Index

A¶

• absolute continuity: 19, 20, 25, 58, 116, 130, 141, 255

• absolutely continuous path: 316, 319

• action density: 320

• active mass: 206, 216, 218

• additively homogeneous map: 163

• admissible potential: 82, 84, 86

• ADMM: 318

• adversarial

  • loss: 119

  • perturbation: 74

• affine

  • closure: 410, 411, 415, 417

  • constraint: 160, 272

  • gradient: 410, 416

  • map: 36, 84, 393

  • marginal constraint: 159, 161

• affine-difference cost: 59

• affine-invariant metric: 39

• Alexandrov solution: 24

• algorithm

  • Birkhoff–von Neumann decomposition: 49

• algorithmic convergence: 159

• alternating

  • optimization: 88

  • projection: 125, 159, 161, 164

  • tree: 8, 10

• alternating minimization: 301

• antisymmetric flux: 375

• antisymmetric velocity: 327

• assignment problem: 1, 2, 7, 8, 41, 46, 48, 50, 51, 92, 283

• Assouad lemma: 187

• atom-preserving map: 78

• atomic measure: 326

• atomless measure: 17, 28

• attention: 397, 404, 410, 417, 418

  • causal: 79

  • Gaussian kernel: 405

  • linear: 204

  • Lipschitz stability: 80

  • map: 79

  • masking: 79

  • non-causal: 79

  • operator: 80, 404

  • self-attention: 79

  • single-head: 404

• auction algorithm: 7, 92, 94, 167

  • coordinate ascent: 92

• augmented Lagrangian: 335

B¶

• Baker-Campbell-Hausdorff formula: 308

• Bakry–Émery criterion: 361

• balance equation: 331, 332, 378

• balanced

  • transport between relaxed marginals: 205

• balanced interpolation: 332

• balanced OT: 143

• Banach dual: 12, 13

• Banach fixed-point theorem: 212

• Banach space: 12

• Banach-space embedding: 30

• barrier

  • entropy: 123

  • logarithmic: 53, 123, 126

  • LP: 123

  • parameter: 53

• barycenter: 241, 401

  • fixed support: 241

  • free support: 242

  • law of large numbers: 250

  • mean: 243

  • support: 242, 243, 258

• barycentric

  • attention: 405

  • constraint: 273

  • mixture: 250, 281, 282

  • projection: 43, 192, 194, 234, 271–273, 276

  • transport: 273

• basic feasible solution: 45, 258

• Beckmann

  • flow: 109

  • formulation: 91, 108, 109, 323

  • problem on graphs: 109, 110

• belief propagation: 260

• Benamou–Brenier: 139, 141, 277–280, 313, 315, 316, 319, 321, 324, 332

  • distance: 321

  • dual: 317

  • formula: 313, 315

  • formulation: 139, 315, 319

  • kernelized: 325, 397

  • viscous: 141

• Bernstein–Schnabl operator: 76

• Berry-Esseen

  • bound: 182

  • theorem: 71, 182

• Bessel function: 113

• bias: 195

• bias-variance tradeoff: 181, 188

• biconvex

  • relaxation: 299

• bilevel problem: 266

• bilinear cost: 269

• Birkhoff

  • contraction theorem: 167, 169, 170, 406

  • polytope: 46

  • von Neumann decomposition: 48

  • von Neumann theorem: 46–48, 58

• birth-death dynamics: 377, 379

• bistochastic matrix: 46, 48

• block-coordinate

  • ascent: 249

• block-coordinate descent: 262

• block-coordinate method: 231

• block-stationary point: 262

• Bobkov-Ledoux formula: 30

• Bochner spectral measure: 115

• Bochner theorem: 198

• Boltzmann entropy: 121

• Boltzmann equation: 384

• book-shifting: 4, 19, 55

• Borel measure: 11, 12, 20

• Borel set: 11, 12, 14, 23

• bounded confidence: 405

• box constraint: 264

• Bregman

  • divergence: 60, 115, 118, 127, 150, 159, 160

  • iteration: 306

  • penalty: 160

  • projection: 123, 152, 159–162, 165, 264, 306, 307

  • regularization: 150–152

  • three-point formula: 161

• Brenier

  • factor: 21

  • map: 20, 22, 23, 29, 33, 57, 82, 87, 88, 130, 178, 193, 232–234, 243, 347, 350, 351, 359, 360, 393, 404, 413

  • perturbation: 222

  • potential stability: 194

  • source hypothesis: 20

  • theorem: 19–21, 24, 25, 34, 36, 57, 58, 62, 87, 98, 232, 255

• Brenier theorem: 299

• Brenier–Strassen projection: 275

• bridge

  • Brownian: 139, 141–143

  • linear: 394

  • OU: 392, 395

  • overshooting: 394, 395

  • speed: 394

  • variance-preserving: 392, 394, 395

• Brownian bridge: 105, 139, 142

• Brunn-Minkowski inequality: 359, 360

• Bures

  • covariance: 36

  • metric: 34, 36–39, 252

  • Procrustes representation: 38

  • smoothed term: 177

  • soft Procrustes limit: 215

  • Wasserstein geometry: 34, 37, 177, 245, 282, 410, 413

• Burg entropy: 118

• Burgers equation: 148, 149

C¶

• c-concavity: 89, 138

• c-convexity: 27

• c-cyclical monotonicity: 61

• c-splitting set: 255

• c-transform: 82, 84–89, 91, 96, 106, 107, 129, 136

• Caffarelli regularity: 23, 24

• capacity-constrained OT: 263–265

• capacity-constrained OT feasibility: 264

• capacity-constrained Sinkhorn: 264

• capped transportation polytope: 264

• Carathéodory theorem: 48

• Catalan number: 7

• causality: 404

• CDF formula: 30

• cell mass: 96, 98

• centered fluctuation: 195

• centered measure: 66

• central limit theorem: 71, 181, 182, 251

  • optimal transport: 195, 196

  • quantitative: 71

• central path: 53

• centroidal Voronoi: 104

• centroidal Voronoi tessellation: 102

• chain graph: 259

• change of variables formula: 15, 24, 25, 39, 179, 244

• characteristic kernel: 342

• Choi matrix: 308

• Cholesky factorization: 146

• circle

  • transport: 5, 6

  • unfolded interval: 5, 6

• classical convexity: 369

• clustering: 405

• co-motion function: 256

• co-motion map: 256

• collapsed mixture: 250, 281, 282

• column normalization: 125

• commutator: 308

• compact sublevel set: 207

• compactness: 328

• complementary slackness: 7, 9, 82–85, 94, 271, 318

• complete graph: 259

• complete metric space: 12

• completely positive cone: 200

• completely positive kernel: 200

• completely positive matrix: 200

• complex

  • epsilon: 156

  • Gibbs kernel: 157

  • matrix scaling: 157

  • measure: 157

  • Sinkhorn coupling: 158

  • Sinkhorn iteration: 157

  • Sinkhorn scaling: 157

• complexity

  • Birkhoff–von Neumann decomposition: 49

• composition constraint: 347

• computational complexity: 167

• computer vision: 51

• concave

  • minimization: 290

• concave cost: 2

• concave envelope: 153

• concavity

  • discrete OT: 43

  • Kantorovich value: 56

• conditional

  • coupling: 239

  • expectation: 171, 274, 387–389, 391, 393

  • geodesic: 239

  • gradient flow: 381

  • independence: 261

  • Jensen inequality: 209

  • law: 32, 33, 141, 271–273, 319

  • mean: 272

  • neuron law: 380

  • optimal transport: 238, 379

  • probability: 12, 65, 281, 388

  • quantile: 232

  • transport: 238

  • variance: 273

  • Wasserstein distance: 238, 239, 379

  • Wasserstein flow: 379

• conditionally negative definite kernel: 113, 156, 290

• conditionally positive definite kernel: 113

• cone

  • formulation: 210

  • lifting: 209, 332

  • metric: 209, 332

• congestion: 263, 333, 344

• congruence normalization: 308

• conic lifting: 209

• conjugate moment measure: 404

• consensus: 405

  • Dobrushin contraction: 407

• constant-speed geodesic: 23, 67, 288, 347

• contact set: 27

• continuity equation: 141, 277, 279, 280, 313, 315–318, 331, 336, 339, 373, 387–389, 391, 396, 399, 400, 410

• continuous

  • bounded function: 70

  • coupling: 53, 54

  • depth: 381, 404

  • dual: 83, 136

  • path: 12

  • Sinkhorn dual: 150

• continuous DTW: 310

• continuous-depth neural network: 404

• contraction ratio: 171

• contractive Gaussian projection: 418

• controlled transport: 410

• convergence

  • in law: 31, 70, 156

  • in probability: 70

  • law: 69

  • mode: 70

• convergence-determining class: 181

• convex

  • action density: 321, 323

  • cone of functions: 59

  • conjugate: 87, 150, 164, 404

  • envelope: 88, 148

  • function: 20, 21, 25, 29, 34, 57, 58, 69, 74, 83, 87, 88, 119, 149, 150, 159, 341, 401

  • neural network: 138

  • order: 274–276

  • potential: 19–25, 61, 62, 85, 91, 139, 392, 401

  • potential generator: 404

  • regularizer: 149, 270

  • relaxation: 68

• convex envelope

  • Kantorovich cost: 57

• convex hull: 243, 407

• convexity

  • along geodesics: 368

  • classical: 368, 369

  • constraint: 25

  • discrete OT: 43

  • geodesic: 336, 347, 348, 350–352, 354, 356, 359

  • Kantorovich value: 56

  • perspective transform: 321

  • strong geodesic: 356

• convolution: 81

  • Wasserstein contraction: 81

• convolutional Sinkhorn: 146

• coordinate ascent: 92, 165

• copositive certificate: 202

• copositive matrix: 200

• correspondence: 43, 277, 284, 288, 304

• cost

  • assignment: 50

  • bilinear: 27

  • derivative: 266

  • learning: 266

  • matrix: 1, 42, 45, 46, 63, 83, 124, 126, 166, 242, 290

  • reduced: 8, 9, 52, 110

  • regularized optimal transport: 293

  • transport: 56

  • unbalanced: 209

• cost-sectional curvature: 27

• Coulomb cost: 255–257

• countable dense subset: 12, 281

• counterexample: 256

• coupling measure: 64

• covariance

  • commuting: 393

  • cone: 37

  • diagonal: 39

  • Loewner order: 276

  • matrix: 29, 36, 39, 418

  • normalization: 373

  • operator: 234

  • subblock: 415

  • term: 36, 37

• covering number: 184, 185

• Cramér condition: 183

• Cramer-von Mises distance: 31

• Cramer-Wold theorem: 220, 253

• created mass: 206

• cross-covariance: 414

• cross-curvature: 90

• cross-moment: 295

• crowd motion: 344, 346

• Csiszár divergence: 350

• cumulant

  • fourth: 183

• cumulative distribution function: 28–30

• cumulative function: 4, 28, 31, 32

• curse of dimensionality: 100, 101, 184, 197

• curve of maximal slope: 339, 370

• cut locus: 27, 28

• cycle graph: 259

• cyclic

  • Bregman projection: 160

  • co-motion plan: 256

  • monotonicity: 61, 62, 272

  • projection: 160

  • shift: 5

• cylindrical Gaussian: 251

D¶

• Dacorogna–Moser

  • construction: 314

  • inversion: 314

• damped iteration: 246

• damped second-order equation: 382

• Danskin theorem: 266

• data-processing inequality: 119

• DCA algorithm: 293

• de la Vallée–Poussin criterion: 182

• Delaunay graph: 110

• delta method: 196, 197

• demographic parity: 242

• denoising score matching: 387

• dense linear system: 123

• density

  • closure: 385

  • constraint: 334, 344, 346, 347, 351

  • estimation: 187, 188

  • formula: 15

  • matrix: 305, 306, 308

  • ratio: 12, 115, 116, 118–120, 149–153, 263

  • relative: 13

• density-functional theory: 255–257

• depth evolution: 404

• depth variable: 380

• derivative

  • horizontal: 338

  • vertical: 336

• destroyed mass: 206

• detailed balance: 376, 377

• deterministic interpolant: 387

• diagonal coupling: 274

• diameter

  • contraction: 407

• difference-of-convex programming: 293

• diffusion model: 78, 81, 387, 391, 394

  • Gaussian noising: 132

• diffusive map: 81

• dimension dependence: 186

• Dirac mass: 11, 14, 17, 29, 34, 41, 63, 68, 78, 243, 261, 336, 340, 347, 388

• Dirac measure: 255

• directional derivative: 266

• Dirichlet energy: 341

• discrepancy minimization: 395

• discrete

  • barycenter: 242, 257

  • c-transform: 86

  • coupling: 41

  • Laplacian: 146

  • measure: 11, 67, 70, 121

  • space: 73

  • target: 96

  • Wasserstein geometry: 329, 330, 377

• discretization error: 247, 388

• discriminator class: 111, 112, 120

• disintegration: 12, 32, 65, 141, 210, 238, 255, 271–273, 389, 396

• displacement convexity: 22, 26, 349, 359, 365, 368, 401, 402

• displacement covariance: 234, 236, 238

• displacement interpolation: 22, 23, 31, 32, 238, 351

• distance profile: 289

• distance residual: 288

• distortion: 283, 286, 288, 290, 303, 304

• distributional robustness: 73

• divergence-free field: 411

• Dobrushin coefficient: 406

• domain adaptation: 51

• Doob transform: 140

• doubly nonnegative cone: 200

• doubly nonnegative matrix: 200

• doubly positive kernel: 200

• doubly stochastic normalization: 410

• Douglas-Rachford algorithm: 318

• Douglas-Rachford splitting: 318, 319

• drifting

  • field: 399, 410

  • model: 397, 399, 400

• dual

  • alternate maximization: 152

  • ascent: 144

  • attainment: 84–86, 306

  • block maximization: 138

  • certificate: 7, 89

  • formula: 119

  • gap: 126, 159, 166, 270, 271

  • gradient: 98

  • Hessian: 172

  • norm: 111–114, 120

  • objective gap: 165

  • pairing: 12

  • potential: 82, 83, 86, 88, 96, 125, 126, 129, 138, 149, 151, 154–156, 165, 170, 177, 184, 192, 196, 266, 270, 306

  • price: 8

  • problem: 82, 86, 91, 109, 122, 135, 138, 249

  • rate: 165

  • Sinkhorn: 144

  • slack: 85

  • twist condition: 26

  • weight: 96–98

• duality

  • entropic OT: 137

  • Fenchel-Rockafellar: 149, 208, 272

  • strong: 109, 249, 305

  • weak: 10, 83, 272

• Dudley entropy: 192

• Dudley metric: 111, 112

• dyadic cube: 186

• dyadic partition: 185

• Dykstra algorithm: 162, 264

• dynamic

  • action: 316

  • distance: 322, 324

  • formulation: 315, 319, 322

  • optimal transport: 313

  • unbalanced OT: 331

  • value: 316

  • Wasserstein flow: 370

• dynamic programming: 309

• dynamic time warping: 309

  • algorithm: 310

  • numerics: 312

• dynamical optimal plan: 68, 358

• dynamical plan: 239

E¶

• earth mover’s distance: 51

• edge flux: 330

• edge imbalance: 31

• Edgeworth expansion: 183

• effective cost: 262

• effective dimension: 200

• effective rank: 199

• eikonal equation: 419

• electron density: 256

• elliptical law: 36

• elliptically contoured distribution: 36

• embedding

  • alignment: 230

• empirical

  • context: 404

  • distribution: 344

  • dual potential: 193

  • energy: 385

  • law: 73, 75, 181, 344

  • measure: 3, 11, 18, 44, 48, 70, 76, 104, 155, 181, 187, 190, 227, 265, 268, 287, 313, 342, 343, 374, 380, 382, 396, 404

  • measure over measures: 250

  • Monge map: 16

  • OT: 101, 185, 186

  • OT rate: 105, 185, 288

  • process: 192, 196

  • token measure: 79

  • Wasserstein metric: 382

• endpoint coupling: 140–143

• energy

  • decay: 354

  • dissipation: 353–355, 383

  • interaction: 154, 341–343, 348, 413

• energy distance: 113, 156, 342, 343, 349, 384

• energy-dissipation identity: 353

• energy-dissipation inequality: 340, 353

• entropic

  • barycenter: 244, 249

  • barycentric map: 194

  • bias: 126, 133, 153, 154

  • map estimation: 194

  • OT: 121–123, 132, 135, 154, 165, 176, 196, 267, 271, 301, 306, 399

  • path: 123

  • plan: 203

  • potential: 139, 154

  • regularization: 88, 89, 96, 121, 123, 154, 159, 184, 190, 196, 211, 258, 262, 264, 268, 299, 306

  • smoothing: 122, 165, 192, 248

• entropy: 402

  • closure: 385

  • decay: 361

  • function: 115–117, 149, 205

  • penalty: 121

  • score: 385

• envelope theorem: 122, 266, 267

• $$-scaling: 95

• equal marginal: 256

• equality constraint: 52, 109

• essential continuity: 401

• Euclidean group: 230

• Eulerian

  • description: 313

  • velocity: 314, 316, 317

• exact OT: 24, 184–186, 192

• explicit Euler scheme: 382, 404

• extreme minimizer: 47

• extreme point: 46–48, 52

F¶

• factored coupling: 261

• fairness: 242

• farthest-point sampling: 385

• fast diffusion equation: 372

• fast Legendre transform: 147, 148

• feasible coupling: 52, 55, 233, 288

• feature cost: 302

• feature sketch: 197

• feature term: 303

• Fenchel duality: 144, 149, 150, 208, 272

• Fenchel inequality: 139

• Fenchel-Rockafellar duality: 317, 318

• Fenchel-Young loss: 268

• FFT: 148

• fiber: 239

• fiberwise optimal plan: 239

• fiberwise transport: 238

• fill-in edge: 259

• filtered back-projection: 253

• finite difference scheme: 24, 386

• finite measure: 12, 111, 112, 207, 277

• finite metric length: 353

• finite space: 196

• finite-action component: 322, 326

• finite-dimensional linear program: 107, 109

• finite-sample bias: 396

• finite-state transport: 329

• first variation: 150, 266, 267, 307, 336, 338, 354, 367–369, 381, 399, 400, 410, 413, 415, 417

• Fisher information: 134, 141, 360, 362, 365, 413

• Fisher-Rao metric: 34, 35, 39, 40

• fixed-point

  • Gaussian barycenter: 246

• flattening map: 250

• flow

  • conservation: 52

  • continuous normalizing: 388

  • cross-covariance: 414

  • map: 313, 393

  • minimum-cost: 52, 110

  • momentum: 382

  • normalizing: 388

  • one-step: 396

  • parameter-domain: 395

  • phase-space: 383

  • probability ODE: 387, 388, 390–392, 394

  • rectified: 387, 388, 392

  • reverse: 395

  • second-order: 382

• flow matching: 315, 387–389, 391–393, 417

• flux: 277, 279, 314, 339

• Fokker–Planck

  • equation: 343–345, 361, 366, 372, 376, 378, 379, 390, 393, 394

  • fractional equation: 376

• Fokker–Planck particle closure: 399

• Fortet iteration: 164

• Fourier features: 197

• Fourier multiplier: 113

• Fourier random features: 198

• Fourier transform: 252

• Fourier-slice theorem: 253

• fractional diffusion: 376

• fractional heat equation: 376

• fractional Laplacian: 375, 376

• fractional PDE: 375

• Frechet inception distance: 115

• Frechet mean: 241, 243

• Frobenius inner product: 37

• Frobenius norm: 159, 236

• frozen surrogate: 400

• functional

  • derivative: 336

  • on Wasserstein space: 75

• functional inequality: 359, 360, 363

• fused Gromov-Wasserstein: 302, 303

G¶

• Gamma convergence: 132

• GAN: 119

• gauge fixing: 178

• Gaussian

  • barycenter: 245, 246

  • blurring: 132

  • closure: 179, 215, 397, 410, 415, 417

  • complex Sinkhorn: 158

  • convex order: 275

  • convolution: 147, 148

  • covariance: 177, 412

  • denoiser: 391

  • flow matching: 392, 393

  • kernel: 198

  • kernel attention: 405

  • kernel drifting: 400, 417

  • kernel rank: 200

  • manifold: 410–412, 415

  • marginals: 139, 176

  • mean: 412

  • measure: 18, 22, 34, 35, 245, 276, 362, 401, 410, 416, 417

  • mixture: 3, 97, 103, 125, 126, 129, 153, 206, 244, 263, 265, 282, 392, 394

  • one-dimensional: 415

  • optimal map: 29

  • preservation: 410, 411, 417, 418

  • preserving flow: 417

  • projection: 415, 418

  • push-forward: 35

  • scale mixture: 202

  • Sinkhorn: 176, 177, 414

  • Sinkhorn divergence: 177

  • sliced Wasserstein: 224

  • submanifold: 410, 416

  • unbalanced: 213

• Gaussian Hellinger-Kantorovich: 213

• Gaussian measure

  • Gromov–Wasserstein: 295

• Gelbrich inequality: 418

• Gelbrich theorem: 418

• generalized

  • inverse: 17, 28

  • Wasserstein: 205

• generalized Gaussian kernel: 201, 202

• generalized quantile: 32

• generalized Sinkhorn algorithm: 143, 145

• generalized Wasserstein flow: 370

• generative model: 111, 225, 313, 352, 387, 395, 404

• generator manifold: 396

• geodesic: 327

  • convexity: 347, 351, 354

  • distance: 25, 32, 146, 320

  • semiconcavity: 352

  • space: 67, 358

• geodesics in heat: 145, 146

• Gibbs

  • coupling: 306, 307

  • formula: 306

  • kernel: 125–127, 130, 144, 156, 160, 162, 171, 199

    • push-forward: 131

    • sketching: 202

  • path law: 311

  • reference: 160

• Girsanov theorem: 140

• global

  • minimizer: 368, 369

  • optimality: 369

• global invariance: 230

• gluing lemma: 63–66, 211, 286

• gradient

  • field: 20

  • flow: 22, 284, 288, 313, 315, 336, 340–342, 344, 347, 353, 354, 367, 368, 387, 397, 410, 411, 417, 418

  • inequality: 354

  • kernelized flow: 397

  • noise: 376

  • projection: 400

  • structure: 278, 315, 409

  • velocity: 314, 315, 411

• graph

  • augmenting path: 10, 49

  • bipartite: 45, 48

  • coupling: 55, 68

  • equality: 8–10

  • geodesic distance: 108

  • matching: 283

  • support: 45, 49

  • transport: 109

• graphical model: 258

• Grassmann manifold: 225

• gravity interaction: 384

• greedy sweep: 45

• Gromov-Hausdorff distance: 284, 303, 304

• Gromov–Monge: 298, 304

• Gromov–Wasserstein: 230, 277, 283–286, 288, 302, 304

  • compactness: 286

  • completeness: 286

  • conditional concavity: 290

  • distance: 277, 283

  • geodesic: 288

  • metric: 286

  • sample complexity: 287

• Gromov–Wasserstein

  • Gaussian measures: 295

• ground cost: 63, 106, 250, 266, 268, 283, 289

  • adversarial: 293

• ground metric learning: 66

• group action: 229

• growth term: 331

• Gurvits scaling: 307, 308

H¶

• Hall theorem: 49

• Hamilton-Jacobi equation: 141, 148, 317, 318

• Hamilton-Jacobi-Bellman equation: 334

• hard congestion: 334

• Hausdorff distance: 284, 303, 304

• Hausdorff measure: 20

• heat equation: 148, 340–342, 385

• heat flow: 81, 329

• heat kernel: 145–147

• heavy-ball method: 382, 383

• heavy-tailed noise: 376

• Hegselmann–Krause model: 405

• Hellinger

  • divergence: 118

  • geodesic: 215

  • geometry: 39

• Hellinger-Kantorovich: 332

• Helmholtz decomposition: 411

• Hermitian matrix: 305, 307

• hidden convexity: 402

• Hilbert

  • diameter: 170

  • embedding: 224

  • metric: 167

  • projective metric: 167, 169, 170

• Hilbert space: 352

• Hilbertian

  • embedding: 233

  • limit: 156

• Hilbertian tangent norm: 320

• Hilbertian Wasserstein space: 30

• histogram: 4, 11, 30, 32, 63, 64, 122, 157, 205, 242, 277, 283

• Hoeffding inequality: 199

• Holder curve: 353

• Hölder inequality: 236

• holomorphic continuation: 156

• holomorphic implicit function theorem: 157

• homogeneous action: 322

• homogeneous formulation: 208, 209

• homogeneous Sobolev norm: 156

• homogenization: 209

• Hopf-Cole transform: 145, 146, 148, 149

• Hopf-Lax formula: 146, 148

• Horn matrix: 202

• Hungarian primal-dual method: 1, 7–9, 167

• HWI inequality: 365, 366

• hyperbolic geodesic: 26

• hyperbolic metric: 34

• hypersurface: 20

I¶

• ice-cream cone: 37

• imitation learning: 120

• implicit Euler scheme: 336, 370

• in-context map: 404

• incompressible Euler: 260

• increasing rearrangement: 60

• induced width: 259

• inertia: 384

• inertial particle method: 383

• infinite-depth limit: 404

• infinite-dimensional linear programming: 56

• inner product

  • Gromov–Wasserstein: 295

• input-convex neural network: 90, 138, 139, 404

• integer multiplicity: 51

• integral probability metric: 110–112, 118–120, 396

• interacting particle system: 366, 369

• interaction energy: 358

• interaction kernel: 384

• interaction polynomial: 76

• interior-point method: 52, 123, 126, 167

• intermediate measure: 261

• internal energy: 349

• interpolant: 394

• intrinsic dimension: 181, 184–186

• intrinsic length metric: 223

• inverse OT: 266, 268, 270

• inverse OT gap loss: 268

• inverse square root: 308

• irreducibility: 328

• isometric embedding: 304

• isoperimetric inequality: 359, 360

• isotonic regression: 60

• iterative closest point: 230

• iterative proportional fitting: 123, 124

J¶

• Jacobian determinant: 15, 350

• Jacobian equation: 27, 179

• Jensen inequality: 69, 116, 128, 182, 190, 194, 273, 316, 325

• Jensen-Shannon divergence: 118, 120

• JKO scheme: 143, 336, 339, 340, 346, 370, 373, 377, 378, 382, 416, 418

• joint

  • law: 53, 56, 257

  • measure: 53

• Jordan decomposition: 13

• jump kernel: 326

• jump process: 326, 375

• junction tree: 259

K¶

• Kantorovich

  • axiomatic characterization: 57

  • coupling: 58

  • dual: 318

  • duality: 7, 20, 27, 82–84, 87, 270, 272, 305, 402

  • finite equal-mass measures: 205

  • interpolation: 22

  • potential: 83, 85, 255, 266

  • problem: 43, 48, 50, 52, 55–58, 62, 68, 82, 129, 139, 142, 263, 272, 280

  • relaxation: 1, 16, 20, 41, 43, 48, 49, 53, 57, 68, 205

• Kantorovich–Rubinstein

  • duality: 71, 80, 112, 120, 182

  • formula: 107, 109

  • norm: 31, 107

• kernel

  • attention: 203

  • characteristic: 114

  • density estimate: 385, 397, 405

  • drifting: 400

  • energy: 113, 114

  • Gaussian: 113, 127, 399, 409, 417

  • Laplacian: 113

  • logarithmic accuracy: 198

  • low-rank approximation: 197

  • Matérn: 113

  • mean embedding: 114, 190, 233

  • norm: 113, 114, 120, 156, 348, 399

  • positive: 113, 154, 167, 399

  • positive definite: 156, 190, 224, 342

  • positivity: 198, 199

  • quadrature: 342

  • regression: 405, 409

  • ridge regression: 61, 198

  • Riesz: 113

  • sample complexity: 197

  • sketching: 197

  • smoothing: 188

  • universal: 114, 156

• kernelized self-interaction: 342

• kernelized Stein discrepancy: 397

• kernelized transport distance: 325, 326

• kinetic action: 141

• kinetic equation: 384

• kinetic mean-field limit: 383

• KL

  • projection: 110, 144, 159–161, 165, 248, 264, 308

  • proximal map: 144

  • relaxation: 211

• KL barycenter: 246

• KL proximal operator: 143–145

• Knothe–Rosenblatt rearrangement: 32, 33

• Kolmogorov forward equation: 377

• Kolmogorov-Smirnov distance: 31

• Kullback-Leibler divergence: 34, 39, 117, 127, 128, 130, 143, 356

• Kurdyka–Łojasiewicz inequality: 354, 356–358

L¶

• L2 attention: 405

• L2 self-attention: 405

• Lagrange multiplier: 345

• Lagrangian

  • description: 313

  • flow: 313

• Lagrangian velocity: 383

• Laguerre cell: 18, 87, 96–98, 104

  • discrete: 92

• Langevin

  • dynamics: 140, 345

  • free energy: 343

• Laplace method: 179

• Laplace-Beltrami operator: 145

• large-temperature collapse: 154

• large-temperature limit: 156

• latent

  • atom: 261

  • mass: 262

  • variable: 261, 388, 390

• latent measure: 261

• lattice distribution: 183

• Lavenant criterion: 418

• law of large numbers: 181, 250

• law over measures: 250

• layer normalization: 404, 410

• layer-cake formula: 30

• layerwise matching: 381

• least squares

  • Radon reconstruction: 252

• least-square

  • inversion: 315

  • velocity: 315

• Lebesgue decomposition: 115, 208

• Lebesgue measure: 13, 17, 20, 21, 57, 128

• Legendre function: 159

• Legendre transform: 88, 119, 249, 402

• Legendre-Fenchel transform: 147, 148

• length metric: 223

• length space: 320

• Lévy process: 376

• linear

  • attention: 203

  • convergence: 126, 162, 164, 167, 169, 212, 355

  • form: 12, 13, 47, 369

  • functional: 340

  • linearization: 25, 301

  • objective: 48

  • operator: 44

  • OT: 232, 233, 235

  • potential: 340

  • programming: 43, 44, 47, 52, 123, 241, 264

  • tilt: 160

• linear-time Sinkhorn: 197

• linearized Wasserstein metric: 363

• Liouville equation: 383

• Lipschitz

  • constant: 106

  • function: 19, 91, 106, 107, 109, 111, 113

  • map: 78

  • stability: 86

  • test function: 71

• Lloyd

  • algorithm: 101, 352

  • flow: 102, 103

• local action: 320, 370

• local distance distribution: 283

• local matching indicator: 2

• local metric slope: 339

• local profile: 283

• local tangent action: 320

• localization: 24

• Loewner order: 275, 325

• log-concave

  • density: 401

  • measure: 401

• log-concave generator: 404

• log-concavity: 361

• log-domain Sinkhorn: 122

• log-sum-exp: 96, 129, 136, 138, 146, 154, 260

• logarithmic error: 199

• logarithmic kernel approximation: 199

• logarithmic mean: 327, 329, 330

• logarithmic Sobolev inequality: 360, 362, 364–366

• LogDet divergence: 213

• look-ahead point: 382

• Lorentz cone: 37

• low-rank

  • coupling: 203, 261

  • optimal transport: 260, 261, 263

• lower semicontinuity: 56, 115, 116, 155, 207, 208, 250, 272, 277, 328, 332

• lower semicontinuous envelope: 68, 69

• lower semicontinuous relaxation: 322

• lower-dimensional set: 20

• Lyapunov equation: 411

M¶

• Ma-Trudinger-Wang condition: 25, 27

• Maas distance: 329

• manifold support: 184, 185

• many-token regime: 79

• map estimation: 194

• map extrapolation: 192

• marginal

  • constraint: 42, 45, 53, 56, 110, 124, 125, 128, 130, 142, 149, 151, 159, 161, 165, 171, 210, 272, 274, 305

  • divergence: 206–208

  • penalty: 143, 145, 207, 208

  • relaxation: 206–208

  • relaxed: 205

  • residual: 122, 166

• marginal-dependent problem: 143, 144

• market clearing: 164

• Markov

  • averaging: 405

• Markov chain: 329, 377

• Markov kernel: 32, 81, 168, 283

  • Wasserstein stability: 81

• Markov semigroup: 375, 377

• Markov-chain Wasserstein geometry: 375, 377

• martingale coupling: 273, 274

• martingale OT: 273–276

• masking: 404

• mass

  • balance: 2, 97

  • conservation: 41, 124, 205

  • creation: 205

  • free: 101

  • shape separation: 206

  • splitting: 26, 41, 69, 81

  • variation: 207

• mass flux: 334

• matching

  • algorithm: 7

  • non-crossing: 6, 7

  • optimal: 1, 62

  • partial: 9

  • perfect: 7, 9, 49

  • uniform: 46, 48, 50, 51, 57, 62

• matrix

  • cone: 279

  • incidence: 110

  • permutation: 44, 46–48, 283

  • perspective: 324

  • polar decomposition: 22

  • scaling: 121, 123, 124, 156, 157, 159

  • square root: 176

  • subproblem: 280

• matrix-valued measure: 277, 279, 280

• matrix-vector iteration: 123

• matrix-vector product: 199

• max-flow min-cut theorem: 264

• maximal correlation: 171, 402

• maximal density constraint: 344

• maximum mean discrepancy: 112–115, 119, 120, 184, 190, 233, 342, 348, 349, 363, 384, 396, 412, 413

• McCann

  • displacement convexity: 349

  • interpolation: 23, 26, 32, 33, 67, 238, 242, 319, 347

• McKean-Vlasov equation: 343, 344

• McKean-Vlasov limit: 343, 344

• mean

  • Wasserstein decomposition: 66

• mean field game: 333

• mean shift: 405

  • blurring: 405

  • iteration: 405

  • vector: 405

• mean-field limit: 80, 344, 359, 404

• mean-field momentum: 383

• mean-field neural network: 336, 359, 367, 381

• mean-preserving splitting: 273

• mean-shift PDE: 405

• measurable function: 12, 111

• measurable selection: 139, 140, 239, 283

• measure

  • atomless: 17, 68

  • evolution: 313, 314, 340

  • isomorphism: 17

• measure convolution: 71

• measure-function duality: 83

• measure-preserving

  • isometry: 284, 286

  • map: 19, 21, 22, 298

• measure-to-measure map: 77

• measure-to-vector map: 75

• measure-valued Radon transform: 219

• mechanical energy: 383

• Mémoli profile: 283, 289

• mesh size: 146

• message passing: 259

• metric

  • derivative: 222, 339, 353

  • entropy: 184, 186

  • equivalence: 236

  • learning: 74, 266, 267

  • length: 353

  • length bound: 353

  • pull-back: 219

  • slope: 354, 357

  • space: 12

  • triangle inequality: 66

• metric-measure space: 277, 283, 284, 286, 288, 289, 358, 359

• min-plus algebra: 89

• Min-SW: 227, 229

• minibatch: 395

• minibatch bias: 395

• minibatch noise: 377

• minimal velocity: 340

• minimax: 234, 237

• minimax lower bound: 187

• minimax optimality: 187

• minimizer

  • sparse: 44

• minimizing movement: 336, 340, 373, 418

• minimizing movement scheme: 336

• Minkowski inequality: 19, 64, 66, 79, 220, 286

• Minkowski problem: 401

• mixed Hessian: 27

• MMD-GAN: 395

• mobility: 323, 371

  • concave: 323, 324, 371

  • convexity mechanism: 323

  • nonlocal: 327

• modulo isometry: 286

• modulus of continuity: 86

• moment

  • convergence: 72

• moment closure: 419

• moment condition: 181

• moment energy: 358

• moment functional: 411, 413

• moment matching: 410, 418

• moment measure: 401, 402, 404

• momentum: 277, 278, 316–318

• momentum formulation: 316, 320, 321, 334

• momentum variable: 320

• Monge

  • distance: 18, 19, 23, 68, 69

  • geodesic: 22, 348

  • interpolation: 22, 30

  • map: 11, 16, 17, 19, 255

  • problem: 1, 11, 16, 17, 19, 20, 22, 28, 29, 33, 34, 41, 55, 57, 66, 68, 315, 316, 347

• Monge gap: 59

  • convexity: 59

  • isotonicity: 60

  • kernel regression: 61

  • one-dimensional formula: 60

• Monge map

  • Gromov–Wasserstein: 299

• Monge–Ampère

  • equation: 23–25, 233

  • parabolic equation: 178, 180

  • solver: 24

• Monge-Kantorovich equivalence: 57, 255

• monotone

  • alignment: 309

  • convergence: 163

  • field: 20

  • fixed point: 164

  • matching: 1, 5, 227

  • rearrangement: 3, 4, 20, 29, 32, 33, 58, 62

  • spectral gauge: 234, 324

  • transport map: 397

• Moore-Penrose pseudo-inverse: 252

• MTW condition: 27

• multi-marginal

  • cost: 257

  • formulation: 257

  • Monge map: 254, 255

  • OT: 241, 242, 254, 257, 258

  • Sinkhorn: 257

• multi-omics: 303

• multi-species

  • flow: 346

  • gradient flow: 346, 347

• multi-valued subdifferential: 20

• multinomial distribution: 197

• multiplicative scaling: 144

• multiscale coupling: 185

• Muon algorithm: 234, 237, 325, 372, 374, 375

• Muon polar factor: 373

• mutual information: 130

N¶

• Nash equilibrium: 333

• natural gradient: 396

• natural language processing: 67

• near-linear Sinkhorn: 199

• nearest neighbor: 230

• negative Shannon entropy: 159–161, 341

• negative Sobolev norm: 364

• Nesterov acceleration: 382

• network simplex: 109, 110

• neural

  • dual solver: 138, 139

  • Sinkhorn solver: 138

• neural optimal transport: 90

• neuron: 366–369, 380, 415

• neuron law: 367, 414, 415

• Newton dynamics: 384

• Newton step: 53, 123

• Newtonian dynamics: 384

• Newton–Schulz iteration: 374

• node feature: 303

• noising path: 394

• noising schedule: 392

• noisy gradient descent: 369

• non-convex domain: 24

• non-convexity: 352

• non-variational scaling: 164

• nonconvex

  • optimization: 242

• nonlinear flattening: 250

• nonlocal

  • action: 326

  • continuity equation: 327

  • gradient: 327

• nonlocal Wasserstein

  • distance: 326, 327, 375

  • flow: 375

• nonnegative

  • orthant: 161

• nonnegative factorization: 200

• nonnegative measure: 150, 151, 205, 210

• nonnegative rank: 261

• nonnegative transport matrix: 49

• norm

  • flat: 111, 112

  • RKHS: 114

  • Wasserstein-1: 106

• normal approximation: 182

• normal cone: 344

• normalized dynamics: 372

• normalized gradient flow: 371, 372

• normalized SGD: 372

• north-west corner

  • plan: 45

  • rule: 45, 52

• Nystrom approximation: 200

O¶

• ODE limit: 382

• one-dimensional

  • Monge map: 28

  • projection: 18, 205, 219, 221, 414

  • transport: 21, 28, 31, 45, 62, 348

• one-step generative model: 387, 395, 404

• Onsager operator: 320, 329, 330

• operator bound: 237

• operator gauge: 325, 372

• operator norm: 13

• operator scaling: 305, 307, 308

• operator-valued coupling: 308

• opinion dynamics: 405

• optimal coupling: 28–30, 44, 48, 50, 55, 57, 58, 62, 65, 67, 82, 83, 88, 101, 243, 248, 257, 266, 268, 271, 286, 288, 316, 347

• optimal plan: 20, 27, 42, 44, 45, 62, 84, 88, 125, 149, 271, 286, 347

• optimal transport map

  • consistency: 193

  • estimation: 192

• optimality

  • first-order: 98, 161, 307

• orbit space: 229

• order-constrained coupling: 274

• order-preserving map: 163

• Ornstein-Uhlenbeck flow: 362, 366

• Ornstein-Uhlenbeck process: 385, 392, 394

• OT map estimation: 192

• OT on trees: 31

• OU bridge: 394

• overshooting bridge: 394

P¶

• pair space: 327

• pair-space action: 326

• pairwise distance: 284

• parabolic optimal transport: 178

• parameter space: 395

• partial matching: 8

• partial transport: 207, 210, 216, 218

  • active region: 219

  • fixed mass: 218

  • TV penalization: 217

• particle method: 78

• particle ODE: 313, 343, 366

• particle polynomial: 337

• particle splitting: 326

• particle Wasserstein descent: 382

• path action: 320

• path entropy: 311

• path measure: 316, 325

• path metric: 223

• path space: 12, 68, 139–143

• path-space formulation: 121, 139–142, 319

• path-space problem: 140

• path-space transport: 139, 142

• Pearson divergence: 118

• penalized minimization oracle: 370

• permutation

  • equivariance: 79

• permutation matrix: 46

• Perron-Frobenius theorem: 163, 167, 168

• perspective function: 316–319, 321

• perspective recession: 277

• perspective transform: 209, 321

• perturbation response: 396

• phase space: 383

• phase-space density: 386

• phase-space equation: 385

• phase-space measure: 383

• phi-divergence: 111, 115, 116, 118, 119, 127, 149–151, 351, 363

• phi-divergence regularized OT: 149–152, 163

• Pinsker inequality: 117, 166

• PL convergence: 354

• plan

  • interpolation: 67

  • lifted: 227, 229, 238

  • optimal: 67

  • sparse: 42, 44

  • transport: 41, 50, 61, 67, 83, 238, 268, 277

• plug-in estimator: 181, 188

• PMO: 370

• Poincaré disk: 26

• Poincaré inequality: 363, 364

• Poisson equation: 146

• polar factor: 374

• polar factorization: 21, 22

• polar formula: 237

• polar set: 234, 236, 237, 372

• Polish space: 12, 17, 64, 65, 75, 272, 281

• Polyak momentum: 382

• Polyak-Lojasiewicz inequality: 354

• population law: 333

• porous medium equation: 341, 372

• Portmanteau theorem: 56

• positional encoding: 79

• positive

  • autocorrelation: 202

  • cone: 277, 278, 307

  • feature map: 200

  • features: 200

  • linear map: 169

  • matrix-valued measure: 279

  • measure: 378

  • operator: 277, 305

  • random features: 197, 201–203

  • semidefinite cone: 37

  • semidefinite kernel: 197, 198

  • semidefinite matrix: 34, 36, 37, 234, 277, 278, 280, 306

  • sketch: 202

  • vector-valued measure: 277

• potential energy: 358

• potential game: 333

• power diagram: 96, 98, 99

• power divergence: 350

• power-law jump kernel: 376

• Prékopa

  • inequality: 403

  • Leindler inequality: 360

• pressure field: 345–347

• primal-dual

  • method: 92

• principal component analysis: 233–235

• principal value: 376

• probabilistic coupling: 14, 15

• probability integral transform: 28

• probability measure: 11, 12, 14, 17, 19, 30, 33, 53, 56, 63, 70, 75, 84, 112–114, 118, 151, 219, 224, 232, 241, 254, 281, 285, 289, 315, 316, 342, 368

• probability path: 388

• probability simplex: 11, 123, 151, 168

• Procrustes

  • alternating minimization: 231

  • Bures metric: 38

  • orthogonal: 230

  • problem: 230

  • rigid update: 230

  • soft: 215

• product

  • coupling: 42, 54, 56, 65, 68, 154, 263, 392

  • measure: 125, 130, 152, 258

• product Wasserstein metric: 346

• profile lower bound: 230, 283, 289

• projected formulation: 234

• projected measure: 219

• projection direction: 219

• projection-free attention: 405

• projection-robust Wasserstein: 225

• projective

  • cone: 167

  • geometry: 163

  • metric: 167

• Prokhorov theorem: 56, 75

• propagation of chaos: 344

• proximal

  • operator: 318

  • splitting: 318

• pullback: 14, 15

• push-forward: 11, 14–18, 28, 29, 35, 36, 41, 314, 338, 387, 389

  • measure: 79

  • stability: 78

• Pythagorean identity: 159, 161

Q¶

• quadratic

  • action: 320

  • assignment problem: 283

  • barycenter: 243, 257

  • closure: 176

  • confinement: 385

  • cost: 3, 17, 20, 22, 23, 25, 27, 29, 33, 36, 83–85, 87, 96, 98, 101, 138, 171, 176, 205, 234, 241, 271, 316

  • energy: 384

  • form: 290

  • perspective: 316, 321

  • potential: 20, 35, 36, 176, 411, 413

  • regularization: 162

  • scaling: 158

  • tangent norm: 320

• quantile: 28

  • barycenter: 244

  • coupling: 58

  • formula: 282

  • function: 4, 28–31, 219, 224, 232–235, 244, 252

  • map: 16, 29, 30, 278

  • process: 105

  • push-forward: 28

  • quantization: 104

  • semi-discrete OT: 97

• quantization

  • energy: 101, 103

  • fixed weights: 104

  • high-resolution limit: 102

  • mean-field limit: 102

  • non-uniform weights: 100

  • optimal: 100, 352

  • rate: 100

• quantum

  • Bregman projection: 307

  • coupling: 305

  • Kantorovich duality: 305

  • OT: 277, 305, 306

  • OT duality: 306

  • Sinkhorn: 307

• quotient

  • metric property: 229

  • Wasserstein distance: 229, 230

R¶

• Rademacher bound: 192

• radial coupling: 21

• radial generator: 36

• radial measure: 20

• Radon barycenter: 252

• Radon measure: 12, 13, 61, 84

• Radon pseudo-inverse: 252

• Radon range condition: 220, 252

• Radon sinogram: 252

• Radon structure: 221

• Radon transform: 219, 252

• Radon Wasserstein barycenter: 252

• Radon Wasserstein distance: 220

• Radon-Nikodym derivative: 389

• ramp filter: 253

• random features: 197, 198

• random measure: 250

• random probability measure: 250, 281

• random variable: 14, 16, 56, 70, 182, 411

• random walk: 330

• rank: 44

• rank constraint: 234

• rank-deficient covariance: 39

• rate

  • parametric: 181, 190, 192

  • sublinear: 99, 165

• rational weights: 49–51, 281

• raw second moment: 37

• reaction term: 378

• reaction–transport action: 331

• reaction-transport geometry: 331

• recession convention: 207, 208, 277, 317

• rectifiable set: 20

• reference

  • distribution: 387

  • measure: 13, 22, 119, 127, 128, 150, 151, 208, 232, 233, 247, 387

• reflow: 388

• regression loss: 387

• regular conditional distribution: 32

• regular vector field: 326

• regularity theory: 23, 24, 27, 28

• regularization

  • Monge gap: 60

• regularization bias: 181, 190, 195

• reinforcement learning: 120

• relative

  • entropy: 117, 121, 127, 130, 141–143, 176, 343, 350, 351, 356, 358, 359, 375, 378, 413, 415

  • error: 199

  • Fisher information: 361, 364, 415

• relative entropy

  • regularity: 76

• reproducing kernel Hilbert space: 113, 114

• rescaled convolution: 71

• residual attention: 404

• residual block: 379

• residual network: 381

• ResNet: 379

  • continuous-depth: 380

  • finite-depth: 379

  • finite-depth mean-field: 380

  • finite-width: 379

  • infinite-depth: 379, 381

  • particle discretization: 381

• resolvent: 146

• reverse formulation: 208

• reverse KL divergence: 118

• reversible jump kernel: 326

• reversible Markov chain: 329

• Ricci curvature: 359

• Richardson–Romberg extrapolation: 337

• Riemann sum: 70

• Riemannian manifold: 25, 359

• Riemannian metric: 320

• Riemannian tensor: 320

• Riesz potential: 113

• Riesz-Markov-Kakutani theorem: 12, 13

• rigid motion: 230

• rigid registration: 230

• rigid update: 231

• RKHS: 112–115, 120, 190, 233

  • cost regularization: 294

  • vector field: 325

  • velocity: 325

• robust

  • envelope: 75

  • finance: 273

  • optimal transport: 293

  • representation: 236

  • Wasserstein: 234, 237

  • Wasserstein distance: 234, 237

• rotational component: 29, 393

• row normalization: 125

• running-intersection property: 259

S¶

• Sakoe–Chiba band: 309

• sample complexity: 101, 181, 184, 251, 287

• sampling: 78

• scalar bridge: 394

• scaling

  • algorithm: 121, 258

  • column: 125, 159, 161

  • form: 123, 124, 135, 161, 248, 308

  • potential: 126

  • row: 125, 159, 161

  • vectors: 162, 167

• Schatten norm: 236

• Schoenberg theorem: 202

• Schrödinger

  • bridge: 139–142

  • interpolation: 141

  • problem: 121, 134, 139–142, 156

  • system: 164

• score function: 341, 391, 392, 394, 399

• score-based generative modeling: 387

• score-SDE: 387

• second boundary condition: 24, 25

• second moment: 37, 413, 414

• second-moment quotient: 37

• second-order limit: 197

• sectional curvature: 28

• self Sinkhorn: 156

• self-concordance: 123

• self-corrected field: 400

• semi-coupling: 209

• semi-discrete

  • closed form: 97

  • dual: 97, 98

  • Monge map: 17, 18

  • one-dimensional formula: 97

  • OT: 18, 55, 87, 91, 96–100, 137

• semi-dual: 91, 96, 97, 248

  • discrete: 91

  • entropic: 174

• semi-relaxed problem: 86

• semidefinite program: 305

• separable space: 12

• separator: 260

• Shampoo optimizer: 372

• Shannon entropy: 116, 117, 121, 306, 329, 341, 342, 347, 351

• Shannon-Boltzmann entropy: 121

• shape space: 229

• shearing component: 29, 393

• signed

  • measure: 13, 31, 70, 106, 107, 113, 114, 336, 338, 342

  • negative part: 109

  • positive part: 109

• simplex: 330

  • projection: 162

• simplex network: 52

• single-cell genomics: 51, 52, 66, 207, 303, 340, 396

• Sinkhorn

  • acceleration: 171

  • algorithm: 88, 96, 110, 121, 123, 124, 126, 136, 169, 171

  • barycenter: 234, 247, 333

  • closed-form coupling: 131

  • continuous dual iteration: 138

  • continuous flow: 178, 180

  • continuous iteration: 130

  • continuous limit: 179

  • convergence: 159, 163, 165, 167

  • coupling: 142

  • divergence: 121, 153–156, 177, 184, 187, 190, 197, 344, 396, 397, 414, 415

  • dual: 135, 150, 165

  • fixed point: 157, 159

  • full-cycle map: 170

  • GAN: 395

  • half-step: 126, 127, 161, 165

  • holomorphic continuation: 156–158

  • iteration: 126, 154, 162, 176, 248, 260, 271, 301

  • Jacobian: 172

  • kernel: 199

  • linear-time: 197

  • local convergence: 171

  • low-rank: 261, 262

  • over-relaxation: 173

  • potential: 178

  • scaling: 121, 123, 125, 143, 149, 152, 160, 170, 212, 308, 399

  • sketching: 197

  • statistical convergence: 181

  • update: 307

• skip connection: 404

• slack variable: 8

• Slater condition: 305, 306

• sliced barycenter: 252

• sliced length metric: 223

• sliced tangent norm: 222

• sliced Wasserstein

  • barycenter: 252

  • CDF estimator: 30

  • compact comparison: 221

  • directional aggregation: 225

  • distance: 194, 219, 220, 225, 233, 414

  • flow: 397

  • Gaussian: 224

  • length: 223

  • max: 225

  • Monte Carlo directions: 195

  • reverse bound: 221

  • sample complexity: 194

  • tangent norm: 222

• slope

  • metric: 339

  • Wasserstein: 354, 356

• small-jump limit: 329

• smooth density: 188

• smooth OT estimation: 189

• smoothed plug-in estimator: 188

• smoothing estimator: 188

• smoothness: 188

• Sobolev norm: 156

• Sobolev smoothness: 188

• Sobolev space: 189

  • learned cost: 294

• soft

  • barycentric projection: 192

  • c-transform: 135–138, 147, 152–154, 163, 165, 176, 192, 211, 212, 267

  • conjugation: 193

  • minimum: 136

  • transform: 121, 136–139, 159, 176, 177, 211

• soft-DTW: 310

  • expected alignment: 312

• softmax attention: 204

• source term: 331

• spanning tree: 52, 110

• sparse linear program: 109

• sparsity: 44

• spectral

  • dynamics: 237

  • gap: 364

  • gauge: 205, 234, 236, 238, 372

  • steepest descent: 372

  • tangent action: 324, 372

• spectral Wasserstein

  • distance: 234, 236, 237, 324, 325

  • dynamic distance: 324

  • flow: 372, 373, 375

• sphere

  • c-concave potential: 90

• spherical average: 219

• spherical geodesic: 26

• splitting obstruction: 17, 20, 24

• splitting set: 254, 255

• squared local action: 320

• stable process: 376

• static-dynamic equivalence: 325, 332

• stationarity: 88, 301, 306, 368, 369

• stationary condition: 368

• stationary density: 368

• stationary point: 290

• stationary potential: 179

• statistical bias: 195

• statistical optimal transport: 181, 251

• Stein force: 397

• Stein geometry: 325

• Stein method: 182

• Stein operator: 397

• Stein variational gradient descent: 325, 397, 399

• step-size normalization: 374

• Stiefel manifold: 225

• stochastic

  • control: 140

  • differential equation: 376

  • gradient: 99, 120, 389, 396

  • gradient descent: 376

  • interpolant: 387, 390

  • optimization: 99

  • order: 274

  • particle: 343

• Strang splitting: 307

• Strassen theorem: 274

• strict

  • convexity: 24, 27, 122, 127, 129, 138, 161

  • positivity: 156, 164

• strictly correlated electrons: 256

• strong law of large numbers: 181

• strong MTW condition: 27

• strong upper gradient: 340

• structured data: 283

• subdifferential: 20, 24, 57, 59

• subgaussian measure: 190

• subgradient: 267

• subgradient inequality: 369

• sublinear convergence: 357

• subspace

  • projection: 225

  • robust Wasserstein: 237

• subspace-sliced Wasserstein: 225

• successive over-relaxation: 173

• sum-product algorithm: 259

• supergradient: 267

• superposition principle: 316, 325

• support: 12, 14, 27, 48, 57, 83, 84, 108

• SVGD: 397

• symmetric plan: 256

• synthetic Ricci curvature: 359

T¶

• tail index: 377

• Talagrand inequality: 365, 366

• tangent action: 320, 370

• tangent coordinate: 251

• tangent norm: 320

• teacher distribution: 342

• teacher kernel mean: 342

• temporal noise: 387

• tensor product coupling: 54

• Thompson metric: 212

• tightness: 10, 12, 31, 57, 65, 221, 281

• time change: 394

• time reparametrization: 394

• time series

  • alignment: 309

• time-dependent vector field: 313, 387

• token limit: 80, 404

• token measure: 79, 387, 404

• Toland duality: 293, 403

• tomography: 253

• topical map: 163

• topology

  • strong: 70, 72, 116, 117

  • weak: 56, 69, 70, 72, 73, 84, 117, 118

• total unimodularity: 50

• total variation: 11, 13, 63, 70, 73, 111, 112, 116–118, 206, 207, 330

  • vector-valued measure: 107

• trace

  • entropy: 306

  • gauge: 236, 238

  • norm: 237

  • partial: 305, 306, 308

  • partial constraint: 305

• trace-class operator: 308

• trajectory inference: 51, 52

• transfer learning: 51

• transformer: 78, 79, 204, 404, 410, 417, 418

• translation-invariant kernel: 113, 115

• transport

  • cost: 283

  • coupling: 309

  • equation: 410

  • inequality: 359

  • map: 4, 17, 22, 32, 41, 83, 271, 389, 393

  • metric: 375

  • PDE: 409

  • representation: 78

• transport-entropy inequality: 365

• transportation polytope: 41, 42, 45, 50, 52–54, 122, 128, 160, 161, 166, 264, 283

• transportation simplex: 45, 52

• tree

  • decomposition: 259

  • earth mover distance: 31

  • embedding: 32

  • geodesic: 32

  • interpolation: 32

  • metric: 31

  • transport: 31

• tree-sliced Wasserstein: 32

• tree-Wasserstein distance: 31

• treewidth: 259

• triangle inequality: 6, 19, 23, 39, 64, 65, 107, 167, 186, 190, 211, 220, 237, 284–286, 288, 328

• triangular map: 32

• triangular rearrangement: 32, 33

• trivial coupling: 388

• tropical

  • barycenter: 89

• truncated marginal: 218

• Tweedie identity: 391, 394

• twist condition: 25, 26, 254, 255

  • learned bilinear cost: 299

• twist on splitting sets: 254, 255

• two-layer neural network: 367–369, 375, 414

• two-sample testing: 115

U¶

• ultrafast diffusion: 102

• unbalanced

  • barycenter: 333

  • gradient flow: 377

  • interpolation: 332

  • JKO step: 377

  • OT: 128, 143, 205–208, 210, 211, 268, 331, 332

  • OT dual: 208

  • Sinkhorn: 145

  • Sinkhorn convergence: 212

  • tangent space: 331

• uniform integrability: 72, 182

• upper hemisphere: 26

• upper matrix: 264

V¶

• Varadhan formula: 145

• variable elimination: 259

• variable projection: 174

• variance: 195

• variation seminorm: 163, 165, 167, 406

• variational

  • mean field game: 333

• variational dual formula: 119

• vector quantile: 232

• vector-valued measure: 107, 277, 278, 346

• velocity covariance: 324

• velocity field: 313, 314, 319, 338, 339, 387, 404, 410, 417

• Vlasov equation: 384

• von Neumann entropy: 306

• Voronoi cell: 18, 97, 101, 104

W¶

• Waddington landscape: 340

• Waddington-OT: 51

• warping path: 309

• Wasserstein

  • barycenter: 143, 233, 241, 242, 244, 248, 250, 251, 257, 258

    • consistency: 250

    • population: 250

  • convergence: 181

  • coordinate: 232

  • distance: 30, 63–67, 69–73, 118, 119, 182, 205, 211, 219, 220, 234, 236, 237, 281–283, 285, 289, 315, 319, 323, 354, 402, 418

  • empirical rate: 185

  • flow: 354

  • flow of discrepancy: 396

  • formula: 29, 30

  • GAN: 395

  • Gaussian geometry: 415

  • geodesic: 22, 67, 234, 242, 288

  • gradient: 315, 336, 338–340, 342, 344, 354, 367, 368, 382, 387, 397, 399, 400, 410, 411, 416–418

  • gradient flow: 22, 104, 150, 315, 336, 339–345, 354, 357, 368, 379, 381, 387, 396, 397, 410, 418

  • Hölder functional: 75

  • infinity distance: 74

  • infinity robustness: 74

  • Kurdyka–Łojasiewicz inequality: 356

  • Lipschitz functional: 338

  • Lipschitz map: 78, 80, 81

  • minimax rate: 187

  • over Wasserstein: 250, 281, 283, 289

  • p-action: 323

  • p-distance: 25

  • PL inequality: 354–356, 358, 359, 361, 363, 366

  • polynomial: 76

  • Procrustes: 229–231, 285, 290

  • space: 63, 75, 77, 281, 282, 340, 417

  • space CLT: 251

  • squared distance: 352

  • Taylor regularity: 337

  • topology: 72, 182, 281

  • tower: 90

  • training: 367

  • W1 non-uniqueness: 4, 19, 55

• Wasserstein–Fisher–Rao: 332, 377–379

• weak

  • C-transform: 272

  • convergence: 12, 56, 69–72, 112, 114, 115, 118, 181, 182, 220, 221, 237, 281

  • cost: 272, 273

  • Kantorovich duality: 272

  • limit: 17, 34, 70

  • map: 24

  • optimal transport: 271, 272

  • transport: 272, 273

• weight clipping: 120

• weighted

  • Laplacian: 364

  • Poisson equation: 25, 315

  • sweep: 45

• wide stencil: 24

• word mover’s distance: 67

Z¶

• zero mass: 34, 97, 113, 123

• zero mean: 315