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Optimal Transport for Imaging and Learning

Optimal Transport for Imaging and Learning

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

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

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  2. https://optimaltransport.github.io

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  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
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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
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    Parametric model: ✓ 7! ↵✓
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
    Observations:
    def.
    = 1
    n
    Pn
    i=1 xi
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    Unsupervised learning
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    D
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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
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    min

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
    Density fitting:
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    Parametric model: ✓ 7! ↵✓
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
    Observations:
    def.
    = 1
    n
    Pn
    i=1 xi
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    Unsupervised learning
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    D
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