Index

A¶

• absolute continuity: 20, 21, 26, 58, 113, 126, 137, 250

• absolutely continuous path: 296, 299

• absorbed log-scaling: 152

• absorptive matching: 163

• action density: 300

• active mass: 200, 210, 212

• additively homogeneous map: 160

• admissible potential: 80, 82, 86

• ADMM: 298

• adversarial

  • loss: 115

  • perturbation: 77

• affine

  • closure: 388, 389, 394–396

  • constraint: 156, 267

  • gradient: 389, 395

  • map: 38, 82, 371

  • marginal constraint: 155, 156

• affine-difference cost: 58

• affine-invariant metric: 40

• Alexandrov solution: 24

• algorithmic convergence: 155

• alternating

  • optimization: 87

  • projection: 121, 155, 156, 158

  • tree: 8, 10

• angular covering: 190

• antisymmetric flux: 353

• antisymmetric velocity: 307

• assignment problem: 1, 2, 7, 8, 43, 48, 51, 52, 89, 278

• Assouad lemma: 182

• atom-preserving map: 75

• atomic measure: 306

• atomless measure: 17, 30

• attention: 375, 382, 388, 396

  • Gaussian kernel: 383

  • linear: 199

  • Lipschitz stability: 76

  • map: 76

  • mechanism: 382

  • operator: 382

  • self-attention: 76, 382

  • single-head: 382

• auction algorithm: 7, 89, 91

  • coordinate ascent: 89

• augmented Lagrangian: 315

B¶

• Baker-Campbell-Hausdorff formula: 288

• Bakry–Émery criterion: 339

• balance equation: 311, 312, 356

• balanced interpolation: 312

• balanced OT: 139

• Banach dual: 12, 13

• Banach fixed-point theorem: 206

• Banach space: 12

• Banach-space embedding: 32

• barrier

  • entropy: 119

  • logarithmic: 53, 119, 122

  • LP: 119

  • parameter: 54

• barycenter: 236, 379

  • fixed support: 236

  • free support: 237

  • law of large numbers: 245

  • mean: 238

  • support: 237, 238, 253

• barycentric

  • attention: 383

  • constraint: 268

  • mixture: 245, 276, 277

  • projection: 45, 187, 189, 229, 266–268, 271

  • transport: 268

• basic feasible solution: 47, 253

• Beckmann

  • flow: 105

  • formulation: 88, 104, 105, 303

  • problem on graphs: 105, 106

• belief propagation: 255

• Benamou–Brenier: 135, 137, 272–275, 293, 295, 296, 299, 301, 304, 312

  • distance: 301

  • dual: 297

  • formula: 293, 295

  • formulation: 135, 295, 299

  • kernelized: 305, 375

  • viscous: 137

• Berry-Esseen

  • bound: 177

  • theorem: 72, 177

• Besov smoothness: 183

• Besov space: 184

• Bessel function: 110

• bias: 190, 191

• bias-variance decomposition: 191

• bias-variance tradeoff: 176, 183

• bilevel problem: 261

• bilinear cost: 264

• Birkhoff

  • contraction theorem: 166, 167, 169, 384

  • polytope: 48

  • von Neumann theorem: 48, 49, 51, 58

• birth-death dynamics: 356, 357

• bistochastic matrix: 48, 50

• block-coordinate

  • ascent: 244

• block-coordinate descent: 257

• block-coordinate method: 226

• block-quotient norm: 165

• block-stationary point: 257

• Bobkov-Ledoux formula: 31

• Bochner spectral measure: 111

• Bochner theorem: 194

• Boltzmann entropy: 117

• Boltzmann equation: 362

• book-shifting: 4, 19, 56

• Borel measure: 11, 12, 21

• Borel set: 11, 12, 14, 23

• bounded confidence: 384

• box constraint: 259

• Bregman

  • divergence: 59, 112, 114, 123, 146–148, 155, 156

  • iteration: 286

  • penalty: 155

  • projection: 119, 149, 155–158, 163, 259, 286, 287

  • regularization: 146, 148, 149

  • three-point formula: 156

• Brenier

  • factor: 22

  • map: 21–24, 31, 35, 57, 80, 86, 126, 172, 187, 226, 227, 229, 238, 325, 328, 330, 337, 338, 371, 382, 392

  • perturbation: 215

  • potential stability: 188

  • source hypothesis: 21

  • theorem: 20–22, 25, 26, 36–38, 57, 58, 62, 86, 94, 226, 250

• Brenier–Strassen projection: 270

• bridge

  • Brownian: 135, 137–139

  • linear: 372

  • OU: 370, 373

  • overshooting: 372, 373

  • speed: 372

  • variance-preserving: 370, 372, 373

• Brownian bridge: 102, 135, 138

• Brunn-Minkowski inequality: 337, 338

• Bures

  • covariance: 38

  • metric: 36–41, 247

  • smoothed term: 171

  • Wasserstein geometry: 36, 39, 171, 174, 240, 277, 389, 392

• Burg entropy: 114

• Burgers equation: 144, 145

C¶

• c-concavity: 134

• c-convexity: 29

• c-cyclical monotonicity: 61, 62

• c-splitting set: 250

• c-transform: 80, 82–88, 93, 103, 125, 132

• Caffarelli regularity: 24, 25

• capacity-constrained OT: 258–260

• capacity-constrained OT feasibility: 259

• capacity-constrained Sinkhorn: 259

• capped transportation polytope: 259

• Catalan number: 7

• causality: 382

• CDF formula: 31

• cell mass: 93, 94

• centered fluctuation: 191, 193

• centered measure: 65

• central limit theorem: 72, 176, 177, 246

  • empirical process: 179

  • entropic OT: 192

  • optimal transport: 190, 191

  • quantitative: 72

• central path: 53, 54

• centroidal Voronoi: 100

• centroidal Voronoi tessellation: 98

• chain graph: 254

• change of variables formula: 15, 25, 26, 42, 174, 239

• characteristic kernel: 321

• Choi matrix: 288

• Cholesky factorization: 142

• circle

  • transport: 5, 6

  • unfolded interval: 5, 6

• classical convexity: 347

• clustering: 384

• co-motion function: 251

• co-motion map: 251

• collapsed mixture: 245, 276, 277

• column constraint: 162

• column normalization: 121, 159

• commutator: 288

• compact sublevel set: 201

• compactness: 308

• complementary slackness: 7, 9, 80–83, 91, 266, 298

• complete graph: 254

• complete metric space: 12

• completely positive cone: 196

• completely positive kernel: 196

• completely positive matrix: 196

• complex

  • epsilon: 152

  • Gibbs kernel: 152

  • matrix scaling: 152

  • measure: 152

  • potential: 152

  • Sinkhorn iteration: 152

  • Sinkhorn potential: 154

  • Sinkhorn scaling: 153

• composition constraint: 325

• computer vision: 57

• concave cost: 2

• concave envelope: 146

• concavity

  • discrete OT: 45

  • Kantorovich value: 56

• conditional

  • coupling: 234

  • expectation: 269, 365–367, 369, 371

  • geodesic: 234

  • gradient flow: 359

  • independence: 256

  • Jensen inequality: 203

  • law: 33–35, 137, 266–268, 299

  • mean: 267

  • neuron law: 358

  • optimal transport: 233, 357

  • probability: 12, 65, 276, 366

  • quantile: 227

  • transport: 233

  • variance: 268

  • Wasserstein distance: 233, 234, 357

  • Wasserstein flow: 357

• conditionally negative definite kernel: 151

• conditionally positive kernel: 109

• cone

  • formulation: 204

  • lifting: 203, 312

  • metric: 204, 312

• congestion: 258, 313, 323

• congruence normalization: 288

• conic lifting: 203

• conjugate moment measure: 382

• consensus: 384

  • Dobrushin contraction: 385

• constant-speed geodesic: 23, 68, 281, 325

• contact set: 29

• continuity equation: 137, 272, 274, 275, 293, 295–298, 311, 316, 318, 352, 365–367, 369, 374, 377, 378, 389

• continuous

  • bounded function: 71

  • coupling: 54

  • depth: 359, 382

  • dual: 81, 132

  • path: 12

  • Sinkhorn dual: 146

• continuous DTW: 290

• continuous-depth neural network: 382

• contraction ratio: 169

• contractive Gaussian projection: 396

• controlled transport: 388

• convergence

  • in law: 32, 71, 152

  • in probability: 71

  • law: 70

  • mode: 71

• convergence-determining class: 176

• convex

  • action density: 301, 303

  • conjugate: 86, 147, 159, 382

  • envelope: 87, 144

  • function: 20–22, 26, 30, 31, 36, 57, 61, 70, 78, 81, 86, 87, 114, 145, 146, 155, 320, 379

  • neural network: 134

  • order: 269–271

  • potential: 20–22, 24–26, 61, 62, 84, 88, 135, 370, 379

  • potential generator: 382

  • regularizer: 145, 265

  • relaxation: 70

• convex hull: 238, 385

• convexity

  • along geodesics: 346

  • classical: 346, 347

  • constraint: 25

  • discrete OT: 45

  • geodesic: 316, 325, 326, 328–330, 332, 335, 337

  • Kantorovich value: 56

  • perspective transform: 301

  • strong geodesic: 334

• convolution: 75

• convolutional Sinkhorn: 142

• coordinate ascent: 89, 163

• coordinate clearing: 161, 162

• copositive certificate: 198

• copositive matrix: 196

• correspondence: 45, 272, 279, 282, 284

• cost

  • assignment: 52

  • bilinear: 28

  • derivative: 261

  • learning: 261

  • matrix: 1, 44, 47, 48, 63, 81, 120, 122, 165, 237, 282

  • reduced: 8, 9, 53, 106

  • transport: 56

  • unbalanced: 203

• cost-sectional curvature: 29

• Coulomb cost: 250–252

• countable dense subset: 12, 276

• counterexample: 251

• coupling measure: 65

• covariance

  • commuting: 371

  • cone: 40

  • diagonal: 41

  • Loewner order: 271

  • matrix: 31, 38, 41, 397

  • normalization: 352

  • operator: 229

  • subblock: 394

  • term: 38, 39

• covering number: 180, 182

• Cramér condition: 178

• Cramer-von Mises distance: 32

• Cramer-Wold theorem: 214, 248

• created mass: 200

• cross-covariance: 393

• crowd motion: 323, 324

• Csiszár divergence: 328

• cumulant

  • fourth: 178

• cumulative distribution function: 29–31

• cumulative function: 4, 29, 32, 33

• curse of dimensionality: 96, 97, 180, 191

• curve of maximal slope: 318, 348

• cut locus: 28, 29

• cycle graph: 254

• cyclic

  • Bregman projection: 157

  • co-motion plan: 251

  • monotonicity: 61, 62, 267

  • projection: 156

  • shift: 5

• cylindrical Gaussian: 246

D¶

• Dacorogna–Moser

  • construction: 294

  • inversion: 294

• damped iteration: 241

• damped second-order equation: 360

• Danskin theorem: 192, 261

• data-processing inequality: 114, 164

• de la Vallée–Poussin criterion: 177

• Delaunay graph: 106

• delta method: 190, 192

• demographic parity: 237

• denoising score matching: 365

• dense linear system: 119

• density

  • closure: 363

  • constraint: 314, 323–325, 329, 330

  • estimation: 182, 183

  • formula: 15

  • matrix: 285, 286, 288

  • ratio: 13, 111, 112, 114, 116, 145–149, 258

  • relative: 13

• density-functional theory: 250–252

• depth evolution: 382

• depth variable: 358

• destroyed mass: 200

• detailed balance: 355

• deterministic interpolant: 365

• diagonal coupling: 269

• diagonal dominance: 162

• diagonal isotonicity: 161

• diameter

  • contraction: 385

• diffusion model: 74, 75, 365, 369, 372

  • Gaussian noising: 128

• diffusive map: 75

• dimension dependence: 182

• Dirac mass: 11, 14, 18, 31, 36, 43, 63, 69, 75, 238, 256, 316, 319, 325, 366

• Dirac measure: 250

• directional derivative: 261

• Dirichlet energy: 320

• discrepancy minimization: 373

• discrete

  • barycenter: 237, 252

  • c-transform: 84

  • coupling: 43

  • Laplacian: 142

  • measure: 11, 68, 71, 117

  • space: 74

  • target: 93

  • Wasserstein geometry: 309, 310, 355

• discretization error: 242, 366

• discriminator class: 107, 108, 116

• disintegration: 12, 34, 65, 85, 137, 204, 233, 250, 266–268, 367, 374

• displacement convexity: 23, 26, 327, 337, 344, 346, 379, 380

• displacement covariance: 230, 233

• displacement interpolation: 23, 24, 32, 34, 233, 330

• distance profile: 283

• distance residual: 282

• distortion: 278, 280, 282, 284

• distributional robustness: 77

• divergence-free field: 389

• Dobrushin coefficient: 384

• domain adaptation: 57

• Doob transform: 136

• doubly nonnegative cone: 196

• doubly nonnegative matrix: 196

• doubly positive kernel: 196

• doubly stochastic normalization: 388

• Douglas-Rachford algorithm: 298

• Douglas-Rachford splitting: 298, 299

• drifting

  • field: 377, 388

  • model: 375, 377, 378

• dual

  • ascent: 140, 146, 159

  • attainment: 82, 84, 86, 286

  • block maximization: 134

  • certificate: 7, 87

  • formula: 115

  • gap: 122, 155, 165, 166, 265, 266

  • gradient: 94

  • norm: 107–110, 116

  • objective gap: 163

  • pairing: 12

  • potential: 80, 81, 84, 85, 87, 93, 121, 122, 125, 134, 145, 148–151, 165, 169, 172, 180, 187, 190, 192, 193, 261, 265, 286

  • price: 8

  • problem: 80, 84, 88, 105, 118, 131, 134, 244

  • rate: 164

  • Sinkhorn: 140

  • slack: 83

  • twist condition: 28

  • weight: 93, 94

• duality

  • entropic OT: 133

  • Fenchel-Rockafellar: 145, 202, 267

  • strong: 105, 244, 285

  • weak: 10, 81, 267

• Dudley entropy: 187

• Dudley metric: 107, 108

• Dvoretzky-Kiefer-Wolfowitz inequality: 190

• dyadic cube: 182

• dyadic partition: 180

• Dykstra algorithm: 259

• dynamic

  • action: 296

  • distance: 302, 304

  • formulation: 295, 299, 302

  • optimal transport: 293

  • unbalanced OT: 311

  • value: 296

  • Wasserstein flow: 348

• dynamic programming: 289

• dynamic time warping: 289

  • algorithm: 290

  • numerics: 292

• dynamical optimal plan: 69, 337

• dynamical plan: 234

E¶

• earth mover’s distance: 57

• edge flux: 310

• edge imbalance: 33

• Edgeworth expansion: 178

• effective cost: 257

• effective dimension: 195

• effective rank: 195

• electron density: 251

• elliptical law: 39

• elliptically contoured distribution: 39

• embedding

  • alignment: 224

• empirical

  • context: 382

  • distribution: 322, 382

  • dual potential: 187

  • energy: 363

  • law: 77, 78, 176, 322

  • measure: 3, 11, 18, 46, 51, 71, 101, 150, 176, 180, 184, 221, 260, 263, 280, 293, 321, 322, 352, 358, 360, 374, 383

  • measure over measures: 245

  • Monge map: 17

  • OT: 97, 181, 182

  • OT rate: 101, 181, 281

  • process: 179, 187, 190, 193

  • Wasserstein metric: 360

• endpoint coupling: 136–139

• energy

  • decay: 332

  • dissipation: 331, 333, 361

  • interaction: 150, 320–322, 326, 392

• energy distance: 109, 151, 320, 321, 327, 362

• energy-dissipation identity: 331

• energy-dissipation inequality: 318, 331

• entropic

  • barycenter: 239, 244

  • barycentric map: 188

  • bias: 122, 129, 149

  • map estimation: 189

  • OT: 117–119, 128, 131, 150, 163, 170, 190, 192, 262, 266, 283, 286, 377

  • path: 119

  • plan: 199

  • potential: 135, 149, 158

  • regularization: 87, 93, 117, 119, 149, 155, 180, 185, 190, 205, 253, 257, 259, 263, 283, 286

  • smoothing: 118, 163, 189, 243

  • temperature: 173

• entropy: 380

  • closure: 363

  • decay: 340

  • function: 112, 113, 145, 200

  • penalty: 117

  • score: 363

• envelope theorem: 118, 192, 261, 262

• $$-scaling: 92

• equal marginal: 251

• equality constraint: 53, 105

• essential continuity: 379

• Euclidean group: 224

• Eulerian

  • description: 293

  • velocity: 294, 296, 297

• exact OT: 25, 180, 182, 189

• explicit Euler scheme: 360

• extreme minimizer: 49

• extreme point: 49–51, 53

F¶

• factored coupling: 256

• fairness: 237

• farthest-point sampling: 363

• fast diffusion equation: 350

• fast Legendre transform: 143, 144

• feasible coupling: 53, 55, 227, 281

• feature cost: 283

• feature sketch: 193

• feature term: 284

• Fenchel duality: 140, 145, 147, 202, 267

• Fenchel inequality: 135

• Fenchel-Rockafellar duality: 297, 298

• Fenchel-Young loss: 263

• FFT: 144

• fiber: 234

• fiberwise optimal plan: 234

• fiberwise transport: 233

• fill-in edge: 254

• filtered back-projection: 248

• finite difference scheme: 25, 364

• finite measure: 12, 107, 108, 201, 272

• finite metric length: 331

• finite space: 190, 191

• finite-action component: 302, 306

• finite-dimensional linear program: 103, 105

• finite-sample bias: 374

• finite-state transport: 309

• first variation: 147, 261, 262, 287, 316, 317, 332, 345, 347, 359, 377, 378, 388, 392, 393, 395

• Fisher information: 130, 137, 339, 340, 344, 392

• Fisher-Rao metric: 36, 37, 40

• fixed-point

  • Gaussian barycenter: 241

• flattening map: 245

• flow

  • conservation: 53

  • continuous normalizing: 366

  • cross-covariance: 393

  • map: 293, 371

  • minimum-cost: 53, 106

  • momentum: 360

  • normalizing: 366

  • one-step: 374

  • parameter-domain: 373

  • phase-space: 361

  • probability ODE: 365, 366, 368–370, 372

  • rectified: 365, 366, 370

  • reverse: 373

  • second-order: 360

• flow matching: 295, 365–367, 369–371, 396

• flux: 272, 274, 294, 317

• Fokker–Planck

  • equation: 321–323, 340, 344, 350, 354, 357, 368, 371, 372

  • fractional equation: 355

• Fokker–Planck particle closure: 377

• Fortet iteration: 158

• Fourier features: 193

• Fourier multiplier: 109, 110

• Fourier random features: 194

• Fourier transform: 247

• Fourier-slice theorem: 248

• fractional diffusion: 354

• fractional heat equation: 354

• fractional Laplacian: 353, 354

• fractional PDE: 353

• Frechet inception distance: 111

• Frechet mean: 236, 238

• Frobenius inner product: 40

• Frobenius norm: 230

• frozen surrogate: 378

• functional inequality: 337, 338, 341

• fused Gromov-Wasserstein: 283, 284

G¶

• Gamma convergence: 128

• GAN: 115

• gauge direction: 161

• gauge fixing: 173

• gauge invariance: 162

• Gaussian

  • barycenter: 240, 241

  • blurring: 128

  • closure: 174, 209, 375, 389, 394–396

  • complex Sinkhorn: 153

  • convex order: 270

  • convolution: 143, 144

  • covariance: 171, 390

  • denoiser: 369

  • flow matching: 370, 371

  • kernel: 194

  • kernel attention: 383

  • kernel drifting: 378, 396

  • kernel rank: 195

  • manifold: 388–390, 394

  • marginals: 135, 170

  • mean: 390

  • measure: 18, 22, 36, 37, 240, 271, 340, 379, 388, 395

  • mixture: 3, 93, 99, 121, 122, 125, 147, 200, 239, 258, 260, 277, 370, 372

  • one-dimensional: 394

  • optimal map: 31

  • preservation: 389, 390, 395, 397

  • preserving flow: 395

  • projection: 395–397

  • push-forward: 37

  • scale mixture: 198

  • Sinkhorn: 170–172, 174, 393

  • Sinkhorn divergence: 171

  • Sinkhorn flow: 174

  • Sinkhorn problem: 174

  • sliced Wasserstein: 218

  • submanifold: 389, 395

  • unbalanced: 207

• Gaussian Hellinger-Kantorovich: 207

• Gelbrich inequality: 396, 397

• Gelbrich theorem: 397

• generalized

  • inverse: 17, 29, 30

  • Wasserstein: 200

• generalized Gaussian kernel: 197, 198

• generalized quantile: 34

• generalized Sinkhorn algorithm: 139, 141

• generalized Wasserstein flow: 348

• generative model: 107, 220, 293, 330, 365, 373, 382

• generator manifold: 374

• geodesic: 307

  • convexity: 325, 329, 332

  • distance: 26, 33, 142, 300

  • semiconcavity: 330

  • space: 69, 337

• geodesics in heat: 141, 142

• Gibbs

  • coupling: 286, 287

  • formula: 286

  • kernel: 121–123, 126, 140, 152, 156, 158, 169, 194

    • push-forward: 127

    • sketching: 198

  • path law: 291

  • reference: 155, 156

• Girsanov theorem: 136

• global

  • minimizer: 347

  • optimality: 347

• global invariance: 224

• gluing lemma: 63–65, 68, 205, 280

• gradient

  • field: 21

  • flow: 23, 279, 281, 293, 295, 316, 318–323, 325, 331, 332, 345–347, 365, 375, 388–390, 395, 397

  • inequality: 333

  • kernelized flow: 375

  • noise: 355

  • projection: 378

  • structure: 273, 295, 388

  • velocity: 294, 295, 390

• graph

  • augmenting path: 10

  • bipartite: 47, 50

  • coupling: 56, 69

  • equality: 8–10

  • geodesic distance: 105

  • matching: 278

  • support: 47

  • transport: 105

• graphical model: 253

• Grassmann manifold: 220

• gravity interaction: 362

• greedy sweep: 47

• Gromov-Hausdorff distance: 279, 284

• Gromov–Wasserstein: 225, 272, 278–281, 283, 284

  • distance: 272, 278

  • geodesic: 281

  • metric: 280

  • sample complexity: 280

• gross substitutes: 161

• ground cost: 63, 103, 245, 261, 263, 278, 282

• ground metric learning: 66

• group action: 223

• growth term: 311

• Gurvits scaling: 287, 288

H¶

• Hadamard differentiability: 193

• Hall theorem: 50

• Hamilton-Jacobi equation: 137, 144, 297, 298

• Hamilton-Jacobi-Bellman equation: 314

• hard congestion: 314

• Hausdorff distance: 279, 284

• Hausdorff measure: 21

• heat equation: 144, 319, 320, 363

• heat flow: 75, 309

• heat kernel: 141–143

• heavy-ball method: 360, 361

• heavy-tailed noise: 355

• Hegselmann–Krause model: 384

• Hellinger

  • divergence: 114

  • geodesic: 209

  • geometry: 41

• Hellinger-Kantorovich: 312

• Helmholtz decomposition: 390

• Hermitian matrix: 285, 287

• hidden convexity: 380

• Hilbert

  • diameter: 169

  • embedding: 217

  • metric: 166

  • projective metric: 166, 167, 169

• Hilbert space: 330

• Hilbertian

  • embedding: 227

  • limit: 151

• Hilbertian tangent norm: 300

• Hilbertian Wasserstein space: 32

• histogram: 4, 11, 32, 34, 63, 65, 118, 152, 165, 200, 237, 272, 278

• Hoeffding inequality: 195

• Holder curve: 331

• Hölder inequality: 231

• holomorphic continuation: 152

• holomorphic implicit function theorem: 153

• homogeneous action: 302

• homogeneous formulation: 202, 203

• homogeneous Sobolev norm: 151

• homogenization: 203

• Hopf-Cole transform: 141, 142, 144, 145

• Hopf-Lax formula: 142, 144

• Horn matrix: 198

• Hungarian primal-dual method: 1, 7–9

• HWI inequality: 344, 345

• hyperbolic geodesic: 27

• hyperbolic metric: 36

• hypersurface: 21

I¶

• ice-cream cone: 40

• imitation learning: 116

• implicit Euler scheme: 316, 348

• in-context map: 383

• incompressible Euler: 255

• increasing rearrangement: 59

• induced width: 254

• inertia: 362

• inertial particle method: 361

• infinite-depth limit: 382

• infinite-dimensional linear programming: 56

• input-convex neural network: 134, 135, 382

• integer multiplicity: 52

• integral probability metric: 106–108, 114–116, 374

• interacting particle system: 344, 347

• interaction energy: 337

• interaction kernel: 362

• interior-point method: 53, 119, 122

• intermediate measure: 256

• internal energy: 327

• interpolant: 372

• intrinsic dimension: 176, 180, 182

• intrinsic length metric: 217

• inverse isotonicity: 161

• inverse OT: 261, 263, 265

• inverse OT gap loss: 263

• inverse square root: 288

• irreducibility: 308

• isometric embedding: 284

• isoperimetric inequality: 337, 338

• isotonic regression: 59

• iterative closest point: 224

• iterative proportional fitting: 119, 120

J¶

• Jacobi iteration: 161

• Jacobian determinant: 15, 328

• Jacobian equation: 28, 29, 174

• Jensen inequality: 70, 113, 124, 177, 185, 188, 268, 296, 305

• Jensen-Shannon divergence: 114, 116

• JKO scheme: 139, 316, 317, 319, 325, 348, 352, 355, 356, 360, 395, 397

• joint

  • law: 54, 57, 252

  • measure: 54

• Jordan decomposition: 13

• jump kernel: 306

• jump process: 306, 353

• junction tree: 254

K¶

• Kantorovich

  • coupling: 27, 60

  • dual: 298

  • duality: 7, 20, 28, 80–82, 86, 265, 267, 285, 380

  • interpolation: 23

  • potential: 81, 83, 250, 261

  • problem: 45, 51–53, 56, 57, 60, 62, 69, 80, 125, 135, 138, 258, 267, 275

  • relaxation: 1, 16, 20, 43, 45, 51, 54, 57, 69, 200

• Kantorovich–Rubinstein

  • duality: 72, 77, 108, 116, 177

  • formula: 103–105

  • norm: 32, 103

• kernel

  • attention: 199

  • characteristic: 110

  • density estimate: 363, 375, 383

  • drifting: 378

  • energy: 109, 110

  • Gaussian: 110, 123, 377, 388, 396

  • Laplacian: 110

  • logarithmic accuracy: 195

  • low-rank approximation: 193

  • Matérn: 109, 110

  • mean embedding: 110, 184, 228

  • norm: 110, 116, 151, 326, 377

  • positive: 109, 150, 166, 377

  • positive definite: 151, 184, 217, 321

  • positivity: 195

  • quadrature: 321

  • regression: 383, 388

  • ridge regression: 60, 194

  • Riesz: 109

  • sample complexity: 193

  • sketching: 193

  • smoothing: 183

  • universal: 110, 111, 152

• kernelized self-interaction: 321

• kernelized Stein discrepancy: 375

• kernelized transport distance: 305, 306

• kinetic action: 137

• kinetic equation: 362

• kinetic mean-field limit: 361

• KL

  • projection: 106, 140, 155–157, 163, 165, 243, 259, 288

  • proximal map: 140

  • relaxation: 205

• KL barycenter: 241

• KL proximal operator: 139–141

• Knothe–Rosenblatt rearrangement: 33–35

• Kolmogorov forward equation: 355

• Kolmogorov-Smirnov distance: 32

• Kullback-Leibler divergence: 36, 40, 113, 123, 124, 126, 139, 335

• Kurdyka–Łojasiewicz inequality: 333, 335, 336

L¶

• L2 attention: 383

• L2 self-attention: 383

• Lagrange multiplier: 324

• Lagrangian

  • description: 293

  • flow: 293

• Lagrangian velocity: 361

• Laguerre cell: 18, 84, 93–95, 100

  • discrete: 89

• Langevin

  • dynamics: 136, 323

  • free energy: 322

• Laplace method: 173

• Laplace-Beltrami operator: 141

• large-temperature collapse: 149

• large-temperature limit: 151

• latent

  • atom: 256

  • mass: 257

  • variable: 256, 366, 368

• latent measure: 256

• lattice distribution: 178

• Lavenant criterion: 397

• law of large numbers: 176, 245

• law over measures: 245

• layer normalization: 382, 388

• layer-cake formula: 32

• layerwise matching: 359

• least squares

  • Radon reconstruction: 247

• least-square

  • inversion: 295

  • velocity: 295

• Lebesgue decomposition: 112, 203

• Lebesgue measure: 13, 17, 18, 20, 22, 57, 124

• Legendre transform: 87, 115, 244, 380

• Legendre-Fenchel transform: 143, 144

• length metric: 217

• length space: 300

• Lévy process: 354

• linear

  • attention: 199

  • convergence: 122, 158, 161, 166, 168, 206, 333

  • form: 12, 13, 49, 347

  • functional: 319

  • linearization: 25, 283

  • objective: 51

  • operator: 46

  • OT: 226–230

  • potential: 319

  • programming: 45, 46, 49, 53, 119, 236, 259

  • tilt: 155

• linear-time Sinkhorn: 193

• linearized Wasserstein metric: 341

• Liouville equation: 361

• Lipschitz

  • constant: 103

  • function: 20, 88, 103, 105, 107, 109

  • map: 74

  • stability: 85

  • test function: 72

• Lloyd

  • algorithm: 97, 330

  • flow: 98, 99

• local action: 300, 349

• local distance distribution: 278

• local matching indicator: 2

• local metric slope: 318

• local profile: 278

• local tangent action: 300

• localization: 24

• Loewner order: 270, 305

• log-concave

  • density: 379

  • measure: 379

• log-concave generator: 382

• log-concavity: 339

• log-domain Sinkhorn: 118

• log-scaling: 161

• log-sum-exp: 93, 125, 132, 134, 142, 149, 255

• logarithmic error: 195

• logarithmic kernel approximation: 195

• logarithmic mean: 307, 309, 310

• logarithmic Sobolev inequality: 339, 340, 342–345

• look-ahead point: 360

• Lorentz cone: 40

• loss factor: 163

• lossy Sinkhorn clearing: 163, 164

• low-rank

  • coupling: 199, 256

  • optimal transport: 255, 256, 258

• lower semicontinuity: 57, 112, 113, 151, 201, 202, 245, 267, 272, 308, 312

• lower semicontinuous envelope: 69, 70

• lower semicontinuous relaxation: 302

• lower-dimensional set: 21

• Lyapunov equation: 389

M¶

• M-function: 155, 161, 164

• M-matrix: 162

• M0-function: 161, 162

• Ma-Trudinger-Wang condition: 26, 28, 29

• Maas distance: 309

• manifold support: 180, 182

• many-token regime: 76

• map estimation: 189

• map extrapolation: 187

• marginal

  • constraint: 44, 47, 53, 54, 57, 106, 119, 121, 124, 126, 138, 145, 148, 155, 156, 163, 170, 204, 267, 269, 285

  • divergence: 200–202

  • penalty: 139, 141, 201, 202

  • relaxation: 200–202

  • residual: 118, 165

• marginal-dependent problem: 139, 140

• market clearing: 161

• Markov

  • averaging: 384

• Markov chain: 309, 355

• Markov kernel: 34, 75, 167, 278

• Markov semigroup: 353, 355

• Markov-chain Wasserstein geometry: 353, 355

• martingale coupling: 268, 269

• martingale OT: 268–271

• masking: 382

• mass

  • balance: 2, 94

  • conservation: 43, 120, 200

  • creation: 200

  • free: 97

  • splitting: 28, 43, 70, 75

  • variation: 201

• mass flux: 314

• matching

  • algorithm: 7

  • non-crossing: 6, 7

  • optimal: 1, 62

  • partial: 9

  • perfect: 7, 9

  • uniform: 48, 51, 52, 58, 62

• matrix

  • cone: 274

  • incidence: 106

  • permutation: 46, 48, 50, 51, 278

  • perspective: 304

  • polar decomposition: 22

  • scaling: 117, 119, 120, 152, 155

  • square root: 170

  • subproblem: 275

• matrix-valued measure: 272, 274, 275

• matrix-vector iteration: 119

• matrix-vector product: 194

• max-flow min-cut theorem: 259

• maximal correlation: 380

• maximal density constraint: 323

• maximum mean discrepancy: 108–111, 115, 116, 180, 184, 185, 228, 321, 326, 327, 342, 362, 374, 390, 392

• McCann

  • displacement convexity: 327

  • interpolation: 24, 27, 33, 34, 68, 233, 237, 299, 325

• McKean-Vlasov equation: 321, 322

• McKean-Vlasov limit: 321, 322

• mean

  • Wasserstein decomposition: 65

• mean field game: 313

• mean shift: 383

  • blurring: 383

  • iteration: 383

  • vector: 383

• mean-field limit: 322, 337, 382

• mean-field momentum: 361

• mean-field neural network: 316, 337, 345, 359

• mean-preserving splitting: 268

• mean-shift PDE: 383

• measurable function: 12, 107

• measurable selection: 85, 135, 136, 234, 278

• measure

  • atomless: 17, 69

  • evolution: 293, 294, 319

  • isomorphism: 17

• measure convolution: 72

• measure-function duality: 81

• measure-preserving

  • isometry: 279, 280

  • map: 19, 22

• measure-to-measure map: 74

• measure-valued Radon transform: 213

• mechanical energy: 361

• Mémoli profile: 278, 282, 283

• mesh size: 142

• message passing: 254

• metric

  • derivative: 215, 318, 331

  • entropy: 180, 182

  • equivalence: 231

  • learning: 77, 261, 262

  • length: 331

  • length bound: 331

  • pull-back: 213

  • slope: 333, 335

  • space: 12

  • triangle inequality: 68

• metric-measure space: 272, 278–282, 337

• Min-SW: 221, 223

• minibatch: 373

• minibatch bias: 373

• minibatch noise: 355

• minimal velocity: 318

• minimax: 230, 231

• minimax lower bound: 182

• minimax optimality: 182

• minimizer

  • sparse: 46

• minimizing movement: 316, 319, 352, 397

• minimizing movement scheme: 316

• Minkowski inequality: 20, 64, 67, 76, 214, 279, 280

• Minkowski problem: 379

• mixed Hessian: 28

• MMD-GAN: 373

• mobility: 303, 350

  • concave: 303, 304, 350

  • convexity mechanism: 303

  • nonlocal: 307

• modulo isometry: 280

• modulus of continuity: 85

• moment

  • convergence: 73

• moment condition: 176

• moment energy: 336

• moment functional: 390, 392

• moment matching: 389, 397

• moment measure: 379, 380, 382

• momentum: 272, 273, 296–298

• momentum formulation: 296, 300, 301, 314

• momentum variable: 300

• Monge

  • distance: 19, 23, 24, 69, 70

  • geodesic: 23, 326

  • interpolation: 23, 32

  • map: 11, 17, 19, 250

  • problem: 1, 11, 16–18, 20, 23, 30, 35, 36, 43, 55, 57, 67, 69, 295, 296, 325

• Monge gap: 58

  • convexity: 58

  • isotonicity: 59

  • kernel regression: 60

  • one-dimensional formula: 59

• Monge–Ampère

  • complex equation: 173

  • equation: 24, 25, 172, 227

  • geometric equation: 172

  • parabolic equation: 172, 175

  • solver: 25

• Monge-Kantorovich equivalence: 57, 250

• monotone

  • alignment: 289

  • convergence: 158

  • field: 21

  • fixed point: 158

  • matching: 1, 5, 221

  • rearrangement: 3, 4, 21, 30, 31, 34, 35, 60, 63

  • spectral gauge: 230, 304

  • transport map: 375

• Moore-Penrose pseudo-inverse: 247

• MTW condition: 28, 29

• multi-marginal

  • cost: 252

  • formulation: 252

  • Monge map: 249, 250

  • OT: 236, 237, 249, 252, 253

  • Sinkhorn: 252

• multi-omics: 284

• multi-species

  • flow: 324

  • gradient flow: 324, 325

• multi-valued subdifferential: 21

• multinomial distribution: 192

• multiplicative scaling: 140

• multiscale coupling: 180

• Muon algorithm: 230, 232, 305, 350, 352, 353

• Muon polar factor: 352

• mutual information: 126

N¶

• Nash equilibrium: 313

• natural gradient: 374

• natural language processing: 66

• near-linear Sinkhorn: 195

• nearest neighbor: 224

• negative Shannon entropy: 155–157, 319

• negative Sobolev norm: 342

• Nesterov acceleration: 360

• network simplex: 106

• neural

  • dual solver: 134, 135

  • Sinkhorn solver: 134

• neuron: 344–347, 358, 394

• neuron law: 345, 393, 394

• Newton dynamics: 362

• Newton step: 53, 119

• Newtonian dynamics: 362

• Newton–Schulz iteration: 353

• node feature: 284

• noising path: 372

• noising schedule: 370

• noisy gradient descent: 347

• non-convex domain: 25

• non-convexity: 330

• non-Gaussian limit: 193

• non-variational scaling: 161, 163

• nonconvex

  • optimization: 237

• nonlinear flattening: 245

• nonlocal

  • action: 306

  • continuity equation: 307

  • gradient: 307

• nonlocal Wasserstein

  • distance: 306, 307, 353

  • flow: 353

• nonnegative factorization: 196

• nonnegative measure: 147, 148, 200, 204

• nonnegative rank: 256

• nonnegative transport matrix: 51

• norm

  • flat: 107, 108

  • RKHS: 110

  • Wasserstein-1: 103

• normal approximation: 177

• normal cone: 323

• normalized dynamics: 350

• normalized gradient flow: 349, 350

• normalized SGD: 351

• north-west corner

  • plan: 47, 48

  • rule: 47, 53

• Nystrom approximation: 195

O¶

• ODE limit: 360

• one-dimensional

  • Monge map: 30

  • projection: 18, 200, 213, 215, 393

  • transport: 22, 29, 32, 47, 63, 326

• one-form: 163

• one-step generative model: 365, 373, 382

• Onsager operator: 300, 309, 310

• operator bound: 231

• operator gauge: 305, 351

• operator norm: 13

• operator scaling: 285, 287, 288

• operator-valued coupling: 288

• opinion dynamics: 384

• optimal coupling: 29–31, 46, 51, 52, 56, 57, 61, 62, 67, 68, 80, 81, 86, 97, 238, 243, 252, 261, 263, 266, 280, 281, 296, 325

• optimal face: 192

• optimal plan: 20, 28, 44, 46, 48, 62, 83, 86, 121, 145, 266, 280, 325

• optimal transport map

  • consistency: 187

  • estimation: 187

• optimality

  • first-order: 94, 156, 157, 287

• orbit space: 224

• order interval: 161

• order-constrained coupling: 269

• order-preserving map: 158–160

• Ornstein-Uhlenbeck flow: 341, 345

• Ornstein-Uhlenbeck process: 363, 370, 372

• OT map estimation: 187

• OT on trees: 33

• OU bridge: 372

• outside option: 163

• overshooting bridge: 372

P¶

• pair space: 307

• pair-space action: 306

• pairwise distance: 279

• parabolic optimal transport: 172

• parameter space: 373

• partial matching: 8

• partial transport: 201, 204, 210, 212

  • active region: 213

  • fixed mass: 212

  • TV penalization: 211

• particle method: 75

• particle ODE: 293, 322, 344

• particle splitting: 306

• particle Wasserstein descent: 360

• path action: 300

• path entropy: 291

• path measure: 296, 305

• path metric: 217

• path space: 12, 69, 135–139

• path-space formulation: 117, 135–138, 299

• path-space problem: 136

• path-space transport: 135, 138

• Pearson divergence: 114

• penalized minimization oracle: 349

• perfect matching: 50

• permutation matrix: 48

• Perron-Frobenius theorem: 160, 166, 167

• perspective function: 296–299, 301

• perspective recession: 272

• perspective transform: 203, 301

• perturbation response: 374

• phase space: 361

• phase-space density: 364

• phase-space equation: 363

• phase-space measure: 361

• phi-divergence: 107, 111–115, 123, 145, 146, 148, 329, 341

• phi-divergence regularized OT: 145–149, 159, 160

• piecewise affine map: 190

• Pinsker inequality: 116, 164

• PL convergence: 333

• plan

  • interpolation: 68

  • lifted: 221, 223, 233

  • optimal: 68

  • sparse: 44, 46

  • transport: 43, 51, 61, 68, 81, 233, 263, 272

• plug-in estimator: 176, 183, 184

• PMO: 349

• Poincaré disk: 27

• Poincaré inequality: 341, 342

• Poisson equation: 142

• polar factor: 353

• polar factorization: 22, 23

• polar formula: 231

• polar set: 230–232, 351

• Polish space: 12, 17, 65, 78, 267, 276

• Polyak momentum: 360

• Polyak-Lojasiewicz inequality: 333

• population law: 313

• porous medium equation: 320, 350

• Portmanteau theorem: 57

• positive

  • autocorrelation: 198

  • cone: 272, 273, 287

  • feature map: 196

  • features: 196

  • linear map: 168

  • matrix-valued measure: 274

  • measure: 356

  • operator: 272, 285

  • random features: 193, 197–199

  • semidefinite cone: 40

  • semidefinite kernel: 193, 194

  • semidefinite matrix: 36–39, 230, 272, 273, 275, 286

  • sketch: 198

  • vector-valued measure: 272

• potential energy: 336

• potential game: 313

• power diagram: 93, 94

• power divergence: 328

• power-law jump kernel: 354

• Prékopa

  • inequality: 381

  • Leindler inequality: 338

• pressure field: 324, 325

• primal-dual

  • method: 89

• principal component analysis: 228–230

• principal value: 354

• probabilistic coupling: 14, 16

• probability integral transform: 29

• probability measure: 11, 12, 14, 17, 20, 32, 35, 54, 57, 63, 71, 78, 82, 108–110, 114, 148, 213, 217, 226, 236, 249, 276, 279, 282, 295, 296, 321, 346

• probability path: 366

• probability simplex: 11, 148, 167

• Procrustes

  • alternating minimization: 226

  • orthogonal: 224

  • problem: 224

  • rigid update: 225

• product

  • coupling: 44, 54, 55, 57, 65, 69, 150, 258, 370

  • measure: 121, 126, 149, 253

• product Wasserstein metric: 325

• profile lower bound: 225, 278, 281

• projected formulation: 230

• projected measure: 213

• projection direction: 213

• projection-free attention: 383

• projection-robust Wasserstein: 220

• projective

  • cone: 166

  • diameter: 166, 167

  • geometry: 160

  • metric: 166

• Prokhorov theorem: 57, 78

• propagation of chaos: 322

• proximal

  • operator: 298

  • splitting: 298

• pullback: 15

• push-forward: 11, 14–18, 30, 37, 38, 43, 294, 317, 365, 367

  • measure: 76

  • stability: 75

• Pythagorean identity: 155, 156, 165

Q¶

• quadratic

  • action: 300

  • assignment problem: 278

  • barycenter: 238, 252

  • closure: 170

  • confinement: 363

  • cost: 3, 18, 20, 21, 23, 24, 26, 28, 31, 35, 38, 81, 83, 86, 93, 94, 97, 134, 170, 171, 200, 230, 236, 266, 296

  • energy: 362

  • perspective: 296, 301

  • potential: 21, 37, 38, 170, 390, 392

  • scaling: 153

  • tangent norm: 300

• quantile: 29

  • barycenter: 239

  • coupling: 60

  • formula: 277

  • function: 4, 29–32, 213, 217, 226, 227, 229, 239, 247

  • map: 17, 30, 31, 273

  • process: 102, 189, 190

  • push-forward: 30

  • quantization: 101

• quantization

  • energy: 97, 99

  • fixed weights: 100

  • high-resolution limit: 99

  • mean-field limit: 99

  • non-uniform weights: 96

  • optimal: 96, 330

  • rate: 96

• quantum

  • Bregman projection: 287

  • coupling: 285

  • Kantorovich duality: 285

  • OT: 272, 285, 286

  • OT duality: 286

  • Sinkhorn: 287

• quotient

  • metric property: 224

  • Wasserstein distance: 223, 224, 282

R¶

• Rademacher bound: 187

• radial coupling: 22

• radial generator: 39

• radial measure: 21

• Radon barycenter: 247

• Radon measure: 12, 13, 61, 82

• Radon pseudo-inverse: 247

• Radon range condition: 214, 247

• Radon sinogram: 247

• Radon structure: 215

• Radon transform: 213, 247

• Radon Wasserstein barycenter: 247

• Radon Wasserstein distance: 214

• Radon-Nikodym derivative: 367

• ramp filter: 248

• random features: 193, 194

• random measure: 245

• random probability measure: 245, 276

• random variable: 14, 16, 57, 71, 177, 390

• random walk: 310

• rank: 46

• rank constraint: 230

• rank-deficient covariance: 40

• rate

  • parametric: 176, 184, 189, 191

  • sublinear: 95, 163

• rational weights: 51, 52, 276

• raw second moment: 38

• reaction term: 356

• reaction–transport action: 311

• reaction-transport geometry: 311

• recession convention: 202, 203, 272, 297

• rectifiable set: 21

• reference

  • distribution: 365

  • measure: 13, 22, 115, 123, 124, 146, 148, 202, 227, 228, 242, 365

• reflow: 366

• regression loss: 365

• regular conditional distribution: 33

• regular vector field: 306

• regularity theory: 24, 28, 29

• regularization

  • Monge gap: 59

• regularization bias: 176, 185, 191, 193

• reinforcement learning: 116

• relative

  • entropy: 113, 117, 123, 126, 137–139, 164, 171, 321, 328, 329, 335, 337, 353, 357, 392, 394

  • error: 195

  • Fisher information: 339, 342, 394

• reproducing kernel Hilbert space: 109, 110

• rescaled convolution: 72

• residual attention: 383

• residual block: 358

• residual network: 359

• ResNet: 357

  • continuous-depth: 358

  • finite-depth: 358

  • finite-depth mean-field: 358

  • finite-width: 358

  • infinite-depth: 357, 359

  • particle discretization: 359

• resolvent: 142

• reverse formulation: 203

• reverse KL divergence: 114

• reversible jump kernel: 306

• reversible Markov chain: 309

• Riccati equation: 174

• Ricci curvature: 337

• Riemann sum: 71

• Riemannian manifold: 26, 337

• Riemannian metric: 300

• Riemannian tensor: 300

• Riesz potential: 109

• Riesz-Markov-Kakutani theorem: 12, 13

• rigid motion: 224, 225

• rigid registration: 224

• rigid update: 226

• RKHS: 108–111, 116, 184, 185, 228

  • vector field: 305

  • velocity: 305

• robust

  • envelope: 78

  • finance: 268

  • representation: 231

  • Wasserstein: 229, 232

  • Wasserstein distance: 229, 232

• rotational component: 31, 371

• row constraint: 162

• row normalization: 121, 159

• running-intersection property: 254

S¶

• Sakoe–Chiba band: 289

• sample complexity: 97, 176, 180, 246, 280

• sampling: 74

• scalar bridge: 372

• scaling

  • algorithm: 117, 253

  • column: 121, 155, 157

  • form: 119, 120, 131, 157, 243, 288

  • potential: 122

  • row: 121, 155, 157

  • vectors: 158, 166

• Schatten norm: 230, 231

• Schoenberg theorem: 198

• Schrödinger

  • bridge: 135–138

  • interpolation: 137

  • problem: 117, 130, 135–138, 151

  • system: 158

• score function: 319, 369, 370, 372, 377

• score-based generative modeling: 365

• score-SDE: 365

• second boundary condition: 24, 25

• second moment: 38, 392, 393

• second-moment quotient: 38

• second-order limit: 193

• sectional curvature: 29

• self Sinkhorn: 151

• self-concordance: 119

• self-corrected field: 378

• self-distance: 191

• semi-coupling: 203

• semi-discrete

  • dual: 94

  • Monge map: 18

  • OT: 18, 55, 85, 88, 93–96, 133

• semi-dual: 88, 93, 243

  • discrete: 88

• semi-relaxed problem: 85

• semidefinite program: 285

• semidefinite programming: 184

• separable space: 12

• separator: 255

• Shampoo optimizer: 351

• Shannon entropy: 112, 113, 117, 286, 309, 320, 325, 329

• Shannon-Boltzmann entropy: 117

• shape space: 223

• shearing component: 31, 371

• signed

  • measure: 13, 32, 72, 103, 109, 110, 316, 317, 320

  • negative part: 105

  • positive part: 105

• simplex: 310

• simplex network: 53

• single-cell genomics: 66, 67, 74, 201, 284, 318, 374

• Sinkhorn

  • algorithm: 87, 93, 106, 117, 119, 120, 122, 132, 168, 170

  • barycenter: 229, 242, 313

  • closed-form coupling: 127

  • continuous dual iteration: 134

  • continuous flow: 172, 173, 175

  • continuous iteration: 126

  • continuous limit: 173

  • convergence: 155, 158, 163, 166

  • coupling: 138

  • divergence: 117, 149–151, 171, 172, 180, 185, 193, 322, 374, 375, 393, 394

  • dual: 131, 146, 164

  • fixed point: 152, 155

  • full-cycle map: 168

  • GAN: 373

  • half-step: 122, 123, 156, 157, 165

  • holomorphic continuation: 152–154

  • iteration: 122, 150, 157, 170, 243, 255, 266, 283

  • kernel: 194

  • linear-time: 193

  • low-rank: 256, 257

  • potential: 173

  • scaling: 117, 119, 121, 139, 145, 149, 156, 158, 161, 162, 169, 206, 288, 377

  • sketching: 193

  • statistical convergence: 176

  • update: 287

• slack variable: 8

• Slater condition: 285, 286

• sliced barycenter: 247

• sliced length metric: 217

• sliced tangent norm: 215

• sliced Wasserstein

  • barycenter: 247

  • CDF estimator: 32

  • compact comparison: 215

  • directional aggregation: 219

  • distance: 189, 213, 214, 219, 220, 228, 393

  • flow: 375

  • Gaussian: 218

  • length: 217

  • max: 219

  • Monte Carlo directions: 190

  • reverse bound: 215

  • sample complexity: 189

  • tangent norm: 215

• slope

  • metric: 318

  • Wasserstein: 333, 334

• small-jump limit: 309

• smooth density: 184

• smooth OT estimation: 184

• smoothed plug-in estimator: 183

• smoothing estimator: 183

• smoothness: 183

• Sobolev norm: 151

• soft

  • barycentric projection: 187

  • c-transform: 131–134, 143, 146, 149, 158–160, 165, 170, 173, 187, 206, 262

  • conjugation: 187

  • minimum: 132

  • transform: 117, 132–135, 146, 152, 155, 158, 160, 170, 172, 206

• soft-DTW: 290

  • expected alignment: 292

• softmax attention: 199

• source term: 311

• spanning tree: 53, 106

• sparse linear program: 106

• sparsity: 46

• spectral

  • dynamics: 232

  • gap: 342, 343

  • gauge: 200, 230, 233, 351

  • steepest descent: 350

  • tangent action: 304, 351

• spectral Wasserstein

  • distance: 229, 230, 232, 304, 305

  • dynamic distance: 304

  • flow: 350, 352, 353

• spherical average: 213

• spherical cap: 190

• spherical geodesic: 27

• splitting obstruction: 18, 21, 25

• splitting set: 249, 250

• squared local action: 300

• stable process: 354

• static-dynamic equivalence: 305, 312

• stationarity: 87, 283, 286, 346–348

• stationary condition: 347

• stationary density: 346

• stationary potential: 173

• statistical bias: 193

• statistical optimal transport: 176, 246

• Stein force: 375

• Stein geometry: 305

• Stein method: 177

• Stein operator: 375

• Stein variational gradient descent: 305, 375, 377

• step-size normalization: 353

• Stiefel manifold: 219

• stochastic

  • control: 136

  • differential equation: 354

  • gradient: 95, 116, 367, 374

  • gradient descent: 355

  • interpolant: 365, 368

  • optimization: 95

  • order: 269

  • particle: 321

• Strang splitting: 287

• Strassen theorem: 269

• strict

  • convexity: 24, 28, 118, 123, 125, 134, 157

  • positivity: 151, 161

• strictly correlated electrons: 251

• strong law of large numbers: 176

• strong MTW condition: 29

• strong upper gradient: 318

• structured data: 278

• subdifferential: 20, 21, 25, 57, 58, 159

• subgaussian measure: 185

• subgradient: 262

• subgradient inequality: 347

• sublinear convergence: 335

• subsolution: 161

• subspace

  • projection: 220

  • robust Wasserstein: 232

• subspace-sliced Wasserstein: 219

• substitutability: 161

• sum of squares: 184

• sum-of-squares relaxation: 184

• sum-product algorithm: 254

• supergradient: 262

• superposition principle: 296, 305

• supersolution: 161

• support: 12, 14, 28, 51, 57, 81, 83, 104

• SVGD: 375

• symmetric plan: 251

• synthetic Ricci curvature: 337

T¶

• tail index: 355

• Talagrand inequality: 343, 345

• tangent action: 300, 349

• tangent coordinate: 246

• tangent norm: 300

• teacher distribution: 320, 321

• teacher kernel mean: 321

• temporal noise: 365

• tensor product coupling: 54

• Thompson metric: 206

• tightness: 10, 12, 32, 58, 65, 215, 276

• time change: 372

• time reparametrization: 372

• time series

  • alignment: 289

• time-dependent vector field: 293, 365

• token limit: 382

• token measure: 76, 365, 382, 383

• Toland duality: 381

• tomography: 248

• topical map: 160, 161

• topology

  • strong: 72, 73, 113

  • weak: 57, 70, 72, 74, 82, 113, 114

• total unimodularity: 52

• total variation: 11, 13, 63, 71, 72, 74, 107, 108, 113, 114, 200, 201, 310

• totally positive kernel: 196

• trace

  • entropy: 286

  • gauge: 230, 233

  • norm: 231

  • partial: 285, 286, 288

  • partial constraint: 285

• trace-class operator: 288

• trajectory inference: 67, 74

• transfer learning: 57

• transformer: 74, 199, 382, 383, 388, 396

• translation-invariant kernel: 109, 111

• transport

  • cost: 278

  • coupling: 289

  • equation: 388

  • inequality: 337

  • map: 4, 17, 23, 33, 43, 81, 266, 367, 371

  • metric: 353

  • PDE: 388

  • representation: 75, 76

• transport-entropy inequality: 343

• transportation polytope: 43, 44, 47, 52–54, 118, 119, 124, 156, 165, 259, 278

• transportation simplex: 47, 53

• tree

  • decomposition: 254

  • earth mover distance: 33

  • embedding: 33

  • geodesic: 33

  • interpolation: 34

  • metric: 33

  • transport: 33

• tree-sliced Wasserstein: 33

• tree-Wasserstein distance: 33

• treewidth: 254

• triangle inequality: 6, 19, 20, 24, 41, 64, 65, 67, 103, 166, 182, 185, 205, 214, 231, 279–281, 308

• triangular map: 33

• triangular rearrangement: 33–35

• trivial coupling: 366

• truncated marginal: 212

• Tweedie identity: 369, 372

• twist condition: 26, 28, 249, 250

• twist on splitting sets: 249, 250

• two-layer neural network: 345–347, 353, 393

• two-sample statistic: 192

• two-sample testing: 111

U¶

• ultrafast diffusion: 99

• unbalanced

  • barycenter: 313

  • gradient flow: 356

  • interpolation: 312

  • JKO step: 356

  • OT: 124, 139, 200–202, 204, 205, 263, 311, 312

  • OT dual: 202

  • Sinkhorn: 141

  • Sinkhorn convergence: 206

  • tangent space: 311

• uniform integrability: 73, 177

• upper hemisphere: 26

• upper matrix: 259

V¶

• Varadhan formula: 141

• variable elimination: 254

• variance: 190

• variation seminorm: 160, 165, 166, 384

• variational

  • mean field game: 313

• variational dual formula: 115

• vector quantile: 226, 227

• vector-valued measure: 272, 273, 324

• velocity covariance: 304

• velocity field: 293, 294, 299, 317, 365, 382, 389, 395

• Vlasov equation: 362

• von Neumann entropy: 286

• Voronoi cell: 18, 93, 97, 100

W¶

• Waddington landscape: 318

• Waddington-OT: 66

• warping path: 289

• Wasserstein

  • barycenter: 139, 228, 236, 237, 239, 243, 245, 246, 252, 253

    • consistency: 245

    • population: 245

  • convergence: 176

  • coordinate: 227

  • distance: 32, 43, 63–65, 67, 68, 70, 72–74, 77, 114, 115, 177, 200, 205, 213, 214, 229, 230, 232, 276–279, 282, 295, 299, 303, 332, 380, 396

  • empirical rate: 180

  • flow: 332

  • flow of discrepancy: 374

  • formula: 31

  • GAN: 373

  • Gaussian geometry: 394

  • geodesic: 23, 68, 229, 237, 281

  • gradient: 75, 295, 316–318, 321, 323, 332, 345, 347, 360, 365, 375, 377, 378, 388–390, 395, 397

  • gradient flow: 23, 100, 147, 295, 316–318, 320, 321, 323, 332, 335, 347, 357, 359, 365, 374, 375, 388, 397

  • infinity distance: 78

  • infinity robustness: 78

  • Kurdyka–Łojasiewicz inequality: 335

  • Lipschitz functional: 75

  • Lipschitz map: 75, 76

  • minimax rate: 182

  • over Wasserstein: 245, 276, 278, 283

  • p-action: 303

  • p-distance: 26

  • PL inequality: 333–337, 339, 341, 344

  • Procrustes: 223–226, 282

  • space: 74, 276, 277, 319, 395

  • space CLT: 246

  • squared distance: 330

  • topology: 73, 177, 276

  • training: 345

  • W1 non-uniqueness: 4, 19, 56

• Wasserstein–Fisher–Rao: 312, 356, 357

• wavelet estimator: 184

• weak

  • C-transform: 267

  • convergence: 12, 57, 70–73, 108–112, 114, 176, 177, 214, 215, 231, 276

  • cost: 267, 268

  • Kantorovich duality: 267

  • limit: 18, 35, 71

  • map: 25

  • optimal transport: 266, 267

  • transport: 267, 268

• weight clipping: 116

• weighted

  • Laplacian: 343

  • Poisson equation: 25, 295

  • sweep: 48

• wide stencil: 25

• word mover’s distance: 66

Z¶

• Z-function: 161

• Z-matrix: 162

• zero mass: 35, 94, 110, 119

• zero mean: 295