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Probability Measures Study Group

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Mathieue edited this page Nov 27, 2018 · 3 revisions

Convergence, Distances, Geometry and applications

Organisers: Adam & Emile ?

Sessions (To be updated):

  • First session:

    • Decide what to cover
    • Plan sessions and format

  • Measure theory & refreshing on algreba and analysis (x2?) (Adam ?)

    • Sigma-algebra, measure, Borel/Lebesgue measures, probability measure
    • Convergence of random variables / measures
    • Compact / complete / separable space, Borel space, (metric / Banach / Hilbert space) ? length space
    • Distributions space / lp Banach space

  • Probability measures on manifold:

    • Random variable
    • Variance
    • Fréchet mean
    • Generalisation of Normal distribution
    • Maximum likelihood estimator ? cf hyperbolic prior for univariate normal ?
    • Sampling ? cf generalised normal distribution

  • Density fitting

    • Weak vs strong convergence
    • Csiszar Divergences - Metrization
    • Loss Functions for Measures
    • Sample complexity
    • Deep Generative Models

  • Introduction to optimal transport & theoretical foundations (Emile ?)

    • Monge problem: static formulation through transport map
    • Kantorovich: linear programming formulation - Kantorovich Duality Theorem
    • Benamou-Brenier variational (dynamic) formulation
    • Equivalence between formulations, existence, uniqueness

  • Geometry of probability measures

    • Refreshing on Riemannian geometry: Gauss/sectional/Ricci curvature (analytic & synthetic), tangent space, metric tensor ?
    • Fisher-Rao (information) geometry of parametric densities - natural gradient
    • Wasserstein space (metric space) - weak topology
    • Wasserstein metric tensor and the density space

  • Gradient flow in probability measures spaces

    • Gradient flows
    • Discretization
    • Fisher-Rao gradient flow
    • Wasserstein gradient flows

Ressources:

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