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An Invitation to Statistics in Wasserstein Space.

Bibliographic Details
Main Author: Panaretos, Victor M.
Other Authors: Zemel, Yoav.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2020.
Edition:1st ed.
Series:SpringerBriefs in Probability and Mathematical Statistics Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Preface
  • Contents
  • 1 Optimal Transport
  • 1.1 The Monge and the Kantorovich Problems
  • 1.2 Probabilistic Interpretation
  • 1.3 The Discrete Uniform Case
  • 1.4 Kantorovich Duality
  • 1.4.1 Duality in the Discrete Uniform Case
  • 1.4.2 Duality in the General Case
  • 1.5 The One-Dimensional Case
  • 1.6 Quadratic Cost
  • 1.6.1 The Absolutely Continuous Case
  • 1.6.2 Separable Hilbert Spaces
  • 1.6.3 The Gaussian Case
  • 1.6.4 Regularity of the Transport Maps
  • 1.7 Stability of Solutions Under Weak Convergence
  • 1.7.1 Stability of Transference Plans and CyclicalMonotonicity
  • 1.7.2 Stability of Transport Maps
  • 1.8 Complementary Slackness and More General Cost Functions
  • 1.8.1 Unconstrained Dual Kantorovich Problem
  • 1.8.2 The Kantorovich-Rubinstein Theorem
  • 1.8.3 Strictly Convex Cost Functions on Euclidean Spaces
  • 1.9 Bibliographical Notes
  • 2 The Wasserstein Space
  • 2.1 Definition, Notation, and Basic Properties
  • 2.2 Topological Properties
  • 2.2.1 Convergence, Compact Subsets
  • 2.2.2 Dense Subsets and Completeness
  • 2.2.3 Negative Topological Properties
  • 2.2.4 Covering Numbers
  • 2.3 The Tangent Bundle
  • 2.3.1 Geodesics, the Log Map and the Exponential Mapin W2(X)
  • 2.3.2 Curvature and Compatibility of Measures
  • 2.4 Random Measures in Wasserstein Space
  • 2.4.1 Measurability of Measures and of Optimal Maps
  • 2.4.2 Random Optimal Maps and Fubini's Theorem
  • 2.5 Bibliographical Notes
  • 3 Fréchet Means in the Wasserstein Space W2
  • 3.1 Empirical Fréchet Means in W2
  • 3.1.1 The Fréchet Functional
  • 3.1.2 Multimarginal Formulation, Existence, and Continuity
  • 3.1.3 Uniqueness and Regularity
  • 3.1.4 The One-Dimensional and the Compatible Case
  • 3.1.5 The Agueh-Carlier Characterisation
  • 3.1.6 Differentiability of the Fréchet Functional and Karcher Means
  • 3.2 Population Fréchet Means.
  • 3.2.1 Existence, Uniqueness, and Continuity
  • 3.2.2 The One-Dimensional Case
  • 3.2.3 Differentiability of the Population Fréchet Functional
  • 3.3 Bibliographical Notes
  • 4 Phase Variation and Fréchet Means
  • 4.1 Amplitude and Phase Variation
  • 4.1.1 The Functional Case
  • 4.1.2 The Point Process Case
  • 4.2 Wasserstein Geometry and Phase Variation
  • 4.2.1 Equivariance Properties of the Wasserstein Distance
  • 4.2.2 Canonicity of Wasserstein Distance in Measuring Phase Variation
  • 4.3 Estimation of Fréchet Means
  • 4.3.1 Oracle Case
  • 4.3.2 Discretely Observed Measures
  • 4.3.3 Smoothing
  • 4.3.4 Estimation of Warpings and Registration Maps
  • 4.3.5 Unbiased Estimation When X=R
  • 4.4 Consistency
  • 4.4.1 Consistent Estimation of Fréchet Means
  • 4.4.2 Consistency of Warp Functions and Inverses
  • 4.5 Illustrative Examples
  • 4.5.1 Explicit Classes of Warp Maps
  • 4.5.2 Bimodal Cox Processes
  • 4.5.3 Effect of the Smoothing Parameter
  • 4.6 Convergence Rates and a Central Limit Theoremon the Real Line
  • 4.7 Convergence of the Empirical Measure and Optimality
  • 4.8 Bibliographical Notes
  • 5 Construction of Fréchet Means and Multicouplings
  • 5.1 A Steepest Descent Algorithm for the Computation of FréchetMeans
  • 5.2 Analogy with Procrustes Analysis
  • 5.3 Convergence of Algorithm 1
  • 5.4 Illustrative Examples
  • 5.4.1 Gaussian Measures
  • 5.4.2 Compatible Measures
  • 5.4.2.1 The One-Dimensional Case
  • 5.4.2.2 Independence
  • 5.4.2.3 Common Copula
  • 5.4.3 Partially Gaussian Trivariate Measures
  • 5.5 Population Version of Algorithm 1
  • 5.6 Bibliographical Notes
  • References.

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