Optimal Transport for Imaging and Learning

Optimal Transport for Imaging and Learning

Talk @UCL, Louvain la Neuve, updated 30 Sept. 2018, talk @Lille

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Gabriel Peyré

August 30, 2017
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  1. Optimal Transport for Imaging and 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. https://optimaltransport.github.io

  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 in Data Sciences Probability distributions and histograms <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 in Data Sciences Probability distributions and histograms <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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✓ ↵✓ <latexit 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  5. 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 in Data Sciences Probability distributions and histograms <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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✓ ↵✓ <latexit 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  6. 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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  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
  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 b <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 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xi yj Couplings: <latexit 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<latexit 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<latexit 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U(a, b) def. = n P 2 Rn⇥m + ; P1n = a, P>1m = b o <latexit 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<latexit 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<latexit 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  9. 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 b <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 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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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xi yj d(xi ,yj ) <latexit 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Couplings: <latexit 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<latexit 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U(a, b) def. = n P 2 Rn⇥m + ; P1n = a, P>1m = b o <latexit 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<latexit 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  10. Optimal Transport Distances 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 sha1_base64="ty3Ki+Ibgkag0xq97PVpv2d8vdk=">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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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  11. Optimal Transport Distances b <latexit sha1_base64="LhlbHqBkrxCDCHirUc/UdkhSooo=">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