An Invitation to Statistics in Wasserstein Space

Description

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