An Invitation to Statistics in Wasserstein Space.

• Superior document: SpringerBriefs in Probability and Mathematical Statistics Series
• Author: Panaretos, Victor M.
• TeilnehmendeR: Zemel, Yoav.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2020.; ©2020.
• Year of Publication: 2020
• Edition: 1st ed.
• Language: English
• Series: SpringerBriefs in Probability and Mathematical Statistics Series
• Physical Description: 1 online resource (157 pages)

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.