An invitation to statistics in Wasserstein space

Preface......Page 7
Contents......Page 11
 1.1 The Monge and the Kantorovich Problems......Page 14
 1.2 Probabilistic Interpretation......Page 18
 1.3 The Discrete Uniform Case......Page 20
 1.4 Kantorovich Duality......Page 21
  1.4.1 Duality in the Discrete Uniform Case......Page 22
  1.4.2 Duality in the General Case......Page 23
 1.5 The One-Dimensional Case......Page 24
  1.6.1 The Absolutely Continuous Case......Page 26
  1.6.2 Separable Hilbert Spaces......Page 29
  1.6.3 The Gaussian Case......Page 30
  1.6.4 Regularity of the Transport Maps......Page 31
 1.7 Stability of Solutions Under Weak Convergence......Page 33
  1.7.1 Stability of Transference Plans and CyclicalMonotonicity......Page 34
  1.7.2 Stability of Transport Maps......Page 37
 1.8 Complementary Slackness and More General Cost Functions......Page 42
  1.8.1 Unconstrained Dual Kantorovich Problem......Page 43
  1.8.2 The Kantorovich–Rubinstein Theorem......Page 44
  1.8.3 Strictly Convex Cost Functions on Euclidean Spaces......Page 45
 1.9 Bibliographical Notes......Page 47
 2.1 Definition, Notation, and Basic Properties......Page 49
  2.2.1 Convergence, Compact Subsets......Page 51
  2.2.2 Dense Subsets and Completeness......Page 54
  2.2.3 Negative Topological Properties......Page 55
  2.2.4 Covering Numbers......Page 56
  2.3.1 Geodesics, the Log Map and the Exponential Mapin W2(X)......Page 58
  2.3.2 Curvature and Compatibility of Measures......Page 60
  2.4.1 Measurability of Measures and of Optimal Maps......Page 64
  2.4.2 Random Optimal Maps and Fubini's Theorem......Page 66
 2.5 Bibliographical Notes......Page 68
3 Fréchet Means in the Wasserstein Space W2......Page 70
  3.1.1 The Fréchet Functional......Page 71
  3.1.2 Multimarginal Formulation, Existence, and Continuity......Page 72
  3.1.3 Uniqueness and Regularity......Page 74
  3.1.4 The One-Dimensional and the Compatible Case......Page 75
  3.1.5 The Agueh–Carlier Characterisation......Page 76
  3.1.6 Differentiability of the Fréchet Functional and Karcher Means......Page 77
 3.2 Population Fréchet Means......Page 79
  3.2.1 Existence, Uniqueness, and Continuity......Page 80
  3.2.3 Differentiability of the Population Fréchet Functional......Page 83
 3.3 Bibliographical Notes......Page 85
4 Phase Variation and Fréchet Means......Page 86
  4.1.1 The Functional Case......Page 87
  4.1.2 The Point Process Case......Page 93
  4.2.1 Equivariance Properties of the Wasserstein Distance......Page 97
  4.2.2 Canonicity of Wasserstein Distance in Measuring Phase Variation......Page 98
  4.3.1 Oracle Case......Page 100
  4.3.2 Discretely Observed Measures......Page 101
  4.3.3 Smoothing......Page 102
  4.3.4 Estimation of Warpings and Registration Maps......Page 104
  4.3.5 Unbiased Estimation When X=R......Page 105
 4.4 Consistency......Page 106
  4.4.1 Consistent Estimation of Fréchet Means......Page 107
  4.4.2 Consistency of Warp Functions and Inverses......Page 110
  4.5.1 Explicit Classes of Warp Maps......Page 112
  4.5.2 Bimodal Cox Processes......Page 114
  4.5.3 Effect of the Smoothing Parameter......Page 117
 4.6 Convergence Rates and a Central Limit Theoremon the Real Line......Page 118
 4.7 Convergence of the Empirical Measure and Optimality......Page 122
 4.8 Bibliographical Notes......Page 126
5 Construction of Fréchet Means and Multicouplings......Page 128
 5.1 A Steepest Descent Algorithm for the Computation of FréchetMeans......Page 129
 5.2 Analogy with Procrustes Analysis......Page 131
 5.3 Convergence of Algorithm 1......Page 132
  5.4.1 Gaussian Measures......Page 137
  5.4.2 Compatible Measures......Page 140
   5.4.2.1 The One-Dimensional Case......Page 141
   5.4.2.2 Independence......Page 142
   5.4.2.3 Common Copula......Page 144
  5.4.3 Partially Gaussian Trivariate Measures......Page 145
 5.5 Population Version of Algorithm 1......Page 148
 5.6 Bibliographical Notes......Page 149
References......Page 151