQLl9tg0xfV8qPchmeaKnt/mdl+qgg2TZFXCgP7tZ94NyjZ+aFs2RKcmdrfXgfrcrmIuXJih5xtAOK4tnuD/UOjHJfoV1yg9Zcm2bJEGbBLI8iTF8pi7zKt5pXkXoY7ez/Qt3quqg1pBgpm8FlDUn5/ZiiNVfKBGY1z98XtJr8Wp+u9KrmM6a5eOKs5a+h9UP4beQ2z2F0ugWrcJPmAFOwT1Yj3BKt9QjjdbPAy8xMQ8s+W352TppebstZ4mu2bjbGXtSmckiz5rXOWpj6WWg8d2g8D9Rhi84arRZNu7FET8XYoqVPK0P51eHWqkF5LlKWPTKDGgVI6cZSYVT7IlU5xjeotyKtTZFWB1arexrgrvkQpH+tr67ur/PdPVK3yLfpkQfG8UufVumIfC7TWh2pMQXkfF3bV3f1t6kFuXfJgiJlvseJK4ZPnXpUTnNJf6l3tpTsvLUI5t7SK93H2Ng21X+7hjyhNZHRujSI69Qj1vK7ckQrFumq43NElPnvkE/Ffkd1zOz2tu8kKvgTNt7kVWV5caQwJv1Lmbe9teh1z4lfI4oJ59q77gKt+m8YKTDGZBL8nmVGbwh3OT5JYI+2S/Zz3U7xKd7YkegqSb1UnwbYGI567Vx355YZsRnbr6Anat2+dV8PmV8SzFHid5YTvQ7taifaR12uPJ+NVkfvcsXnKj3MV/hafcypjxtZ2CiviGmrT4K5sET1uDAmhEu9UdSRv57kdWTm06lQyqa3oVzMNLCNeUbxknQPFBE+7+6y15v7WBhHd41el7AuNW6RKGE2LtX5AdfSYlbq/No+xK3nK3ejxNmJynYKQ93dLaz9ZgsZk/VLlJSz4d6u7O1ClCJnYZhCT/GN3rL40KX5CRT8HSlfdGg4huQOm+Df3lANtfMObkO81HXOaEbUgragvxJ7d/Q4iz2qdfTSoe7SD+EQzmMEupakH9FOWld2pixL7lIPp/+KrMBUxaL0tmf9Mbhc5JGsc6oznhFZNnk0I2W+i1N3LIZDyEiKXML58LmGNIqBMt9pqjcGQ10eQZFDHR7mHkPYO7e96/NyOVXra51LKA/eBcyJi8HhyV95rGL7hVioqfNG3j0HtA6DCupmt/hfx2H4WE71eYVyy+i7Zs8D3jr3i3VGFv3h+mvGcguZzeUcw3mm+eist+Tnx35fVOtNpc5o3j199EftHDC8lorzoLJ0jHdnkZU3lAqeC/hkSNV/1N/Oyd9GeJnTKJOjDiVzTlFOzfSQqZlvXPpGZz4LkcnSKZOpSM3GEU26EdtQe+oW/DRyD7Du7VD+LiX/Raz/+7N9aB2Q9TBZdM4ctKktpuyHPUXr07O9P1smMd7l5bu9LWjBs/ADasV7vnepP971bRXGVv4NEl7rd1Sq+oWIZPV0z66rLoygePLGOSDzPd+I7tJzFotvnp0EnC2a+1OrEi3pE/lmQbcU33Wk7NFcneizejw5wBv2nTw/FKnfUFtH23nccyXOh6WcD1c4Z6SdIofXzmfVd7PKuDQcLv08d7bQ/VKKs+15XnVudLuUC99Br8YPK/BDR8omaf8FRcJTVZ3Nm1fQnGuZ3BPWsTKZSNYDxpmd/H1XR7aLCl6LgPHvl6L3HUl3QZYu5b8jOmGbEr1E62aHpOebjtWZ1NsV0prvUTJNe9fRzgNza7E6S5/oeceZC/O/ESxpPeBuam4mStmTuIROl24ExkSJ70hK5wayPLI0IbJIVOaiJPOAb+8uAmRZBNAZCNIMRApDURK9gp9e3Nha/f8x1ivH165ubV7dun9947Ob+v/OeF/9XP1CXYZ943fqM5ihh+oIOP1B/Un9Wf2l8dfG3xv/bPyLu753TmN+qgr/Gt/8FxIsIkM=</latexit> v (K>u) = b <latexit 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<latexit 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<latexit sha1_base64="UEfo0q7/awH1ZlqoxebmYfYkUMI=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tM9ZoNF+/Iw0I2pZoWiOJ0AgbhI=">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</latexit>
  15. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit 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<latexit

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<latexit 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u (Kv) = a <latexit 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sha1_base64="L0a9yaq7SoDrCaDqUKFsvOMilSM=">AABELnictVxbbxu5FWa2t016y7YvBfoyW2+KbJGmdhqgBRYLbCI7jtdO4kSyk91VEugyUpSMNYpGUi5at3+mry36D/ovCvSh6Ov2qX+h50IOORJnDsdNQ9jmUPzOOTxDHp5zSKU7SUbZbHPzH+fe+9a3v/Pd771//sL3f/DDH/344gc/Oc7S+bQXH/XSJJ0+6nayOBmN46PZaJbEjybTuHPSTeKH3RcN/PzhIp5mo3Tcmr2ZxI9POsPxaDDqdWbQ9PTiz9rzRtRO++ksutzej9qLxsfRp1G703h6cWPz6ib9i9YrW7qyofS/w/SDD/+o2qqvUtVTc3WiYjVWM6gnqqMyKF+pLbWpJtD2WC2hbQq1EX0eq1N1AbBz6BVDjw60voDfQ3j6SreO4RlpZoTuAZcEfqaAjNQlwKTQbwp15BbR53OijK1ltJdEE2V7A3+7mtYJtM7UM2iVcKZnKA7HMlMD9XsawwjGNKEWHF1PU5mTVlDyyBnVDChMoA3rffh8CvUeIY2eI8JkNHbUbYc+/4Z6Yis+93Tfufo3SXkJSqSaevRpTqGjFkQ/orc5h89YngQ4D4FCrMeItVek6xMa/Rj6L6H9LpRTqhmddKEsqfW0EtmA4kM2ROQuFB9yV0QeQPEhD0TkIRQf8lAjETslnfvxTSg+fFPkfB+KD3lfRD6A4kM+EJHHUHzIYxH5JRQf8ksReQuKD3lLRO5D8SH3RWQLig/ZEpFHUHzIIxG5A8WH3NHI8pU6hZISnZGwKm9AvcgDLUUCLTdE+W6SdfRhbwas6V4JVl7V2/DXj90O0Glcgt0JmHeDEqw883bBRvqxsi26TbuJD3tbxO7BDPBj90Ts5+p5CfbzgJX2ogQrr7UD6OfHytb3Djz5sXdE7F2o+bHyHnUPWvzYewE7xqQEeyhi76uXJdgQqz8twcp2vwl2xY+V96kW9PdjQ6zpvAQr29Nj8GD8WHm3egitfuxDEftIvS7BPhKxX4B192O/CNhh35ZgzR57gXaQIfkjMazYKmqdfFVibQLUOgL/JN9bEvKNu9AuYYY5ZkiYExGxmyN2AxEHOeIgWK4st6MZ+bsyl2aOaAYiuvnehLWZ2L+f98daEoDYzhHbK4gqjxTftRnLgrwL0yIhZ/nOhbWQMaW5/cZarOdDteU1iHsFBM/tZzTzr1C0hBEUaqqK2rN8j2dkRM9ViFcUvZlRGh4ybpZbBRf1WkR1PaiuiHrjQb0RUXMPai6iFh7UQkTZle/i2gEzwOof38WSnngGsI9cXiLwCm7ArnMb1mgE8+cQvMAH1HIP/jYp9pZKlWQYzeM+iVmOxwVLPIXaUm1Au40Ktym+TmiFxSAZ97ynY3x8wtzGUq85tsKn+U4e5RmTcDojkmeY00FvMaL1VI/OPrWcknfHtXr42/m6N7V6+B3S+Cl58Vyrh59p6WdnkL2lsa0zYJuwmiZa+7ZelwbnX5iGqV+gXRctLr7VEz1nkN7rmvT39JvZO8N7aVCN9WPr9Whkzviywvjq0LB6zhw916OC3hN7vaYW1R7JWMe9tl5XhpR20bGWwz7VfTPYp6/fjKnXo3EIHleDYu6lU687eyf5aGy9Ho1jxXnPU/LkTb0ejSE9sz5svR4NzLZ0dJxv63UtO2qAY2dbr2vVx5QFxhwQz3lusV7RlPykuaY2Iv+gOlvj+vzr+xjmbJ7kMUI1JevbltPp5ntZtUTGX4jBqs1qyoH+xdzxwYo0luqaGF+xDLPC/r5Ox+7xqPkD0GIEq5/PAKSceQISmpwEWu8EKG6JUVdxZAZ3TcThLBmsoNq6dSZ6i5YvZ42KbU+pVYrL7GitHttkrzOaexPyCQ9Is5IeDkrfcBlFSUMHBQ3J9Oro7q1er0Xtb4q4yQpiks+0Hp0I8UladZzq03rT0fElfcozg8JnPnb+YrZ5oK0Nxjwp2SKUpYqn28/kkdw23FevKJvj5s8ieqNorxZkNUZ0IpWJUajJFrM3vqRnS/uIzuSQB9PowXuMNJWJ4lMzzKJjPj0ii+raW4k36stk6LiekdU19rgaPXTQQw+6fozTgB3jLtRaEDMcwVMrIMq5kOsqJY1P1a/z09GU3mB1RJ8ULKShwfYmLljIqij7WYHKK0DjbOAoPZzGKh2Db69RkqN+nzw2di1a/kt0cmvOtzs0x8tnc3kmpk9crxHXiFYNn+ry0yoHlmDp/eQa+a/Vo0R+dTiiDZW4PnE4s17GdOIfUwQ7Ic84odUmrY5ibzc/tfqJ4XSozNk5nmanZCEjsn8R7E8pzcmIfty7A+YEnS1CQjYyxO6Mcu/G5+uMxDlm/biR4lsNdr7FZMvmxN/QdVdXRnORIwbeB05X5rbRyQH5gjFxnWrrbtd29e6DSHtPwp0lTNHOlcvE/2P6bX7MPNlYmxGoYXwDmbZ1vveRUsyCOurQLl9tg0xfV8qPchmeaKnt/mdl+qgg2TZFXCgP7tZ94NyjZ+aFs2RKcmdrfXgfrcrmIuXJih5xtAOK4tnuD/UOjHJfoV1yg9Zcm2bJEGbBLI8iTF8pi7zKt5pXkXoY7ez/Qt3quqg1pBgpm8FlDUn5/ZiiNVfKBGY1z98XtJr8Wp+u9KrmM6a5eOKs5a+h9UP4beQ2z2F0ugWrcJPmAFOwT1Yj3BKt9QjjdbPAy8xMQ8s+W352TppebstZ4mu2bjbGXtSmckiz5rXOWpj6WWg8d2g8D9Rhi84arRZNu7FET8XYoqVPK0P51eHWqkF5LlKWPTKDGgVI6cZSYVT7IlU5xjeotyKtTZFWB1arexrgrvkQpH+tr67ur/PdPVK3yLfpkQfG8UufVumIfC7TWh2pMQXkfF3bV3f1t6kFuXfJgiJlvseJK4ZPnXpUTnNJf6l3tpTsvLUI5t7SK93H2Ng21X+7hjyhNZHRujSI69Qj1vK7ckQrFumq43NElPnvkE/Ffkd1zOz2tu8kKvgTNt7kVWV5caQwJv1Lmbe9teh1z4lfI4oJ59q77gKt+m8YKTDGZBL8nmVGbwh3OT5JYI+2S/Zz3U7xKd7YkegqSb1UnwbYGI567Vx355YZsRnbr6Anat2+dV8PmV8SzFHid5YTvQ7taifaR12uPJ+NVkfvcsXnKj3MV/hafcypjxtZ2CiviGmrT4K5sET1uDAmhEu9UdSRv57kdWTm06lQyqa3oVzMNLCNeUbxknQPFBE+7+6y15v7WBhHd41el7AuNW6RKGE2LtX5AdfSYlbq/No+xK3nK3ejxNmJynYKQ93dLaz9ZgsZk/VLlJSz4d6u7O1ClCJnYZhCT/GN3rL40KX5CRT8HSlfdGg4huQOm+Df3lANtfMObkO81HXOaEbUgragvxJ7d/Q4iz2qdfTSoe7SD+EQzmMEupakH9FOWld2pixL7lIPp/+KrMBUxaL0tmf9Mbhc5JGsc6oznhFZNnk0I2W+i1N3LIZDyEiKXML58LmGNIqBMt9pqjcGQ10eQZFDHR7mHkPYO7e96/NyOVXra51LKA/eBcyJi8HhyV95rGL7hVioqfNG3j0HtA6DCupmt/hfx2H4WE71eYVyy+i7Zs8D3jr3i3VGFv3h+mvGcguZzeUcw3mm+eist+Tnx35fVOtNpc5o3j199EftHDC8lorzoLJ0jHdnkZU3lAqeC/hkSNV/1N/Oyd9GeJnTKJOjDiVzTlFOzfSQqZlvXPpGZz4LkcnSKZOpSM3GEU26EdtQe+oW/DRyD7Du7VD+LiX/Raz/+7N9aB2Q9TBZdM4ctKktpuyHPUXr07O9P1smMd7l5bu9LWjBs/ADasV7vnepP971bRXGVv4NEl7rd1Sq+oWIZPV0z66rLoygePLGOSDzPd+I7tJzFotvnp0EnC2a+1OrEi3pE/lmQbcU33Wk7NFcneizejw5wBv2nTw/FKnfUFtH23nccyXOh6WcD1c4Z6SdIofXzmfVd7PKuDQcLv08d7bQ/VKKs+15XnVudLuUC99Br8YPK/BDR8omaf8FRcJTVZ3Nm1fQnGuZ3BPWsTKZSNYDxpmd/H1XR7aLCl6LgPHvl6L3HUl3QZYu5b8jOmGbEr1E62aHpOebjtWZ1NsV0prvUTJNe9fRzgNza7E6S5/oeceZC/O/ESxpPeBuam4mStmTuIROl24ExkSJ70hK5wayPLI0IbJIVOaiJPOAb+8uAmRZBNAZCNIMRApDURK9gp9e3Nha/f8x1ivH165ubV7dun9947Ob+v/OeF/9XP1CXYZ943fqM5ihh+oIOP1B/Un9Wf2l8dfG3xv/bPyLu753TmN+qgr/Gt/8FxIsIkM=</latexit> v (K>u) = b <latexit 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<latexit 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<latexit sha1_base64="UEfo0q7/awH1ZlqoxebmYfYkUMI=">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</latexit> Theorem: [Sinkhorn 1964] <latexit 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<latexit 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<latexit sha1_base64="gcU/kHyZLcweAP0wHsiW13HNtBs=">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</latexit> (u, v) converges. <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="c65tFjuRWb0VgfFjoHmUj+uQL6Q=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="tM9ZoNF+/Iw0I2pZoWiOJ0AgbhI=">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</latexit>
  16. Sinkhorn’s Algorithm Proposition: <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> <latexit 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<latexit

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<latexit sha1_base64="CAr6Lw+hs0edXn6cKgSuWJP1qY0=">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</latexit> Only matrix/vector multiplications. K <latexit 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<latexit 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Matrix-vectors <latexit 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<latexit 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parallelization <latexit 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<latexit 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GPU <latexit 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! Convolution on regular grids, separable kernels. <latexit 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<latexit 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<latexit 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sha1_base64="L0a9yaq7SoDrCaDqUKFsvOMilSM=">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</latexit> v (K>u) = b <latexit 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<latexit 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<latexit sha1_base64="UEfo0q7/awH1ZlqoxebmYfYkUMI=">AABEM3ictVzdbxu5EWeuX5f0K9c+FgX26kuRK9LUTgO0wOGAS2TH8dlJnEh2cndKAq20UpSstYq+8qEz+tC/pq8t+ty/o+hb0Yf2oei/0Pkgl1yJu8N10xC2uRR/M8NZcjgzpBKP0+F0trn513PvfeOb3/r2d94/f+G73/v+D3548YMfHU+z+aSbHHWzNJs8ijvTJB2OkqPZcJYmj8aTpHMSp8nD+EUDP3+4SCbTYTZqzd6Mk8cnncFo2B92OzNoenrxp+1FI2pnvWwWXW7vP2nPsnHUnjc+jj6N2nHj6cWNzaub9C9ar2zpyobS/w6zDz78nWqrnspUV83ViUrUSM2gnqqOmkL5Sm2pTTWGtsdqCW0TqA3p80SdqguAnUOvBHp0oPUF/B7A01e6dQTPSHNK6C5wSeFnAshIXQJMBv0mUEduEX0+J8rYWkZ7STRRtjfwN9a0TqB1pp5Bq4QzPUNxOJaZ6qvf0hiGMKYxteDouprKnLSCkkfOqGZAYQxtWO/B5xOodwlp9BwRZkpjR9126PN/Uk9sxeeu7jtX/yIpL0GJVFOPPsspdNSC6Ef0NufwGcuTAucBUEj0GLH2inR9QqMfQf8ltN+Fcko1o5MYypJaTyuRDSg+ZENE7kLxIXdF5AEUH/JARB5C8SEPNRKxE9K5H9+E4sM3Rc73ofiQ90XkAyg+5AMReQzFhzwWkV9C8SG/FJG3oPiQt0TkPhQfcl9EtqD4kC0ReQTFhzwSkTtQfMgdjSxfqRMoGdEZCqvyBtSLPNBSpNByQ5TvJllHH/ZmwJrulmDlVb0Nf/3Y7QCdJiXYnYB51y/ByjNvF2ykHyvbotu0m/iwt0XsHswAP3ZPxH6unpdgPw9YaS9KsPJaO4B+fqxsfe/Akx97R8TehZofK+9R96DFj70XsGOMS7CHIva+elmCDbH6kxKsbPebYFf8WHmfakF/PzbEms5LsLI9PQYPxo+Vd6uH0OrHPhSxj9TrEuwjEfsFWHc/9ouAHfZtCdbssRdoBxmQP5LAiq2i1slXJdbGQK0j8E/zvSUl3ziGdgkzyDEDwpyIiN0csRuIOMgRB8FyTXM7OiV/V+bSzBHNQESc701Ym4n9e3l/rKUBiO0csb2CqPJI8V2bsSzIuzAtEnKW71xYCxlTlttvrCV6PlRbXoO4V0Dw3H5GM/8KRUsYQaGmqqg9y/d4Rkb0XIV4RdGbGaXhIeNmuVVwUa9FVOxBxSLqjQf1RkTNPai5iFp4UAsRZVe+i2sHzACrf3wXS3riGcA+cnmJwCu4AbvObVijEcyfQ/ACH1DLPfjbpNhbKlWSYTSP+yRmOR4XLPEEaku1Ae02Ktym+DqlFZaAZNzzno7x8QlzG0u95tgKn+Y7eZRnTMLpDEmeQU4HvcWI1lM9OvvUckreHdfq4W/n697U6uF3SOOn5MVzrR5+pqWfnUH2lsa2zoBtwmoaa+3bel0anH9hGqZ+gXZdtLj4Vk/0nEF6r2vS39NvZu8M76VBNdaPrdejMXXGNy2Mrw4Nq+epo+d6VNB7Yq/X1KLaIxnpuNfW68qQ0S460nLYp7pvBvv09Jsx9Xo0DsHjalDMvXTqdWfvOB+Nrdejcaw473lKnryp16MxoGfWh63Xo4HZlo6O8229rmVHDXDsbOt1rfqIssCYA+I5zy3WK5qQnzTX1IbkH1Rna1yff30fw5zNkzxGqKZkfdtyOnG+l1VLZPyFBKzarKYc6F/MHR+sSGOpronxFcswK+zv63TsHo+aPwAtRrD6+QxAypmnIKHJSaD1ToHilhh1FUdmcNdEHM6S/gqqrVtnordo+XLWqNj2lFqluMyO1uqxTfZ6SnNvTD7hAWlW0sNB6Rsuoyhp6KCgIZleHd291eu1qP1NETdeQYzzmdalEyE+SauOU31abzo6vqRPeWZQ+MzHzl/MNve1tcGYJyNbhLJU8XT7mTyS24b76hVlc9z8WURvFO3VgqzGkE6kpmIUarLF7I0v6dnSPqIzOeTBNLrwHiNNZaz41Ayz6JhPj8iiuvZW4o36Mhk6rk/J6hp7XI0eOOiBB10/xmnAjnEXai2IGY7gqRUQ5VzIdZWRxifql/npaEZvsDqiTwsW0tBge5MULGRVlP2sQOUVoHE2cJQeTmOVjsG31yjJUb9PHhu7Fi3/JTq5NefbHZrj5bO5PBPTI67XiGtEq4ZPdflplQNLsPR+co381+pRIr86HNGGSlyfOJxZLyM68U8ogh2TZ5zSapNWR7G3m59a/cRwOlTm7BxPszOykBHZvwj2p4zmZEQ/7t0Bc4LOFiElGxlid4a5d+PzdYbiHLN+3FDxrQY73xKyZXPib+i6q2tKc5EjBt4HTlfmttHJAfmCCXGdaOtu13b17oNIe0/CnSVM0c6Vy8T/Y/ptfsw82VibEahhfANTbet87yOjmAV11KFdvtoGmb6ulB/lMjzRUtv9z8r0UUGybYq4UB7crXvAuUvPzAtnyYTknq714X20KpuLlMcresTR9imKZ7s/0Dswyn2FdskNWnNtmiUDmAWzPIowfaUs8irfal5F6mG0p/8X6lbXRa0hxUjZDC5rSMrvJxStuVKmMKt5/r6g1eTX+mSlVzWfEc3FE2ctfw2tH8JvI7d5DqMTF6zCTZoDTME+WY1wS7TWI4zXzQIvMzMNLfts+dk5aXq5LWeJr9m62Rh7UZvKIc2a1zprYepnofHcofE8UIctOmu0WjTtxhI9FWOLlj6tDOVXh1urBuW5SFn2yAxqGCClG0uFUe2JVOUY36DeirQ2RVodWK3uaYC75kOQ/rW+urq/znf3SN0i36ZLHhjHLz1apUPyuUxrdaTGFJDzdW1f3dXfphbkHpMFRcp8jxNXDJ86damc5pL+XO9sGdl5axHMvaVXuo+xsW2q/3oNeUJrYkrr0iCuU49Ey+/KEa1YpKuOzxFR5r9DPhX7HdUxs9vbvpOo4E/YeJNXleXFkcKI9C9l3vbWotc9J36NKCaca+86Blr13zBSYIzJJPg9yym9Idzl+CSBPdqY7Oe6neJTvJEj0VWSeqk+DbAxHPXaue7OLTNiM7ZfQE/Uun3rvh4yvzSYo8TvLCd6HdrVTrSPulx5Phutjt7lis9Vepiv8LX6mFMfN7KwUV4R01afBHNhiepxYUwIl3qjqCN/PcnryMynU6GUTW9DuZhpYBvzjOIl6R4oInze3WWvN/exMI54jV5MWJcat0iUMBuX6fyAa2kxK3V+bR/i1vOVu1Hq7ERlO4Wh7u4W1n6zhUzI+qVKytlwb1f2diFKkbMwTKGr+EZvWXzo0vwECv6OlC86NBxDcodN8G9vqIbaeQe3IV7qOmc0I2pBW9Bbib07epzFHtU6eulQd+mHcAjnMQRdS9IPaSetKztTliV3qYfTf0VWYKISUXrbs/4YXC7ySNY51RnPkCybPJqhMt/FqTsWwyFkJEUu4Xz4XEMaRV+Z7zTVG4OhLo+gyKEOD3OPIeyd2971ebmcqvW1ziWUB+8C5sTF4PDkrzxWsf1CLNTEeSPvngNah34FdbNb/K/jMHwsp/q8QrlN6btmzwPeOvdLdEYW/eH6a8ZyC5nN5RzDeWb56Ky35OfHfl9U601lzmjePX30R+0cMLyWivOgsnSMd2eRlTeUCp4L+GTI1H/UX87J30Z4mdMok6MOJXNOUU7N9JCpmW9c+kZnPguRydIpk6lIzcYRTboR21B76hb8NHIPsO7tUP4uJf9FrP/7sz1o7ZP1MFl0zhy0qS2h7Ic9RevRs70/WyYx3uXlu70taMGz8ANqxXu+d6k/3vVtFcZW/g0SXut3VKZ6hYhk9XTPrqsYRlA8eeMckPmeb0R36TmLxTfPTgLOFs39qVWJlvSJfLMgLsXHjpRdmqtjfVaPJwd4w76T54ci9Stq62g7j3uuxPmwlPPhCucpaafI4bXzWfXdrDIuDYdLL8+dLXS/jOJse55XnRvdLuXCd9Cr8YMK/MCRsknaf0GR8ERVZ/PmFTTnWib3hHWkTCaS9YBxZid/39WR7aKC1yJg/Pul6H1H0l2QJab8d0QnbBOil2rd7JD0fNOxOpN6u0Ja8z1KpmnvOtp5YG4tVmfpUz3vOHNh/jeCJa0H3E3NzUQpe5KU0InpRmBClPiOpHRuIMsjSxMii0RlLkoyD/j27iJAlkUAnb4gTV+kMBAl0Sv46cWNrdX/H2O9cnzt6tbm1a371zc+u6n/74z31U/Uz9Rl2Dd+oz6DGXqojoDT79Uf1B/Vnxp/bvyt8ffGP7jre+c05seq8K/x7/8CZJYkgw==</latexit> Theorem: [Sinkhorn 1964] <latexit 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<latexit 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<latexit sha1_base64="gcU/kHyZLcweAP0wHsiW13HNtBs=">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</latexit> (u, v) converges. <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="c65tFjuRWb0VgfFjoHmUj+uQL6Q=">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</latexit> Ki,j def. = e d(xi,yj )p " <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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min P2U(a,b) P i,j d(xi, yj)pPi,j + "Pi,j log ⇣ Pi,j aibj ⌘ <latexit 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Pi,j = ui Ki,j vj <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  17. 3. Sinkhorn Divergences <latexit 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<latexit 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<latexit 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<latexit

    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<latexit 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x <latexit 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<latexit 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d(x,y) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  18. Sinkhorn Divergences " > 0 : in general, there exists

    a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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<latexit 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min ↵ Wp ",p (↵, ) <latexit 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<latexit 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<latexit sha1_base64="04xOFucC5Ut2/g96/R1TregIzEs=">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</latexit> W"(↵, ↵) 6= 0 <latexit 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<latexit 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<latexit 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Wp ",p (↵, ) def. = min ⇡1=↵,⇡2= Z X2 dp(x, y)d⇡(x, y) + "KL(⇡|⇠) <latexit 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<latexit 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<latexit 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  19. Sinkhorn Divergences " > 0 : in general, there exists

    a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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<latexit 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min ↵ Wp ",p (↵, ) <latexit 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<latexit 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<latexit sha1_base64="04xOFucC5Ut2/g96/R1TregIzEs=">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</latexit> W"(↵, ↵) 6= 0 <latexit 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<latexit 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<latexit 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Wp ",p (↵, ) def. = min ⇡1=↵,⇡2= Z X2 dp(x, y)d⇡(x, y) + "KL(⇡|⇠) <latexit 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<latexit 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<latexit sha1_base64="Dlm1vC7jo1o6VzAcoMH+ypwqwp4=">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</latexit> Wp p," (↵, ) def. = Wp p," (↵, ) 1 2 Wp p," (↵, ↵) 1 2 Wp p," ( , ) <latexit 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</latexit> <latexit 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</latexit> 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</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  20. Kernel norms (MMD): <latexit 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<latexit 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<latexit

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||⇠||2 dp def. = R X2 d(x, y)pd⇠(x)d⇠(y) <latexit 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<latexit 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Sinkhorn Divergences " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not all us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structu of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gauss sample, the positions of the red dots that make up a model distribution ↵ optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, These registrations are performed in the unit square, with an Earth Move cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 " > 0 : in general, there exists a measure di↵ering from such that OT" ( , ) < OT" ( , ). (2) Consequently, minimizing the cost OT" (↵, ) with respect to ↵ does not allow us to recover ↵ = , but rather drives ↵ towards a “median axis”-like structure of "-localized Fr´ echet means constructed from [FCVP17] – see Figure 1. (a) " = 0.01. (b) " = 0.10. (c) " = 1.00. Figure 1: Entropic bias in the OT" loss. Starting from an arbitrary Gaussian sample, the positions of the red dots that make up a model distribution ↵ are optimized to fit an empirical distribution (in blue) as we minimize OT" (↵, ). These registrations are performed in the unit square, with an Earth Mover’s cost C(x, y) = kx yk. 5 Problem: <latexit 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<latexit 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<latexit 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min ↵ Wp ",p (↵, ) <latexit 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<latexit 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<latexit sha1_base64="04xOFucC5Ut2/g96/R1TregIzEs=">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</latexit> W"(↵, ↵) 6= 0 <latexit 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<latexit 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<latexit 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Arthur Gretton Proposition: || · || ||·||p is a norm for 0 < p < 2. <latexit 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<latexit 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<latexit 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Wp ",p (↵, ) def. = min ⇡1=↵,⇡2= Z X2 dp(x, y)d⇡(x, y) + "KL(⇡|⇠) <latexit 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<latexit 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Wp p (↵, ) <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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" ! +1 <latexit 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</latexit> 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</latexit> 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Wp ",p (↵, ) <latexit 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||↵ ||2 dp <latexit 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<latexit 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wCP4d8T+OvSL9fUic7Ds6gXRNGnXxel0BOvJ7euXGmulHBU90jtqMfqCNr4tnrbqI7npadKdumnyr7ximvJpMzV+4QP+a8R6SiW5qpF00e1D/F9/q6zY5AHa33JqiV77qCY+4JshX2g5K+WtYpqW/HcjJxCKNYhx/H1H5SK55B43p3Hwbhbk0+q5G+Gm+Tkk5mQomx2SfNavwdth/Xuatdt4iT9hr2i11VTy9rLJcgjbRnNzRhzWnkpog0Nk7NYV9XW6tJG8Wa8uhZm262sVNW1te+Lpf1/t7Mq+Zcd8n+bFsaRVT1fdqfCwn5tO//btvM2HurOW1nQTX3Vr23p/6ot3dQP2Xbk34XRpRrmG71c500mhVvmWvqWObl7KzQOzHOQPe+pFTM1rFRZ8TcrZHl8Phbx7XG9GT8TXdW9kzlmDM4nDuSuTXMnbl+vQHH81qZ5BrHnOcVzU2odfAdvS8kp2A+U3I1cjRyVUA9y0XER0yJuU9JoEfl+Np7M30eK+6JxBsE+vSGzh4k+iYW306GlJjnk/Yye2FeIbpEa394ts3ejgob5dIe5QTjJlS+fV/bacvSYZBIZGWXn9FUmXeIpN++LEqmEep4/343cpJxI/V5G35bNSNChFcsqXSVk1T5tmhMv64r3nveojvkEykzhrtqZg2M1p7fn0CSd8DyW7Udl3qCoYZO/7HmZIvvRIXmGGIomf5li1SxSsdzYPta1d04qUW5t2thqjlV1urDi5xDnGBqiRTedUG0hpbUoSqFaylNyeeNE1yPK0qVW3iar5H17UxozTLSf5R1DqN3wPeqLynzHOZ/Id/0PSI+4G6Kly1zFo5iz+pwHn6+YEsq8mzN40sa21Cnt3E30bBTO9V55Stmn2bap6up1HNHsvnqkHqu/BFpdyP21txRTnUewSWk/ln8/xWF2m5+9Qzh8O9KGep7d0Vi1j1DeeZTyXN/z5qe7E6BapruT0fVTDsnrk9hP+ZF6RuOGOtpgqU8jdhQ/prvXfJTxprlTytUo3ZBfTXcLZHhUqntJDe1Ikr1SfBO0SGPfBniLThDyvQtoh9Na0rUoLY6LyXkTTgd65tZ1fst8F9LlntZlR98C5Mt9pXdFhvZCXlEpOOcrJftyx0H6EyXnMWWMfUVns2VEbeKFMJX8mSFzkpLLYNOUcXuI7lUmn8+mOQ/feTZXxfvEXJYew1PuRGJ95KkWNRxDbTdHj+2W92YXJY6nzrQPspIb2nmKKxm3OZUrrj4P1MiiuwLI25H2JP6taAMxXPlckf15xdLVNd3WVaW7sHzym0FDuj3RWGuVbeHti67WUMdrGJ6LkgzzzM6KVhtH8aGTpq2vhdfWwrtJq2WW/shH4Qj6Gr7x+Cii7+Kcpp/Cu0TzrSE81hAqG3Tm4thB4yRI5WFGBS2gqynl6ZxElJ7HpvlzR+4z2f7Tqq3srlW8/fSUauBu7do8UPJbDeYTfu+j56fIO7d5TCttvmOd8l3TERLHN+aUFJ7/uEdpL+FhKz3VZ8ET9RclSWz0jEYer9QdlWQ+xV8HfCeU9KR22opFCcdXt6Psy7aup5HWZGzJn3czy7un4m44RIRg7NsG8hTipd2HUZ2bHksfT+lI0wm1/P2MX35ENSJuTwPoz1XVXbOfq4WzNxSL4fki+QUg+9d5RoQwp1mYdz4//mIQpvTBkgcU/85UvV9vQgq8kwdR2/TW1bt+2DcMafSMEZrcCWF+f4hPn2NLw1kr/9k0PHU+VK5Z4uJvkfC8D5/NQSn5nNkxlGlIZ90/1B7rJMI7DIhC8XYcm0OL5ph4ZwGOnR+pvw1SfU26aRZOZoXkf5qhkkIZQmcg0FvzWYMQP7s0x9AubWyZq9z+tAb2swt+r96pqY7Wq8QQ4ROr5zn5jS82t0Ccam/ViqjZqvusY36v4SBr7+W7X6pujF4BTGjUelDyIzaNGAozVbw/mtvHLFgqucWwiAzvOuyo4h3sXC+dCOvs0biuV2gLeQkeUtlX9SgWa/puFtH5qL+k+6fZP18E+62XVn58O4VexH3mNf99zOyBRAMctRet1Ie+1rHjCo1vXLGCH3up0fXij3BcJb/hUYwFXX2VO3bz28VAyfng4k3OvpNOslq0nXFwzyN2lPmFTabKZ3kblJZSbRsr7NBnffb/9N3lteIvvZZfXqzfX1u9v/bpg+Wf/43+FdgfqT9Rfwa6WFMfqJ+rJ2CTRyDTP6h/Uf+u/uPy7y//+fJfL/+Ns/7wBxrzxyr33+V//g+zBk+l</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[Carlier et al 2017] <latexit 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<latexit 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<latexit 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[L´ eonard 2012] <latexit 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<latexit 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<latexit sha1_base64="FktZ8qWo0YQDE1fIVBPQAYcO+ow=">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</latexit> Theorem: <latexit 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<latexit 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<latexit sha1_base64="SpoJyWCLZt0uGDg27y4FVNG5tpA=">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</latexit> Wp p," (↵, ) def. = Wp p," (↵, ) 1 2 Wp p," (↵, ↵) 1 2 Wp p," ( , ) <latexit 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</latexit> <latexit 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ja77ONjlyak52peuJ56bPF0c63C8jwW3vP0FXDD3Wl8jwDrlWffz5Xc0NKgdS68qWCdcsgqP88BvoJ/j+CvS79cU0c6D8+inhFFn35dlEJPvJ7cunKluVLCUd1Dta0eqUNo41vqbaM6npeeKtmlnyr7xiuuJZMyVx8QPuS/RqSjWJqrFk0f1T7E9/m7zl6BPFjrS1Yt2XMHxdxnZCvsAyV/taxVVNuK52bkFEKxDjmOr/+gVDyHxPPuPA7G3Zp8UiV/M9wkJ5/MhBRls0ua1/pdaDusd1e7bhMn6TfsFb2umlrWXi5BHmnLaG7GmNPKSxFtaJicxbqqtlaXNoo349W1MNtuZaWqrq39UCzt/7udVcm/7JD/u7Qwjqzq+bLbFRb2a9v5ZdvO23io229lQdf1Vb+2pV9VW7quH7LtyL8Lo0s1zDd6uc6bTAq3zLX0LXNy91ZoHJjnIHveUytmalipsuJvVsjy+Hws4tvjej1+JrqqeydzzBicTxzIXZvmTty+XoHi+K1N8wxiz3OK56bUOvgO3paSU7AfKbkbuRo5KqE+zEXHRUyLuE1Jo0XkB9l4Mn8fKe6LxhkE+/SGzB4m+iQW3k6HlprkkPcyemJfIbpFanx7t8zejQoa5tMd5gbhJFe+fF7Za8vRY5JJZGSUndOXmXSJp9y8L0qkEup5/nw3cpNyIvW7GX1bNiNBh1Ysq3SVkFX7tGlOvKwr3nveozrmEygzhbtqZw6O1ZzenkOTdMLzWLYflXmDooZN/rLnZYrsR4fkGWIomvxlilWzSMVyY/tY1945qUS5tWljqzlW1enCip9DnGNoiBbddEK1hZTWoiiFailPyeWNE12PKEuXWnmbrJL37U1pzDDRfpZ3DKF2w/eoLyrzvcr5RL7rf0B6xN0QLV3mKh7FnNXnPPh8xZRQ5t2cwZM29kAd087dRM9G4VzvpaeUfZptm6quXscRze6ph+qR+kug1YXc33hLMdV5BJuU9mP591McZLf52TuEw7cjbahn2R2NVfsI5Z1HKc/0PW9+utsBqmW62xldP+WQvD6J/ZQfqqc0bqijDZb6OGJH8SO6e81HGW+aO6ZcjdIN+dV0H4AMD0t1L6mhHUmyV4pvghZp7NsAb9IJQr53Ae1wWku6FqXFcTE5r8NpX8/cus5vme9CutzVuuzoW4B8uS/1rsjQXshLKgXnfK1kX+44SH+i5DymjLEv6Wy2jKhNvBCmkj8zZE5SchlsmjJuD9G9zOTz2TTn4TvP5qp4n5jL0mN4yp1IrI881aKGY6jt5Oix3fLe7KLE8dSZ9n5WckM7T3El4zancsXV574aWXRXAHkr0p7EvxVtIIYrnyuyP69Yurqi27qqdBeWT34zaEi3JxprrbItvH3R1RrqeA3Dc1GSYZ7ZWdFq4yjed9K09bXw2lp4N2m1zNIf+SgcQl/DNx4fRvRdnNP0U3iXaL41hMcaQmWDzly8ctA4ClK5n1FBC+hqSnk6RxGl57Fp/tyR+0y2/7RqK7trFW8/PaYauFO7NveV/FaD+YTf++j5KfLObR7TSpvvWKd813SExPGNOSWF5z/uUtpLeNhKj/VZ8ET9RUkSGz2jkcdrdVslmU/x1wHfCSU9qZ22YlHC8dWtKPuyretJpDUZW/Ln3czy7qq4Gw4RIRj7toE8hXhp92BU56bH0sdTOtR0Qi1/L+OXH1GNiNuTAPoLVXXX7Bdq4ewNxWJ4vkh+Acj+dZ4RIcxpFuadz4+/GIQpfbDkAcW/M1Xv15uQAu/kQdQWvXX1rh/2DUMaPWOEJndCmN8f4tPn2NJw1sp/Ng1PnQ+Va5a4+FskPO/DZ3NQSj5n9grKNKSz7h9rj3UU4R0GRKF4O47NoUVzTLyzAMfOD9XfBam+Id00CyezQvI/yVBJoQyhMxDorfmsQYifXZpX0C5tbJmr3P60BvazA36v3qmpjtarxBDhE6unOfmNLza3QBxrb9WKqNmq+6xjfq9hP2vv5btfqm6MXgFMaNS6X/IjNo0YCjNVvD+a28csWCq5xbCIDO867KjiHexcL50I6+zRuK5XaAt5Ce5T2Vf1KBZr+k4W0fmov6T7p9k/nwX7rZdWfnw7hl7EfeY1/33M7IFEAxy1F63Uh77SseMKjW9csYIfe67R9eKPcFwlv+FRjAVdfZU7dvPbxUDJ+eDiTc6+k06yWrSVcXDPI3aU+YVNpspneRuUllJtGyvs0Gd99v/43eW14i+9ll+er99bW7239vn68s/+Vv8K7I/Vn6g/A12sqY/Uz9RjsMlD1b7xjzf+5cZ/3PjPi3+4+OeLf734N876oxsa88cq99/Ff/0f3pWsEw==</latexit> 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</latexit> [Ramdas, Garc´ ıa Trillos, <latexit 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<latexit 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<latexit 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Cuturi, 2017] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  21. Sinkhorn Divergences concave <latexit 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</latexit> <latexit 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</latexit> 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</latexit> <latexit

    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</latexit> concave <latexit 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</latexit> 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S" (↵, ) "!+1 ! 1 2 h ↵ , C ? (↵ ) i; (4) which is, if C(x, y) = kx yk, a well-known quantity: the Energy Distance MMD. Interpolating between two standard families of divergences, the Sinkhorn divergences could be expected to share some of their desirable properties. The present paper shows that this is, indeed, what happens: we prove that S" is a convex function of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) Information theoretic Sinkhorn divergences. Before moving on to our contributions, let us mention [AKO17] for the definition of another positive divergence based on Sinkhorn’s iterations, over the discrete probability simplex. The present work opts for a rather di↵erent point of view, as we put forward the geometric structure of our feature space and handle continuous probability densities. min ↵ Wp ",p (↵, ) <latexit 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S" (↵, ) "!+1 ! 1 2 h ↵ , C ? (↵ ) i; (4) which is, if C(x, y) = kx yk, a well-known quantity: the Energy Distance MMD. Interpolating between two standard families of divergences, the Sinkhorn divergences could be expected to share some of their desirable properties. The present paper shows that this is, indeed, what happens: we prove that S" is a convex function of each of its input and that, for any measure ↵, 0 = S" ( , ) 6 S" (↵, ). (5) Information theoretic Sinkhorn divergences. Before moving on to our contributions, let us mention [AKO17] for the definition of another positive divergence based on Sinkhorn’s iterations, over the discrete probability simplex. The present work opts for a rather di↵erent point of view, as we put forward the geometric structure of our feature space and handle continuous probability densities. min ↵ Wp ",p (↵, ) <latexit 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<latexit sha1_base64="nYWupsQXdFV69lfCLDQSS6Yalhc=">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</latexit> Wp p," (↵, ) def. = Wp p," (↵, ) 1 2 Wp p," (↵, ↵) 1 2 Wp p," ( , ) <latexit 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</latexit> <latexit 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</latexit> <latexit sha1_base64="MMvpdcWTyNwp0Jp9lABlK/06LIw=">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</latexit> Theorem: <latexit sha1_base64="WCp6tVB6SGFNLVO3ywKAXDF/o30=">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</latexit> <latexit 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<latexit 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<latexit 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[Feydy, S´ ejourn´ e, P, Vialard, Trouv´ e, Amari 2018] <latexit 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<latexit 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<latexit sha1_base64="+1SAgCrta7UxoRWrAdXNR0bYA1Q=">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</latexit> W",p > 0 and Wp ",p (·, ) is convex. <latexit 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</latexit> <latexit 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</latexit> 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6fxOQKsVx59v1ByQkuD5rnwpIINSiGz/DwGeAT/HsNfl365pI51Gh5FPSeKPv26KIWueD25deV653oTjuoeqB31UB1CHd9WbxvV8bj0RMkq/VTZJ15xKZk3M/Uh4UP+a0Q6iqW5ZtH0Ue1DfF886+wI5MFSX7JKyR47KKc+J1thHyjpq2WtotpWPDYjuxDKZchxfP0LpeIxJB53534wrtbknSrFk+GygnwyElKWzc5pUet3oe6w3l31uk2cpN2wZ/S6amJZ+2IOikhbRnMyxoxmXspoQ8OkLJdVtbW6tFE+Ga+uhdl2KzNVdW3tx2Jpf+h2ViX/skP+79PCOLKq58tuV1jYH23n/9t23sZD3X4rC7qur/qjLf2+2tJ1/ZBtR/5VGF0qYT7Ry7XfJCudMtfSp8zJ2VuhfmCRg6x5T62YqWG9lRl/M0NWxBdjEd8a1+vxM9FV3TOZY/rgvONAzto0Z+L29QwUx29tGmcQe55RPDeh2sFn8LaU7IL9WMnZyNXI0QLqo0J0XMa0iNuENFpGfpj3J4vnkeK6aBxBsHdvyOhhondi4el0aKlJAbma0xP7CtEtU+PTu2X0blTSMO/uMCcIJ4X8FdPKWluOHpNcIiOjrJy+yqVLPPnmdVEilVAv8uezkZuUEqnfzenbshkJOjRjWaWrhKzap02z42VD8drzHpUx70CZKlxVO3VwrOb09hyapBMex7L9qIwblDVs0i96XqbIfnRIniGGokm/SLFqFKmcb6wfG9o7J5UotzZtbDXHqjKdW/FziHMMDdGim06otJDSehSlUCkVKbm8caLLEWXpUi1vk1Xyur0J9Rky7Wd5xRBqN3yO+rwy3VHBJ/JZ/wPSI66GaOk8V/Eop6ze58H7KyaEMvdmD57UsfvqhFbuJno0Csd6rzy57NNo20R19TyOaHZPPVAP1T8CrS6k/tabi4lOI9hkYT2Wfz3FQX6an71COHw60qZ6lp/RWLWOUO65l/JMn/Pmp7sToLpIdyen66ccktcnsZ/yA/WE+g11tMFSn0SsKH5IZ6/5KONJcyeUqrFwQn413fsgw4OFspe3oRVJslaKT4IWaezTAG/SDkI+dwHtcFJLuha9i+NiUl6H074euXXt3zLfQrp8qnXZ0acA+VJf6VWRobWQV5QLTvlKybrccZB+pmQ/pvSxr2hvtvSoTbwQplLcM2R2UnIebJrSbw/Rvcrl89k0p+Ezz2aqfJ6Yy9JjeMqZSKyPItWyhmOo7Rbosd3y2uyyxPHUmfZ+nnNDu0hxJec2o3zFlee+Gll0VwB5K9KexL+VbSCGK+8rsp9XLF29odO6qnQXlk9+M2hIpycaa62yLTx90VUb6ngNw3O+IMMst7Oy1cZRvOekaetr7rW18GrSapmlPfJROIS2hk88PoxouzilaafwLNFibQj3NYTKJu25OHLQOA5SuZdTQQvoakpFOscRuee+aXHfkXtPtn+3ais/axVPPz2hErhTuzT3lfxWg3nC7z56foq8cpv7tFLnO9Yu33UdIXF8Y3ZJ4f6Pu/TuJVxspSd6L3ii/mFBEhs9pZ7HK3VbJblP8ZcBnwklLan9bsWihP2rW1H2ZVvX40hrMrbkT7uVp32q4k44RIRg7NMGihTipd2DXp2bHksfT+lQ0wnV/L2cX7FHNSJujwPoL1XVWbNfqrmzNRSL4fEi+QUg+9d5RoQwu1mYdzE9/mIQvumDJQ8o/p2qer/ehBR4JQ+itumuq1f9sG8YUu8ZIzQ5E8L8/hDvPseahqNW/r1puOt8qFyjxOXfIuFxH96bg1LyPrMjyNOQ9rp/oj3WcYR3GBCF8uk4NocWjTHxygLsOz9Q/xak+pp00yztzArJ/zhHJaU8hPZAoLfmvQYhfnZujqBe2thFrnL60zrYzy74vXq7pjparxJDhHesnhXkN77YnAJxor1VK6Jkq86zjvm9hv28vi+e/VJ1YvQKYEK91v0FP2LTiKEwVeXzo7l+TIO5klMMy8jwqsOOKp/BzuXSibDOHvXreqW6UJTgHuV9TfdisaTv5BGdj/pLOn+a/fN5sN16aaXHuxNoRdx7XovfY0YPJBrgqL1spT70Gx07rlD/xhUr+LEXGl0v/gjHVfIbHuVY0NVWuWM3v10MlOwPLp/k7NvpJLNF2zkH9zhiR5lf2GSqvJe3Qe9SKm1jhR161nv/T95bXi//0uvizfON1fW11fUvNpZ//a/6V2DfUX+j/g50sa4+Vr9Wj8AmD0Gm/7rx0xs/v/GLi99e/O/lzy7f4aQ/uaExf60K/12+/38AEWQr</latexit> 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W",p(↵n, ) ! 0 () ↵n weak⇤ * <latexit 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<latexit sha1_base64="CpmzpF5QAPVTm4deEwUeAEshne0=">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</latexit> If e dp " is positive: <latexit 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<latexit 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</latexit> <latexit sha1_base64="foMVsrwod8b/WdI3pq5gRhQougE=">AAB3C3ic7V1LcxxHcq5dP3YFP1brPfrSNigvyCVpAJIthzYUXhAASVAQAREARQkDYufRAIaal6ZnQECzo3/gg3+Hjz44wvbBPvhH+O7L/gTfnI/Kruqe6qpqUArvhlYdAntq6svMysrKqqzXtEa9bjZZXf3v733/937/D/7wBz98a+mP/vhP/vRHb//4z55nw+m4nR61h73h+EWrmaW97iA9mnQnvfTFaJw2+61e+mnri038/tPLdJx1h4PDyfUoPek3zwfds267OYGk07d/tnOW3Epfzu41zsbN9qzzcjSfNdJR1u0NB/P5raSbJaNh1p10L9MPTt9eXr2/Sv8liy9r+mVZ6f/2hz/+4H9UQ3XUULXVVPVVqgZqAu891VQZPMdqTa2qEaSdqBmkjeGtS9+naq6WADuFXCnkaELqF/D3HD4d69QBfEaaGaHbwKUH/48Bmah3ADOEfGN4R24JfT8lypi6BM87NZ5E7ainalPtqiO1pbbVAfHwP0uE2wS+I3UNPLsg/QWUMFErIM1t+HcdSr+q3oe3RyBji/KkIGOi9uFfxKQkKVP6WG2oQ/WYeK/A+8fwdrtSSzPSTgb/N4HnhUefkvMMJMX6yUiDodwZyNcHmcN0kXuf8mEpYmW4C3mLnExKG2xqnmumiqKUHO2rr668kp6DrpuQekH258+bQXoLJMX6nFId+XUwJtrX3lwTqv8p2eoY+N8l++Y2M6ZypLkGu/BNR7elBFAp0MC0Af09h7QhSJcQjzFxkLJlHglsqli2MdXcXOt4B9IzkmVCtZDAJ6aQ0uczkhBbW0o5WS9SAygzSjSAtJS4dDySXFm0mX91TrvVSzsRNGuDyzwmuVL1mhB9kgzLO4PvWuSNsN11KQ+WG/Ojp8J2MwMvJXIwDSxTSlyxjLPcwmfQPu1v5h7MCN7QMyBqX78PgXeXfCC2Ah8adXhGeWbgj1KqsUEktq31NSPfNCYNst1de3E9XaYZeMFU69CXf5znf6ZtYQx150OkUKN9rZNteG/SJ9MbuFB3qI1l5LVm4BObpIlEp02Jgq2Vams6I0tqaqtvWJ/RIq5JjhlYyxAkS7UHEplM3hnZUUY80f5Qqsf0F3N0F2hLL+bCHZJXkPKHe6wEtPZUPYfW+kztwds26ANTDiP7KykNtpVL8gGoNem3Z+QT5rrNiNcxcvM3Y93LHkPPdl/9LdXpCbWh++pd+oT58Ls19Xf689eUtkS9w4RaAtfPh5SPUfhdl7xMv+LbHmkL27H5HnPjd+wLi+lLGsXScw7bC34IEgpeOBe/t+lk5DFQ+teQq0NWOitIZdKxtPfg+Zp62gbp64x6KZSB9fphBXYJMKzpVHswo3/pE2Pqkfsvf23eou+m5B+wdLe+kzXVIIpX8Bd7fftzRp9XAHebtP/t1Ge4NieBehRb+67V3LfXxsJ1gr0Nxjcn3prx1cnfvEGd8PdV9XJ1Q32uqGUoEdp6TJ0u1dY75m/Ce4dk6hG9c0uqRUoo1bpDB/MoP7kU7FU56tqk/tTXhxqLWBxfYpTQJR9qLOKfwC5Rl1+rf9PxagJl41FqmyzQP2Z10fwlPEjzp/DM1VtE8wikNqVM4BPHUqzdjMZJl/DWhrcORcg83khIW+d6POwr3VN4WM8y2m/BM6PUuRe5CY8LuRlEPoLHhXwURO7C40LuBpH78LiQ+5ZP8NXYATwu/EGQ8yfwuJCfBJHP4HEhnwWRz+FxIZ8HkZ/D40J+HkQ+hMeFfBhEfgSPC/lREHkIjwt5GEQeweNCHgWR2/C4kNsaWd1Sx7qf6ZAn8/HYoNjM5tGGf3uQshGU7wHN6riwDyLadLsCG27VWxRTubBbETpNK7DbEXZ3VoENW96jfAxdxoZ90eN8fFnGPg5idyiSdGF3gtgn6lUF9klES/uiAhtua7vUW7mwYe/7sZ5JWMR+HMQ+1eOERWy4j9qDFDd2L6LHGFVg94PYT9SXFdgYrz+uwIb9/kEe4Zex4X7qMI9DytgYbzqtwIb96XMYwbix4d7qU0h1Yz8NYl/ouYxF7Isg9jOaa3NhP4voYb+qwEofy/Og5zQeSaHF+qg181aJbzhL3Azw7+V9C75hH9UJYs5zzDlhynOHLs8qiEeRiN0csRstV5b70YzGu2EuBzniIBLRyvumFsVbofydPH+HIpowYitHbJUQvhEpr3Yw7pJGF5ISQtozIxdRZRrm/hvfUm0Pfs8riL0Cgm37giwfV4QmFAehpnzULvI+npEJffYhcJ75LC+l8AjjJrlXsFFXQVTLgWoFUdcO1HUQNXWgpkHUpQN1GUSZlm/jGhEWYPT/muZUznLvFp4LL66W7kGPuw29H6bswb8xs+H+mB7nHrCnNDM84ovHClcylmnNSOLCLYqweY0lpTUKzLkH7U1W93A1Y6ZbHfvhed6XJ0rWP+LpdEme85wOjhcT1a9N5yNKmdP4LtWrbHXwj/OWL2/18NukcV4PGuRrLvH4iZZ+cgPZDzX28AbYA2hPI619816XBs/AMA15X1INPRuPtdrXNoP0rmrS39E1s3ODeuGVQ3sVsb6OMqt8WaF8dWgYPWeWnutRwfETj3vlLaldkoGOfM17XRmG1I8OtBzmU92awTwdXTPyXo/GPoy5Ninqnlnvda13lJfGvNej8ZxyclRh3uvR4N0PrA/zXo8Gzrc0daRv3ut6dtQAR8/mva5Xx/n1VJmdA5xixkVjGinJPpIujRD88zX2qH+xH8NZm5d5lOCnZEa31XRaKg3SMRGJrLPVkwNHGFM9CtsEFK4QhOetmjRutnv0Mne7V0dd74Le7H0w/hm5Hsgk8xApzdy/1PtKfBi7LAa3HsShXZyVUA2dOgmOEA1fnikqpp1SaigWM6U1emyQh87I2kY0DtwlzYb0sFvSQ5hiSEO7BQ2F6dXR3Ve6hRa1vxrEjUqIUW5pbVph5H2R/tjUpfUDS8fv6JUdXPXjdZ6qfVxn5HWaJIuPp51P5o7sNOxJ7yozr83fJVSj6KEuyU90aRUqvFdLZoh5/D2jz4b2Ea3+FVfSEk1FdsPhzDnOoSfkQ20PG+KN+pJZOX7PyM+KBw6t2Bn0uQNdN6bZpB0+B/D5GZR7M2qPj1kHHJK+x+pevrtySPXnj+Ht1cZGTkP2zdn+0RdXXxSo8OpvquPyeBplOoJvLFAKx/kueUy0WvT77+Q7Kyc0Euh5bbl67qVDXNeJa0JtZkztcGat2RsOLMHM+c06jVf9pUR+dTiiBw1xfWlxZr0MaPdsShHriEbCPWprobZRzG3PSJW/EU77SnZ34/r1kPxjQt4vgd5pSDaZ0P/2PnBeRxN/0CMPGeN1uvloxjW26QZtzIzbuor3Xhp7c+2XtVsX79HgCIF7gXnJtkUnuzT2490OY+3bTdv29z0p7ZSVHd22lTBFYysrxP82/ZX/xU6WFywCNcz7WtnTuepjSDEK6qhJfbzfB0leW8pbuQwvtdSm9zMy3SpItkURVqL34XaAc1vx3mbkhVYyJrmzhTyyj726j0LKo5IeZfdKQ3v9c93/otx3qY9cpjbXUOakgEQNkjc0b1zm6+dVpB5HO/tWqBtdF7WGFBNl5mxZQ6EZ/ZSiM1vKHp2t4N2+qaa0qPVxKZefz4BssW+15V9B6l/AX5FbPsfRaRW8wgOyAaZgPhmNcEqykCOO14MCL7FMoWU+G37GJiWXnXKTeJq9m4mpL2tT4V3zV3qWQt5vQuOVReNVpA4PaXXRaFHSxROdBiOLQ70+GcuvDrfDGpSnQcrhEZmguhFS2pFUHNVOkGo4whfUV0Faq0FauNfQnv2323wM0t3Wy637V4Xe/SGNbtpKzjLwqLN87sMfqTEF5P2e9rB2+29QCvK3d6va55dwXrJNFETWv9J925A8vfEJslfptc4jXrZB7+8uIPvUKjJqmYJ4j3LI7k1bjqTkk+5bo46E5vqbNKrikYc/ZrZzm1pJCiMKE29yuzK8euqnOl4YUB1M1K+93HYWItgdK4ZNFK9z9vIYA2v517VqGSNLtgyZTXCPLzOqJezreP2Ax7V8umXRW/Hq3QC0L3Z3X/H+2g8jPA1Hvu7d0PjdVMdNWLY7kBM1b2relSPMrxfNMcTvJut4vEO5n+9RLn6+Ga2m7uuKn+vR2ijJtfEGcm2U5NpYkKuqfqYlfZh6mlIeO+4xMWgR01A/j+bCEtXjwpgYLhs3KMtG7bJs3KAsG7XLUq8UdeSvJ3kdmXndL5ay5BbKxTkd9uMXFJmG9tgiwjWOXnGOm28HytFaoNcirE2NU0KUcNZzqGdi7B4N5//eWujvOfUtb6/fs3r8qh5ZqNu9suknuRdKqYfhuN/vv4u7KxItq0gQnu9iCm3Fu6WrInGb5s/hwb+JcsXhwjFmlvYAIgk8MxJ3S4B/n8mX+p1njhNKQb/WKc1yNHU5izn8OvrSom7Tj+EQzwNPfoek79Jopa7sTDksuU09nv5r8gJjlQalNznrl8HmEi7JIqc65emSZwuXpqtnOuuXRTjElKTIJZ4Prx+FSnGm5CaQemUQ6uESFDnU4SE7ROLq3OSuz8vm5NfXIpdYHtwLyMqW4HCFtTomNPliPNTYqpFvngN6hzMPdekt3rQcwsdwqs8rlhvfUPIqotY5X6rnvnFsX7/NGG4x1lzNMZ7nMC+dGS25+fG4L6lVU0OrNN88fRyPGhsQXjPFM85h6RhvW5GRN5YKrsC4ZBiq/42SgfFVMsRSkZWgakqSI0xNTrG6SiXfxchk6FTJVKRmnyJO6J6TTLdfM1OX6W+3abVRVh4/UAmtSPHKY6ZkhzO3n8+pFWAO+7R5+US0ILlV2We1YzE4nhBfE4/qK759ZRHN8dZN0DP1nhoFV12EkhnR4/g5KT3VJ9urNVY9V7aczy77+NilCem5mheup55aPN1cq7A8j4X3PH0B3HB3Gt8jwHrl2fdLJTe0NGidC28qWKccssrPc4DH8O8J/HXpl2vqROfhWdQLoujTr4tS6InXk1tXrjRXSjiqe6h21CN1BG18W71pVMfz0hMlu/RTZd94xbVkUmbqXcKH/NeIdBRLc9Wi6aPah/i+eNfZMciDtb5k1ZI9d1DOfUG2wj5Q8lfLWkW1rXhuRk4hlOuQ4/j6D0rFc0g8787jYNytySdVijfDZQX5ZCakLJtd0qLW70HbYb272nWbOEm/Ya/oddXEsvbFEhSRtozmZowZrbyU0YaGyVmuq2prdWmjfDNeXQuz7VZWqura2nfF0n7b7axK/mWH/N+khXFkVc+X3amwsN/Zzv+37byJh7rzRhZ0U1/1O1v6TbWlm/oh2478uzC6VMN8o5frvElWumWupW+Zk7u3QuPAIgfZ855aMVPDSpUVf7NCVsQXYxHfHteb8TPRVd07mWPG4HziQO7aNHfi9vUKFMdvbZpnEHueUTw3odbBd/C2lJyCfV/J3cjVyNEC6r1CdFzGtIjbhDRaRr6bjyeL95HivmicQbBPb8jsYaJPYuHtdGipSQF5P6cn9hWiW6bGt3fL7N2opGE+3WFuEE4K5Svmlb22HD0muURGRtk5fZVLl3jKzfuiRCqhXuTPdyM3KSdSv5fTt2UzEnRoxbJKVwlZtU+b5sTLuuK95z2qYz6BMlW4q3bq4FjN6c05NEknPI9l+1GZNyhr2ORf9LxMkf3okDxDDEWTf5Fi1SxSudzYPta1d04qUW5t2thqjlV1Orfi5xDnGBqiRTedUG0hpbUoSqFaKlJyeeNE1yPK0qVW3iar5H17ExozZNrP8o4h1G74HvV5Zb7jgk/ku/4HpEfcDdHSZa7iUc5Zfc6Dz1dMCGXezRk8aWNb6pR27iZ6Ngrneq88pezTbNtEdfU6jmh2Xz1Uj9RfA60u5P7KW4qJziPYZGE/ln8/xWF+m5+9Qzh8O9KGepbf0Vi1j1DeeZTyTN/z5qe7E6C6SHcnp+unHJLXJ7Gf8kP1lMYNdbTBUp9G7Ch+RHev+SjjTXOnlKuxcEN+Nd0tkOHhQt1LamhHkuyV4pugRRr7NsBbdIKQ711AO5zUkq5FaXFcTM6bcDrQM7eu81vmu5Au97QuO/oWIF/uK70rMrQX8opKwTlfKtmXOw7Sz5Scx5Qx9hWdzZYRtYkXwlSKZ4bMSUoug01Txu0hule5fD6b5jx859lMle8Tc1l6DE+5E4n1UaRa1nAMtd0CPbZb3ptdljieOtM+yEtuaBcpruTcZlSuuPo8UCOL7gogb0fak/i3sg3EcOVzRfbnFUtX13RbV5XuwvLJbwYN6fZEY61VtoW3L7paQx2vYXjOF2SY5XZWtto4ig+cNG19zb22Ft5NWi2z9Ec+CkfQ1/CNx0cRfRfnNP0U3iVabA3hsYZQ2aAzF8cOGidBKg9yKmgBXU2pSOckovQ8Ni2eO3KfyfafVm3ld63i7aenVAN3a9fmgZLfajCf8HsfPT9F3rnNY1pp8x3rlO+ajpA4vjGnpPD8xz1KewEPW+mpPgueqJ8tSGKjpzTyeKnuqCT3Kf464DuhpCe101YsSji+uh1lX7Z1PYm0JmNL/rybed49FXfDISIEY982UKQQL+0+jOrc9Fj6eEpHmk6o5e/n/IojqhFxexJAf6aq7pr9TM2dvaFYDM8XyS8A2b/OMyKEOc3CvIv58ReDMKUPljyg+Heq6v16E1LgnTyI2qa3rt71w75hSKNnjNDkTgjz+0N8+hxbGs5a+c+m4anzoXLNEpd/i4TnffhsDkrJ58yOoUxDOuv+gfZYJxHeYUAUyrfj2BxaNMfEOwtw7PxQ/X2Q6ivSTbN0Misk/5MclZTKEDoDgd6azxqE+NmlOYZ2aWMXucrtT2tgP7vg9+qdmupovUoMET6xel6Q3/hicwvEqfZWrYiarbrPOub3Gg7y9r5490vVjdErgAmNWg8W/IhNI4bCVJXvj+b2MQ2WSm4xLCPDuw47qnwHO9dLJ8I6ezSu65XaQlGCB1T2VT2KxZq+m0d0Puov6P5p9s8XwX7rhZUf306hF3GfeS1+HzN7INEAR+1lK/Whr3XsuELjG1es4MdeanS9+CMcV8lveJRjQVdf5Y7d/HYxUHI+uHyTs++kk6wWbecc3POIHWV+YZOp8lneBqWlVNvGCjv0WZ/9P317ea38S6+LL8/X76+t3l/7ZH35Fw/0r8D+UP25+kvQxZp6X/1CPQabPAKZ/lH9i/oP9Z+X/3D5z5f/evnvnPX739OYn6jCf5f/9X/jL0oX</latexit>
  22. Sample Complexity Theorem: n d = 2 d = 3

    Optimal transport: su↵ers from curse of dimensionality. ! Adapt to support dimensionality [Weed, Bach 2017] E(|||ˆ ↵ ˆ||k ||↵ ||k |) = O(n 1 2 ) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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↵ <latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit 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log(n) log(n) E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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log(Wp(ˆ ↵, ˆ ↵0)) <latexit 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<latexit 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  23. Sample Complexity Theorem: n d = 2 d = 3

    Optimal transport: su↵ers from curse of dimensionality. ! Adapt to support dimensionality [Weed, Bach 2017] E(|||ˆ ↵ ˆ||k ||↵ ||k |) = O(n 1 2 ) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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↵ <latexit 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<latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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ˆ ↵ <latexit 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<latexit 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<latexit 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log(n) log(n) E(|Wp(ˆ ↵, ˆ) Wp(↵, )|) = O(n 1 d ) <latexit 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<latexit 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log(Wp(ˆ ↵, ˆ ↵0)) <latexit 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log(n) d = 5 log(n) d = 2 log(W",p(ˆ ↵, ˆ ↵0)) <latexit 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E(|W",p(ˆ ↵, ˆ) W",p(↵, )|) = O(" d 2 n 1 2 ) <latexit 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<latexit 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Theorem: [Genevay, Bach, P, Cuturi] <latexit 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<latexit 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<latexit 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  24. 3. Sinkhorn Divergences <latexit 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<latexit 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<latexit 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<latexit

    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<latexit 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x <latexit 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<latexit 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d(x,y) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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4. Application to <latexit 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<latexit 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Generative Models <latexit 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<latexit 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g✓ µ✓ X ⇣ 2. Entropic <latexit 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<latexit 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Regularization <latexit 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1. Optimal Transport <latexit 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<latexit 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  25. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">AAA/fHiczVvddhu3EUbSv1j9s9vLXnRbxY2dOj6y6tO0J6d1bEmWFdOWbIm2kzB2+bOiNl5x6V1Ssswyb9Lb9qKP0CfpG9SP0Kt2ZgAssCR2B3DTcyIcSVgQ38xgAMwPsOyN06SYrK398623v/Xt73z3e++cW/n+D374ox+fv/CTR0U2zftxu5+lWf6k1y3iNBnF7UkySeMn4zzuHvfS+HHv+QZ+/vgkzoskGx1MzsbxF8fd4Sg5TPrdCTQ9O39h1iEis146jeedXjx/dn517eoa/UTLlWuqsirUz1524efvi44YiEz0xVQci1iMxATqqeiKAsrn4ppYE2No+0LMoC2HWkKfx2IuVgA7hV4x9OhC63P4O4Snz1XrCJ6RZkHoPnBJ4TcHZCQuAiaDfjnUkVtEn0+JMrbW0Z4RTZTtDP73FK1jaJ2II2jlcLqnP24ArYcgG2qmecwT6Pc7GmsCvcfUglroK25T0h6OMLJGPwEKY2jD+gA+z6HeJ6Sej4gwBekI56BLn/+LemIrPvdV36l4TaP5pujvm6eXiwElEjvivtgQLdEWm2JL7JO8zWWFcBswjjHoOAfJh6AhlPYSSHMZ/q/DnloTH0JtG2TsUZ8YxhKJPfh/RvKvKFkjcU/cFAfiDvG+BPV7ULscME9cT722C5oN/3WzwqyynGbkrFGCCY1/ShYgFy8b++LaGajZPKT1UKgd2TynUoMb4qHYbZw/5DwC+qdkiY5Jg8hxBu1DkAlXaAr80CbiWv67+EqsQv0r8Q+1jiOQK1cr+4joySd/mn+CgjTfgzIX54hmG6ReKUcZwZOcgaxc/V1xQnsjIks+hc/kXkpp7Y2oV9Po7kOZU02vmh6UGbXOG5EbUFzIDRa5DcWF3GaRLSguZItF7kFxIfcUsgn7AIoL+4Dl+hCKC/mQRR5AcSEPWOQjKC7kIxb5GRQX8jMWeRuKC3mbRd6F4kLeZZFtKC5km0VuQXEhtxSyfq+hXcuITsLsq5tQr/JAP5VCy01WvltkY13YWx67sl+D5fflJvx3Yzc9dBrXYLc8Vs9hDZZfP9tg5dxY3prcIR/pwt5hsTuwAtzYHRb7ifiyBvuJx355XoPld0yL/I0Ly9vPe/Dkxt5jsfeh5sbyXmYXWtzYXQ+bP67B7rHYB+JFDdbH6uc1WN7u71Ms5sLue/iMSQ2W9xptiB3cWN6ePoIYxI3lfc5jaHVjH7PYJxQ1urBPWOynFJ26sJ96+MlXNVjtKWX0PqQYMIYd20StW+5KrI2BWpfhn5a+BWvoowYsZlhihoQ5ZhHbJWLbE9EqES1vuYrSjhYUsfJc9kvEvieiV/omrE3Y/oOy/4ByMx6xWSI2FxBNWQDOtR7LCUUXuoVDTkrPhTWfMWWl/cZarNZDs+XViN0KQq7tI1r5VyhXx0xmQHlrPbWj0sdLZETPTYhTyvX0KDUPHjcprYKNesmieg5Uj0WdOVBnLGrqQE1Z1IkDdcKizM63cR2PFWD0j3Mxoyed+3NnKdWzi13wuFvg/bBlF/77nKU0Z+V4IoCeUmfPxhbnUJtRJm0yu03KkeVJQwySyZ676owJn/BscqZ2nbTD89KXR0KfePrTSUieYUkH48WIdlQYnbvUMqf4TtbC8HfKna9rYfgt0vic4nhZC8NPlPSTN5D9QGEP3gC7D/tprLRv6qE05BmKpKHrnG3eKe0mnle+pJ0j20L5b1BN6sDUw2gU1hiKyhhCaBhdFpYuw6hgjCRjW12LgkcyUtmtqYfKkJGvHCk5zFPozGCfgZoZXQ+jsQdx1QZl1jOrHrpCx+VoTD2MxiMhz9bnFK/rehiNIT1LfZh6GA08U+mqbN7UQ603akBmyKauY5acohh94pyQ924+S7Ej8mUfgycqT8sIvpmSiTzr6fRKP9Ms0aJ1CZEDvf/UipCqNGZinc1+pAyTiu9dpmP8L2q+BVqMYNfKc33uTDoFCfWJQUyn5E+JWjOmOjKNW2dxaEkOF1Ad1TphYznDV57pVNueUSuXNZnRGj12yM4WtPbGFLG1SLOcHlq1M1xHkdNQq6Ihnl6I7l7RDs4WtL/G4sYLiHG50vp0OyRvqZuzSJfW9y0dX1S3KBMo8k7FrF88Cz4kXE4ZSUbWBmVp4mn306c8dhv6wyvCnEDLzyKaUbRXJ2Q1ErrxKdg4RJ/lykh5Rs+Gdpvu2aq3VpGiMhbyRhXPuPG0O6L7TNvWcrxRX/r8TNYLsrrmHr35dsyghw50aPaxAb7iPtQOIJpvw9OBR/5h7twy0ncuPijvPzOav+Zs277Z65Q0pLWJK/axKQM+qlA5BTSuBZlB+9NYpKPxnSVKfEbuksfklVW7f5Hu9PWbI11a4fVruf6UZEBc14lrRHtG3vfLp0UOUoKZ85N1ijqbR4n8QjiiBeW4PrU4S72M6F2amHLLMcWzKe01bm9Ue9tnR4ufaE57Qr9VgXfFGdnHiKxfBN4pozUZ0a/9Vo5+t0Lag5QspI/VScrYxhXpJOwaS2iXyzUi3xcy6y0mSzYl/pquvbsKWosyzpdeYL6wtrVOWhQJxsQ1V7bd7O1m34NI8yaEvUokRbNWLhH/y/RX/+p1srq0IlDDOAOFsnSu+cgo00AddcnHN9sg3deW8t1ShqdKauP9jEzvViTbpDwJ5UFfPQDOfXqWvHCV5CR3sdRHetGmk1akPF7QI44WvWtHWf2h8r8o9xXykau05zrCvGGjY3/dlzvhXeTbzKtK3Y928X+hbnRd1RpSjIQ5XZUa4s7eY8qxbClTeidJvn0TK0rLWs8XejXzGdFaPLb28p+h9RfwV8utn/3o9CpW4RatAUnBPBmNyJZoqYcfr1sVXnplalrm2fAza1L3slveJCuW1s1kxifBVPZo1bxUZw267jP+A7rDMxrQ7dqKPFuIoA/UnZ8v9TDa/pSnLGU+dtKoxENKO+fxozpgqfK5uEa9YmmtsbTwbTz7RN3enbbvvE2xQ58iHJkdDGgXJBTT6NbmPEhSQF7Xlf2yd1eHWnAv9chCIeXqW4V4dtcX8h1SKeOvlOfIyI6aHaff2TlVfbQN61D9N0vIY8pmC9o7GnGdesRKfluOaGHHX7V8ekRn3l2KWaRfb85I7d5mFqKKvzbZnNwLhlcq3lPR+IjmYCJeN3LbWcoPd6wMMRLyvi8tI3ic5ddBs4x5m1wZOld3R28FzRJ6kpzyeRk19shGLZ8yyFusEWhfr7urJPVM/MHDOsi80uwUe33hZ1OVleDY3oeeqHkz864ePL/UmyPH703us7rkOY5VHDhbeH4zWl3lSarPTXqYLvA1+piKkcoJdfRuMqkqpiM+8uYiJQrjIjE+XMJGESJ/mOQhMst7G1/KuremXM3mpY05opyEew8SEa4I6pIzYrrMjKO3RK9HWJuabOEo4XlXpnJw29riyc+5JV8kW881eqTU8kZ13kJTtz2GseHSQsZk/VLBnYvI3n0h30ety6Ds3OAjKPg3Eq78SXsTn9O1fYgi8b16v29FNN/kv1B1eeIXUQvu5MFCdtpV46z2aJ7lFxZ1m74PB38eCeiakz4hPxgqu6TMS25T96d/Sns4FzErvekZPgabCz+SZU4h40nILvGjSYT+HlPoWDQHn5FUufjzkef+3CgOhf4+XdgYNHV+BFUOITz0/bzfnJve4bxsTs36Wubiy0PacH0joXF4M1afbZh+PhYqt2bk6+eA1uGwgbr2Fv/rODQfwymcly+3gr7r9KXHrMt+sTqzxGg2fM8Ybj6ruZ6jP8+sHJ2Jddz8ZNQWBc1UZo3m66eP0aRZA5rXTMiTQl46ibdXkZHXlwqenLtkyMS/vWSQ+DoZfKnoE/x6SroHT01/0881Kv2Zj0yGTp1MVWpIT2baHdrDMZ0WmJudAT2Xb1s+O796bfFb58uV9vrV31+99uD66se31BfS3xE/E7+EQP6a+FB8LO5AgNkGEU7FX8Rfxd/++J8bF2/8+sYHsuvbbynMT0Xl58Zv/wsjoSbD</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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  26. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">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</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Density fitting: Maximum likelihood (MLE) d↵✓(x) = ⇢✓(x)dx <latexit 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min ✓ c KL(↵✓ | ) def. = X i log(⇢✓(xi)) <latexit 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<latexit 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  27. Density Fitting and Generative Models ✓ Parametric model: ✓ 7!

    ↵✓ <latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="/IX3qCorxR4bVe9DcVqsRyK1ko8=">AAA/fHiczVvddhu3EUbSv1j9s9vLXnRbxY2dOj6y6tO0J6d1bEmWFdOWbIm2kzB2+bOiNl5x6V1Ssswyb9Lb9qKP0CfpG9SP0Kt2ZgAssCR2B3DTcyIcSVgQ38xgAMwPsOyN06SYrK398623v/Xt73z3e++cW/n+D374ox+fv/CTR0U2zftxu5+lWf6k1y3iNBnF7UkySeMn4zzuHvfS+HHv+QZ+/vgkzoskGx1MzsbxF8fd4Sg5TPrdCTQ9O39h1iEis146jeedXjx/dn517eoa/UTLlWuqsirUz1524efvi44YiEz0xVQci1iMxATqqeiKAsrn4ppYE2No+0LMoC2HWkKfx2IuVgA7hV4x9OhC63P4O4Snz1XrCJ6RZkHoPnBJ4TcHZCQuAiaDfjnUkVtEn0+JMrbW0Z4RTZTtDP73FK1jaJ2II2jlcLqnP24ArYcgG2qmecwT6Pc7GmsCvcfUglroK25T0h6OMLJGPwEKY2jD+gA+z6HeJ6Sej4gwBekI56BLn/+LemIrPvdV36l4TaP5pujvm6eXiwElEjvivtgQLdEWm2JL7JO8zWWFcBswjjHoOAfJh6AhlPYSSHMZ/q/DnloTH0JtG2TsUZ8YxhKJPfh/RvKvKFkjcU/cFAfiDvG+BPV7ULscME9cT722C5oN/3WzwqyynGbkrFGCCY1/ShYgFy8b++LaGajZPKT1UKgd2TynUoMb4qHYbZw/5DwC+qdkiY5Jg8hxBu1DkAlXaAr80CbiWv67+EqsQv0r8Q+1jiOQK1cr+4joySd/mn+CgjTfgzIX54hmG6ReKUcZwZOcgaxc/V1xQnsjIks+hc/kXkpp7Y2oV9Po7kOZU02vmh6UGbXOG5EbUFzIDRa5DcWF3GaRLSguZItF7kFxIfcUsgn7AIoL+4Dl+hCKC/mQRR5AcSEPWOQjKC7kIxb5GRQX8jMWeRuKC3mbRd6F4kLeZZFtKC5km0VuQXEhtxSyfq+hXcuITsLsq5tQr/JAP5VCy01WvltkY13YWx67sl+D5fflJvx3Yzc9dBrXYLc8Vs9hDZZfP9tg5dxY3prcIR/pwt5hsTuwAtzYHRb7ifiyBvuJx355XoPld0yL/I0Ly9vPe/Dkxt5jsfeh5sbyXmYXWtzYXQ+bP67B7rHYB+JFDdbH6uc1WN7u71Ms5sLue/iMSQ2W9xptiB3cWN6ePoIYxI3lfc5jaHVjH7PYJxQ1urBPWOynFJ26sJ96+MlXNVjtKWX0PqQYMIYd20StW+5KrI2BWpfhn5a+BWvoowYsZlhihoQ5ZhHbJWLbE9EqES1vuYrSjhYUsfJc9kvEvieiV/omrE3Y/oOy/4ByMx6xWSI2FxBNWQDOtR7LCUUXuoVDTkrPhTWfMWWl/cZarNZDs+XViN0KQq7tI1r5VyhXx0xmQHlrPbWj0sdLZETPTYhTyvX0KDUPHjcprYKNesmieg5Uj0WdOVBnLGrqQE1Z1IkDdcKizM63cR2PFWD0j3Mxoyed+3NnKdWzi13wuFvg/bBlF/77nKU0Z+V4IoCeUmfPxhbnUJtRJm0yu03KkeVJQwySyZ676owJn/BscqZ2nbTD89KXR0KfePrTSUieYUkH48WIdlQYnbvUMqf4TtbC8HfKna9rYfgt0vic4nhZC8NPlPSTN5D9QGEP3gC7D/tprLRv6qE05BmKpKHrnG3eKe0mnle+pJ0j20L5b1BN6sDUw2gU1hiKyhhCaBhdFpYuw6hgjCRjW12LgkcyUtmtqYfKkJGvHCk5zFPozGCfgZoZXQ+jsQdx1QZl1jOrHrpCx+VoTD2MxiMhz9bnFK/rehiNIT1LfZh6GA08U+mqbN7UQ603akBmyKauY5acohh94pyQ924+S7Ej8mUfgycqT8sIvpmSiTzr6fRKP9Ms0aJ1CZEDvf/UipCqNGZinc1+pAyTiu9dpmP8L2q+BVqMYNfKc33uTDoFCfWJQUyn5E+JWjOmOjKNW2dxaEkOF1Ad1TphYznDV57pVNueUSuXNZnRGj12yM4WtPbGFLG1SLOcHlq1M1xHkdNQq6Ihnl6I7l7RDs4WtL/G4sYLiHG50vp0OyRvqZuzSJfW9y0dX1S3KBMo8k7FrF88Cz4kXE4ZSUbWBmVp4mn306c8dhv6wyvCnEDLzyKaUbRXJ2Q1ErrxKdg4RJ/lykh5Rs+Gdpvu2aq3VpGiMhbyRhXPuPG0O6L7TNvWcrxRX/r8TNYLsrrmHr35dsyghw50aPaxAb7iPtQOIJpvw9OBR/5h7twy0ncuPijvPzOav+Zs277Z65Q0pLWJK/axKQM+qlA5BTSuBZlB+9NYpKPxnSVKfEbuksfklVW7f5Hu9PWbI11a4fVruf6UZEBc14lrRHtG3vfLp0UOUoKZ85N1ijqbR4n8QjiiBeW4PrU4S72M6F2amHLLMcWzKe01bm9Ue9tnR4ufaE57Qr9VgXfFGdnHiKxfBN4pozUZ0a/9Vo5+t0Lag5QspI/VScrYxhXpJOwaS2iXyzUi3xcy6y0mSzYl/pquvbsKWosyzpdeYL6wtrVOWhQJxsQ1V7bd7O1m34NI8yaEvUokRbNWLhH/y/RX/+p1srq0IlDDOAOFsnSu+cgo00AddcnHN9sg3deW8t1ShqdKauP9jEzvViTbpDwJ5UFfPQDOfXqWvHCV5CR3sdRHetGmk1akPF7QI44WvWtHWf2h8r8o9xXykau05zrCvGGjY3/dlzvhXeTbzKtK3Y928X+hbnRd1RpSjIQ5XZUa4s7eY8qxbClTeidJvn0TK0rLWs8XejXzGdFaPLb28p+h9RfwV8utn/3o9CpW4RatAUnBPBmNyJZoqYcfr1sVXnplalrm2fAza1L3slveJCuW1s1kxifBVPZo1bxUZw267jP+A7rDMxrQ7dqKPFuIoA/UnZ8v9TDa/pSnLGU+dtKoxENKO+fxozpgqfK5uEa9YmmtsbTwbTz7RN3enbbvvE2xQ58iHJkdDGgXJBTT6NbmPEhSQF7Xlf2yd1eHWnAv9chCIeXqW4V4dtcX8h1SKeOvlOfIyI6aHaff2TlVfbQN61D9N0vIY8pmC9o7GnGdesRKfluOaGHHX7V8ekRn3l2KWaRfb85I7d5mFqKKvzbZnNwLhlcq3lPR+IjmYCJeN3LbWcoPd6wMMRLyvi8tI3ic5ddBs4x5m1wZOld3R28FzRJ6kpzyeRk19shGLZ8yyFusEWhfr7urJPVM/MHDOsi80uwUe33hZ1OVleDY3oeeqHkz864ePL/UmyPH703us7rkOY5VHDhbeH4zWl3lSarPTXqYLvA1+piKkcoJdfRuMqkqpiM+8uYiJQrjIjE+XMJGESJ/mOQhMst7G1/KuremXM3mpY05opyEew8SEa4I6pIzYrrMjKO3RK9HWJuabOEo4XlXpnJw29riyc+5JV8kW881eqTU8kZ13kJTtz2GseHSQsZk/VLBnYvI3n0h30ety6Ds3OAjKPg3Eq78SXsTn9O1fYgi8b16v29FNN/kv1B1eeIXUQvu5MFCdtpV46z2aJ7lFxZ1m74PB38eCeiakz4hPxgqu6TMS25T96d/Sns4FzErvekZPgabCz+SZU4h40nILvGjSYT+HlPoWDQHn5FUufjzkef+3CgOhf4+XdgYNHV+BFUOITz0/bzfnJve4bxsTs36Wubiy0PacH0joXF4M1afbZh+PhYqt2bk6+eA1uGwgbr2Fv/rODQfwymcly+3gr7r9KXHrMt+sTqzxGg2fM8Ybj6ruZ6jP8+sHJ2Jddz8ZNQWBc1UZo3m66eP0aRZA5rXTMiTQl46ibdXkZHXlwqenLtkyMS/vWSQ+DoZfKnoE/x6SroHT01/0881Kv2Zj0yGTp1MVWpIT2baHdrDMZ0WmJudAT2Xb1s+O796bfFb58uV9vrV31+99uD66se31BfS3xE/E7+EQP6a+FB8LO5AgNkGEU7FX8Rfxd/++J8bF2/8+sYHsuvbbynMT0Xl58Zv/wsjoSbD</latexit> Observations: def. = 1 n Pn i=1 xi <latexit 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<latexit 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Density fitting: Maximum likelihood (MLE) d↵✓(x) = ⇢✓(x)dx <latexit 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min ✓ c KL(↵✓ | ) def. = X i log(⇢✓(xi)) <latexit 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<latexit 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<latexit 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g✓ X Z ⇣ Generative model fit: ! MLE undefined. ! Need a weaker metric. ↵✓ = g✓,]⇣ <latexit sha1_base64="W/xAH4ZEbKVL5sYCOuVPDm3rS+Q=">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</latexit> <latexit 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<latexit 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<latexit 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↵✓ <latexit 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min ✓ Wp ",p (↵✓, ) <latexit 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c KL(↵✓ | ) = +1 <latexit 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<latexit 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  28. Deep Discriminative vs Generative Models ✓1 ✓2 ⇢ ⇢ z

    x g✓ ⇢ ⇢ z x Discriminative Generative d⇠ ⇠1 ⇠2 g✓(z) = ⇢(✓K(. . . ⇢(✓2(⇢(✓1(z) . . .) Deep networks: d⇠(x) = ⇢(⇠K(. . . ⇢(⇠2(⇢(⇠1(x) . . .)
  29. Deep Discriminative vs Generative Models ✓1 ✓2 ⇢ ⇢ z

    x g✓ ⇢ ⇢ z x Discriminative Generative z1 z2 g✓ Z x z X d⇠ d⇠ ⇠1 ⇠2 g✓(z) = ⇢(✓K(. . . ⇢(✓2(⇢(✓1(z) . . .) Deep networks: d⇠(x) = ⇢(⇠K(. . . ⇢(⇠2(⇢(⇠1(x) . . .)
  30. Training Architecture g✓ Z X y1 <latexit 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<latexit 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Stochastic gradient descent <latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="ZvDledmv+Gwz+FxUnlfvrUdxKFc=">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</latexit> min ✓ E(✓) def. = Wp ",p (↵✓, ) <latexit 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<latexit 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sha1_base64="PiJWHJFbS74WSpUZvh1mwt8Pg+M=">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</latexit> ˆ E(✓) def. = Wp ",p ( 1 m P i g✓(zi ), ) <latexit 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✓ ✓ ⌧r ˆ E(✓) <latexit 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  31. Training Architecture g✓ Z X y1 <latexit 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<latexit 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Stochastic gradient descent <latexit 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<latexit 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↵✓ <latexit 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<latexit sha1_base64="ZvDledmv+Gwz+FxUnlfvrUdxKFc=">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</latexit> min ✓ E(✓) def. = Wp ",p (↵✓, ) <latexit 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<latexit 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sha1_base64="PiJWHJFbS74WSpUZvh1mwt8Pg+M=">AAA94XictVvbbhy5EaU3l107N2/yGCDoROvADhxDUgwkwELA2pJsa621Zc9I9u62LcylNWq7Z3o8PSNfZucD8hbkNR+TD8gX5A+SpyCvyUuqimSTPcPuIhWvGpLYHJ5TxWqyWEX2dMdZWkzX1/9+4YPvfPd73//wo4uXfvDDH/34J5c//ulRkc8mveSwl2f55Gm3UyRZOkoOp+k0S56OJ0ln2M2SJ92X2/j5k7NkUqT5qD19O06eDTuDUXqS9jpTqDq+nMSzYTqax9PTRRTvJlehcC2Kk1f95CSKc0Ai8TwedvM38yeLxfE8TsZFmuWj6+PF8/HVeUwqzCdJfxF3smPkua4ru9ksWcTdZHHt+PLa+o11+olWCxuqsCbUz0H+8S82RSz6Ihc9MRNDkYiRmEI5Ex1RwPW12BDrYgx1z8Qc6iZQSunzRCzEJcDOoFUCLTpQ+xL+DuDua1U7gnvkLAjdAykZ/E4AGYkrgMmh3QTKKC2iz2fEjLV13HPiRN3ewv+u4hpC7VScQi2H0y19cdiXqTgRf6A+pNCnMdVg73qKZUZWQc0jq1dTYBhDHZb78PkEyj1CajtHhCmo72jbDn3+D2qJtXjfU21n4p+k5RW4ItFSvc9Lho44I/6InuYMPpP6ZCB5AAyJ6iOWXpOth9T7EbSfQ/0DuBZU0jbpwjWn2kUjchsuF3KbRd6Fy4W8yyL34XIh91nkAVwu5IFCInZCNnfjW3C58C1W8iO4XMhHLPIxXC7kYxZ5BJcLecQiv4LLhfyKRd6By4W8wyLvw+VC3meRbbhcyDaLPITLhTxkkbtwuZC7Clk/Uydw5cSTMrPyFpSrMtBTZFBzi9XvNnlHF/a2x5zu1WD5Wb0D/93YHQ+bJjXYXY9xd1KD5UfeXfCRbizvi+7RauLC3mOxezAC3Ng9Fvu5eFGD/dxjpr2swfJzbR/aubG89/0C7tzYL1jsAyi5sfwa9RBq3NiHHivGuAZ7wGIfiVc1WB+vP6nB8n6/BX7FjeXXqTa0d2N9vOmsBsv70yOIYNxYfrV6ArVu7BMW+1S8qcE+ZbFfgnd3Y7/0WGHf1WD1GnuJVpABxSMJzNgmtk45K7E0BrYOIz8r15aMYuMu1HOYQYkZEGbIIu6WiLueiP0Sse+tV1H60YLiXV5Kq0S0PBHdcm3C0pRt3y/bYynzQOyUiJ0lRFNEis9a9+WMogtdwyGn5cqFJZ8+5aX/xlKixkOz59WIhxWEHNunNPKvU7aEGRRaqonttFzjJTKi+ybEa8redC+1DB43Lb2CjXrDoroOVJdFvXWg3rKomQM1Y1FnDtQZizIz38bFHiPA2B+fxZzu5AiQMXL9FUFUcAtWnXswRyMYPwcQBT6mmofwv0W5N3c1aYbZPK6TuMvxrOKJJ1CaizWoN1nhDuXXGc2wBDSTLR+qHB/vcG9jruac9MKLciWPyh0Tf56U9BmUPBgtRjSfwnjuU82CojtZCsPfK+e9LoXhd8niC4riZSkMP1XaT8+he1th2+fAtmA2jZX1TTmUQ+6/SA5dvkSrLnpcfKpDNWaQ700g/556MnvneC7bVJL2MeUwjsLqX1HpXwiHsXNh2TmMBaMnGfXqUhTck5HKe005VIecVtGR0sPchT4ZbNNXT0aXwzgOIOLappx7bpVDR++47I0ph3EcCbnvuaBIXpfDOAZ0L+1hymEcuNvSUXm+KYd6drSAzJ1NOdSrj2gXGPeA5JiXNSYqmlCcNFNsKcUHzbs1dsy/uo7hns3zMkdoZjKxbT1Pt1zLmjXS8UICXm0aqAfGFzMrBqtyzMUmm19JHaaV9X2Vx6zxaPl9sGIEs1+eAXB75hloqPck0HtnwLjBZl3VnmncJovDUXKyhIpV7ZSNFo1cuWtUrTumWi4vM701dozJXxc09sYUE+6TZTk77Nc+4TpGzkL7FQvxfCG2e6fma9X66yxuvIQYlyOtRydC8iStOU91Wb1l2fiKOuWZwiXPfMz4xd3mE+VtMOfJyRehLk0y7XZ6H8muw3X1ujB73PKziJ4o+qsz8hopnUgVbBaqd4tlND6ne8N9SGdyKENy9OA5RoplLOSpGe6i4356RB7V9recbLSX3qGT5YK8rvbHzeiBhR440OE5zjasGA+g1Iac4RDu2h5ZzqXSVjlZfCJ+W56O5vQEmzP6rOIhNYf0N0nFQzZl2acVlteAxtEgs3R/jmUejY9XmPis36WPyV2rnv8Kndzq8+0OjfH60Vy/E9MnqZskNaJZI0915d2yBKnB3PnJJsWvzb1EeSES0YdyUp9bkqVdRnTin1AGO6bIOKPZxs2Oamt7f2r5Ey3pQOizczzNzslDRuT/IlifchqTEf3a7w7oE3TpETLykT5+Jy2jG1esk7JjzMRxqZBvNZjxlpAvm5F8zWvProLGoswY5DqwWBrb2ib7FAsmJHWivLuZ282rDyLNexL2KJGMZqxcJfnX6K/+1eNkbWVEoIXxCRTK17meR045C9qoQ6t8sw/SbW0tPyl1eK60Nuuf0emTimY7lHGhPrha90Fyj+6lLBwlE9K7WGkj19Gm3VxkHi/ZEXt7Qlm89PsDtQKj3tdplVyjORfTKBnAKJiWWYRuy+0iL8ttllVl9+MuvhV2Y+uq1ZAxEmYHV1qI299PKFuztcxgVMvx+5Jmk9vqk6VWzXJGNBaH1lz+Bmp/CX+13vrej6db8Qq3aQxIBnNnLCJropUWfrJuV2Tpkam5zL2RZ8akbmXXnCe/lt7N5NhnwSwHNGreqF0LXT4PxwuL44WnDdt01misqOu1Jzpmc4u2Oq30lRcirR3APGOZ+YhMo1IPLe1cyo+1z7LyOb5GvWO51lmuDsxW+zTAnvM+SPdcX57d35SreyTuUGzTowhM5i99mqUpxVy6tjlTkwwo+abyr/bsj6kGpXfJgyKzfI8TZ4w8derRtSg1/bVa2XLy88Yj6PeWXqs22sfGVP7dCnJIc6KgeakRN6lFovS39YiWPNINK+aIaOe/QzGVjDuac2a7tXkmUSWeMPmmnFVGlswURmR/budtbyV73bPy14hywpmKrrvAFf6EkUFi9E6CO7Is6AnhKidPEmRE2yX/ueqn5CneyNLoBmk9F1sePkZmvWas22NL91j37TfQEq1unrqrBS8v85bIyTvPiV6HVrWhilHnS/fn4+qoVa5632SH2ZJcY48ZtbEzC5PlVTGx+NRbitQoTIrE+EgJ60WI/mGah+gsT6d8mXVrzVzdaZA+5pTyJe49UES4orurzmjuGtOP7gpfl7A2m6zhmHA3Llf7A7anxV2piyvrkKy92LgaZdZKVLdSaHZ7tTD+W3rIhLxfJrg9G9na1j2uZCn8Loxk6An5Rm9dfmhzfgoX/o2EKzvUEn32DlsQ394S22L3PbwN8UqV5Y5mRDXoC/pLuXdH9bPaotlGryx2m99Hgr+MFGzNaZ/SShqqu2TmNbfZ/flfkxeYiITV3rQM74Mthe/JqqSQ/qTk2fjepEJ/Fye0L1qCT0+qUvzlyHMNrhcnQn+nKawPmp3vQVVCiAz9HoPfMzetw2XZkprttSrFV4ZcBfSJi8bhyV99rmLa+XioifVE3r8E9A4nDex6tfh/+6HlGEnhsnylFfRdsxceT122S9SOLMbD4XPGSPMZzfUS/WXmZe9MtOSWJ+O+KOhJ5VZv3j8/xqNmDGhZcyH3QXntJN4eRUZfXxY8F3DpkIt/ib9e4L+N8KrkqNMjhEmfU9Sz6RY8m/7Gpat3+jMfnQxPnU5VNpNHtOiN2G2xJ+7A73YZAYa+HSq/Syn/I9b9/dk+1J6Q99C76HLnIKa6hHY/zClan+7VHuPx5bWN5W8hrxaONm9srN/YeHRz7bPb6hvKH4mfi19BXrIhfi8+E/egv4eg09/Ev8V/xH+3elt/3PrT1p9l0w8uKMzPROVn6y//A3hG/4c=</latexit> ˆ E(✓) def. = Wp ",p ( 1 m P i g✓(zi ), ) <latexit 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✓ ✓ ⌧r ˆ E(✓) <latexit 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C K . . . ✓1 ✓2 . . . 1m 1/· ⇥mK> ⇥nK 1/· . . . . . . e C/" Input data <latexit 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<latexit 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<latexit 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<latexit 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(y1, . . . , yn) <latexit 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sha1_base64="Lz1gCqAeSuy9yYYGVdEU4UDCj/g=">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</latexit> Generative model <latexit 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<latexit 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<latexit 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(x1, . . . , xm) <latexit 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Sinkhorn, ` = 0, . . . , L 1 <latexit 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ˆ E(✓) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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h(C K)b, ai <latexit 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  32. Automatic Differentiation Setup: E : Rn ! R computable in

    K operations. Hypothesis: elementary operations (a ⇥ b, log(a), p a . . . ) and their derivatives cost O(1). Question: What is the complexity of computing rE : Rn ! Rn? function y = E(x) a = xi1*x b = rho(a) z = xi2*b y = 1/2*norm(z-u)^2 ⇢ z x ⇠1 ⇠2 1 2 || · ||2 <latexit 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<latexit 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<latexit 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  33. Automatic Differentiation Setup: E : Rn ! R computable in

    K operations. Hypothesis: elementary operations (a ⇥ b, log(a), p a . . . ) and their derivatives cost O(1). Question: What is the complexity of computing rE : Rn ! Rn? function y = E(x) a = xi1*x b = rho(a) z = xi2*b y = 1/2*norm(z-u)^2 ⇢ z x ⇠1 ⇠2 1 2 || · ||2 <latexit 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<latexit 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<latexit 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Finite di↵erences: K(n + 1) operations, intractable for large n. rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " n) E(✓)) <latexit 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<latexit 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<latexit 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<latexit 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  34. Automatic Differentiation Setup: E : Rn ! R computable in

    K operations. Hypothesis: elementary operations (a ⇥ b, log(a), p a . . . ) and their derivatives cost O(1). Question: What is the complexity of computing rE : Rn ! Rn? function y = E(x) a = xi1*x b = rho(a) z = xi2*b y = 1/2*norm(z-u)^2 function dx = nablaE(x) dz = z-u db = xi2’*dz da = diag(dphi(a)) * db dx = xi1’*da This algorithm is reverse mode automatic di↵erentiation [Seppo Linnainmaa, 1970] Seppo Linnainmaa Theorem: there is an algorithm to compute rE <latexit 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<latexit 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in O(K) operations. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Finite di↵erences: K(n + 1) operations, intractable for large n. rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " n) E(✓)) <latexit 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<latexit 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<latexit 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<latexit 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  35. Examples of Images Generation (a) WGAN MNIST (b) WGAN CelebA

    (c) WGAN L (d) MMD GAN MNIST (e) MMD GAN CelebA (f) MMD GAN Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and datasets. ⇣ Inputs <latexit 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<latexit 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Generated ↵✓ <latexit 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  36. Examples of Images Generation (a) WGAN MNIST (b) WGAN CelebA

    (c) WGAN L (d) MMD GAN MNIST (e) MMD GAN CelebA (f) MMD GAN Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and datasets. ⇣ (a) WGAN MNIST (b) WGAN CelebA (c) WGAN LSUN (d) MMD GAN MNIST (e) MMD GAN CelebA (f) MMD GAN LSUN Figure 2: Generated samples from WGAN and MMD GAN on MNIST, CelebA, and LSUN bedroom datasets. Inputs <latexit 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Generated ↵✓ <latexit 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<latexit 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h⇠ <latexit 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<latexit 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<latexit 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! Performance evaluation of generative models is an open problem. ! Influence of "? <latexit 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<latexit 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<latexit 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! Need to learn the metric d(x, y) = ||h⇠(x) h⇠(y)|| (GANs) <latexit 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<latexit 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<latexit 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Ian Goodfellow " <latexit 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<latexit 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Inception score <latexit 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  37. Generative Adversarial Networks Progressive Growing of GANs for Improved Quality,

    Stability, and Variation Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018 g✓ X Z ↵✓ <latexit 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<latexit 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  38. Progressive Growing of GANs for Improved Quality, Stability, and Variation

    Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018
  39. Progressive Growing of GANs for Improved Quality, Stability, and Variation

    Tero Karras, Timo Aila, Samuli Laine, Jaakko Lehtinen, ICLR 2018
  40. Open Problems Monge Kantorovic Dantzig Brenier Villani Figalli Toward high-dimensional

    OT: <latexit 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<latexit 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<latexit 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<latexit 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! Scalable geometrical loss functions in high dimension? <latexit 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<latexit 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<latexit 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! Performance quality measures for unsupervised learning? <latexit 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<latexit 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<latexit 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  41. Open Problems Monge Kantorovic Dantzig Brenier Villani Figalli Toward high-dimensional

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<latexit 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<latexit 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! Scalable geometrical loss functions in high dimension? <latexit 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<latexit 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<latexit 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! Performance quality measures for unsupervised learning? <latexit 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<latexit 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<latexit 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<latexit 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Metric learning for OT: <latexit 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<latexit 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<latexit 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z g✓ z x d⇠ ! Adversarial training to <latexit 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<latexit 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<latexit 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<latexit 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leverage multi scale priors? <latexit 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<latexit 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  42. Open Problems Monge Kantorovic Dantzig Brenier Villani Figalli Toward high-dimensional

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<latexit 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<latexit 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<latexit 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! Scalable geometrical loss functions in high dimension? <latexit 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<latexit 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<latexit 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! Performance quality measures for unsupervised learning? <latexit 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<latexit 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<latexit 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<latexit 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Metric learning for OT: <latexit 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<latexit 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<latexit 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z g✓ z x d⇠ ! Adversarial training to <latexit 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<latexit 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<latexit 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<latexit 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leverage multi scale priors? <latexit 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<latexit 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Beyond comparing measures: <latexit 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! Using Gromov-Wasserstein geometry? <latexit 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<latexit 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<latexit 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! Learning for surfaces, graphs, metric spaces? <latexit 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<latexit 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<latexit 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