Index

A¶

• absolute continuity: 14, 15, 35, 68, 79, 81

• absolutely continuous path: 139, 140

• additively homogeneous map: 94, 95

• admissible

  • potential: 49

• adversarial

  • loss: i, 70

  • perturbation: 46

• affine

  • closure: 163--167

  • constraint: 92, 123

  • gradient: 163--165

  • marginal constraint: 91, 92

• affine map: 24, 52, 97, 158

• Alexandrov solution: 18

• alternate optimization: 54

• alternating

  • projection: 74, 91, 92, 94

  • tree: 6, 7

• alternating optimization: 54

• assignment problem: 1, 2, 5--7, 26, 29, 31, 50, 129, 154, 155

• attention: 162, 168

  • mechanism: 162

  • operator: 162

  • self-attention: 162

  • single-head: 162

• auction algorithm: 6, 50, 51

B¶

• Baker-Campbell-Hausdorff formula: 135

• balance equation: 141

• Banach dual: 9, 10

• Banach space: 9

• barrier

  • entropy: 72

  • logarithmic: 32, 72, 75

  • LP: 72

  • parameter: 32

• barycenter mean: 115

• barycentric

  • mixture: 44, 45

  • projection: 27, 122--124

  • transport: 124

• Beckmann

  • flow: 61

  • formulation: 56, 61, 62

• Benamou-Brenier: 79, 80, 125--128, 137, 139--141

  • distance: 140

  • formulation: 79, 139, 140

• Berry-Esseen

  • bound: 48

  • theorem: 47

• Bessel function: 65

• Birkhoff contraction theorem: 97, 98

• Birkhoff polytope: 29

• Birkhoff-von Neumann theorem: 30, 31, 35

• bistochastic matrix: 29, 31

• Boltzmann entropy: 71

• Borel measure: 8, 9, 15

• Borel set: 8--10, 16

• Bregman

  • divergence: 67, 69, 76, 86, 87, 91, 92

  • iteration: 133

  • penalty: 91

  • projection: 72, 88, 91--93, 95, 96, 134, 135

  • regularization: 86--88

  • three-point formula: 92

• Brenier

  • factor: 16

  • map: 16, 17, 20, 22, 35, 49, 54, 82, 101, 111, 115, 147, 149, 158, 166, 167, 170

  • source hypothesis: 15

  • theorem: 14--16, 18, 19, 23, 24, 35, 37, 54, 57, 110

• bridge

  • Brownian: 79, 81, 82

  • OU: 157, 160

  • overshooting: 159, 160

  • variance-preserving: 157, 159, 160

• Bures

  • covariance: 24

  • metric: 22--25

  • smoothed term: 100

• Bures-Wasserstein geometry: 22, 24, 45, 100, 117, 163, 166

• Burg entropy: 69

C¶

• c-concavity: 85

• c-cyclical monotonicity: 35, 36

• c-transform: 49, 51--56, 60, 77, 83, 84

• Caffarelli regularity: 17

• Catalan number: 6

• causality: 162

• cell mass: 57

• central

  • limit theorem: 42, 47

  • path: 32

• centroidal Voronoi: 59

• change-of-variables: 18, 25, 116, 149

• change-of-variables formula: 18, 149

• characteristic kernel: 145

• Choi matrix: 135

• circle

  • transport: 3, 4

  • unfolded interval: 4

• collapsed mixture: 44, 45

• column normalization: 73

• commutator: 135

• compact space: 35

• compact sublevel set: 104

• complete metric space: 9

• conditional

  • expectation: 153--156, 158

  • quantile: 111

• conditional law: i, 21, 22, 79, 81, 122, 123, 140

• conditional probability: 9, 16, 39, 44, 153

• cone

  • formulation: 106

  • lifting: 106, 141

  • metric: 106

• congruence normalization: 135

• conic lifting: 106

• constant-speed geodesic: 16, 40, 130, 147

• continuity equation: 80, 125, 127, 128, 137--139, 141--143, 153--155, 161, 163

• continuous

  • bounded function: 42

  • coupling: 33

  • depth: 162

  • dual: 83

  • Sinkhorn dual: 85

• continuous normalizing flow: 153

• contraction ratio: 98

• controlled transport: 163

• convergence

  • in law: 20, 42, 90

  • in probability: 42

  • law: 41

  • mode: 42

• convex

  • conjugate: 54, 86

  • function: 3, 14--16, 19, 20, 23, 35, 41, 46, 49, 54, 69, 85, 86, 91, 144

  • potential: 14, 15, 17, 18, 35, 37, 52, 56, 157

  • regularizer: 85, 121

• convex envelope: 54

• convexity

  • along geodesics: 151

  • classical: 151, 152

  • geodesic: 115, 142, 147--150, 170

• coordinate ascent: 95

• correspondence: 27, 125, 129, 131, 132

• cost

  • assignment: 5, 51

  • derivative: 120

  • feature: 132

  • quadratic: 2, 13--17, 20, 22, 24, 45, 50, 52, 54, 57, 59, 85, 98, 99, 104, 110, 112, 115, 122, 139, 166

  • reduced: 6, 32

  • transport: 46, 116

  • unbalanced: 106

• cost matrix: 1, 26, 29, 37, 49, 73, 75, 96, 115, 131

• countable dense subset: 9, 44

• coupling measure: 39

• covariance

  • commuting: 158

  • diagonal: 25

  • matrix: 20, 24, 168

  • ODE: 167

  • subblock: 167

  • term: 24

• Cramer-von Mises distance: 20

• created mass: 107

• cumulative

  • distribution function: 13, 18, 19

  • function: 3, 19, 20

• curse of dimensionality: 58, 59, 101

• cyclic

  • monotonicity: 35--37, 122

  • projection: 92

  • shift: 3, 4

D¶

• Dacorogna-Moser inversion: 138

• Danskin theorem: 120

• data processing inequality: 95

• denoising diffusion probabilistic model: 153

• denoising score matching: 153

• dense linear system: 72

• density

  • formula: 11

  • matrix: 133, 134, 136

  • ratio: 9, 66, 67, 69, 70, 85--88

  • relative: 9

• depth evolution: 162

• destroyed mass: 107

• diffusion model: 155, 159

• dimension dependence: 102

• Dirac mass: 8, 11, 13, 20, 23, 26, 37, 41, 115, 142, 144, 147, 153

• discrete

  • barycenter: 119

  • coupling: 26

  • measure: 8, 40, 42, 71

  • space: 44

  • target: 56

• discretization error: 117, 153

• discriminator class: 63, 64, 70

• disintegration: 9, 21, 39, 53, 81, 106, 122, 123, 155

• displacement

  • convexity: 15, 16, 144, 148, 149, 151

  • covariance: 112, 114

  • interpolation: 16, 17, 20, 21, 114

• distance residual: 131

• distortion: 45, 130--132

• doubly stochastic normalization: 163

• drifting

  • field: 160, 163, 166

  • model: 160, 161

• dual

  • attainment: 52, 54, 133

  • certificate: i, 51, 55

  • formula: 69, 70

  • gap: 75, 91, 96, 121

  • norm: 45, 63--66, 70, 170

  • objective gap: 95

  • pairing: 9

  • potential: i, ii, 49--51, 53--56, 74, 77, 84, 85, 87--90, 95, 98, 101, 103, 121, 133, 167

  • price: 50

  • problem: 49, 53, 56, 61, 71, 82--84, 118

  • radius: 96

  • rate: 95

  • weight: 56, 57

• duality

  • Fenchel-Rockafellar: 85, 105, 123

  • strong: 61, 118, 133

  • weak: 7, 52, 123

• Dudley entropy: 103

• Dudley metric: 63

• dyadic cube: 102

• dynamic

  • formulation: 139, 140

  • optimal transport: 137

  • unbalanced OT: 140

• dynamic action: 139

• dynamic value: 139

• dynamical optimal plan: 41, 150

E¶

• elliptic regularity: 111

• empirical

  • distribution: 146, 162

  • law: 46, 47, 146

  • measure: ii, 2, 8, 13, 28, 31, 42, 89, 101, 102, 110, 120, 137, 145, 146, 162, 170

  • Monge map: 12

  • OT: 59, 101, 102

  • OT rate: 101

  • process: 103

• empirical context: 162

• endpoint coupling: 79, 81, 82

• energy

  • decay: 148

  • dissipation: 149

  • distance: 65, 90, 145, 146

  • interaction: 88, 145, 146, 149, 165

• entropic

  • barycenter: 116, 118

  • bias: 75, 88

  • OT: 51, 71, 72, 82, 88, 95, 99, 122, 132, 134, 161

  • path: 72

  • potential: 85, 88, 94

  • regularization: ii, 55, 71, 72, 88, 91, 101, 106, 120, 121, 132, 133, 170

  • smoothing: 71, 95, 103, 117

• entropy

  • function: 67, 68, 85, 104

  • relative: 68, 71, 76, 78, 81, 95, 99, 148--150, 165

• envelope theorem: 71, 103, 120

• equality constraint: 31, 61

• Eulerian

  • description: 137

  • velocity: 137, 139

• exact OT: 18, 101--103

• extreme minimizer: 30

• extreme point: 28, 30, 31

F¶

• feasible coupling: 32, 34, 111, 131

• feature term: 132

• Fenchel duality: 85, 86, 105, 123

• Fenchel-Young loss: 121

• finite measure: 9, 63, 104, 125

• first variation: 86, 135, 142, 143, 148, 151, 152, 161, 163, 165, 167

• Fisher information: 80, 166

• flow

  • conservation: 31

  • cross-moment: 167

  • map: 137, 157, 158

  • matching: 138, 153--158, 167, 170

  • minimum-cost: 32

  • probability ODE: 153, 157, 159

  • rectified: 153, 157

• flux: 125, 127, 138, 143

• Fokker-Planck equation: 145--147, 149, 158, 159, 166

• Fourier multiplier: 65

• Frechet mean: 115

G¶

• GAN: 70

• Gaussian

  • barycenter: 117, 119, 169

  • closure: 163, 165, 167, 168

  • contractive projection: 168

  • covariance: 100, 166

  • energy: 165

  • flow matching: 157, 158

  • kernel drifting: 161, 168

  • manifold: 163, 164, 166

  • marginals: 85, 99

  • measure: 22, 23, 117, 119, 163, 164, 167, 169

  • optimal map: 20

  • preservation: 163--165, 167, 169

  • preserving flow: 167

  • projection: 166, 168, 169

  • push-forward: 23

  • Sinkhorn: 99--101, 166, 167

  • Sinkhorn divergence: 100, 166

  • submanifold: 163, 164

• Gaussian denoiser: 155

• Gaussian mixture: 2, 45, 57, 74, 75, 77, 86, 107, 116, 156, 157, 159

• Gelbrich theorem: 168, 169

• generalized

  • inverse: 13, 19

  • Wasserstein: 104

• generalized quantile: 21

• generative model: i, 63, 109, 137, 153, 160, 170

• geodesic

  • space: 41, 150

• Gibbs

  • coupling: 134, 135

  • formula: 134

  • kernel: 74, 75, 92, 93, 98

  • reference: 91, 92

• global

  • minimizer: 152

  • optimality: 152

• gluing lemma: 37--39, 106, 130

• gradient

  • field: 15

  • flow: i, ii, 16, 129, 130, 137, 138, 142--153, 160, 163, 165, 167, 169, 170

  • projection: 162

  • structure: 126, 138, 163

  • velocity: 138, 164

• graph

  • augmenting path: 7

  • bipartite: 29, 31

  • equality: 6, 7

  • geodesic distance: 61

  • support: 28

  • transport: 62

• greedy sweep: 28

• Gromov-Hausdorff distance: 129, 132

• Gromov-Wasserstein: 45, 125, 128--130, 132

  • distance: 45

  • fused: 132

  • geodesic: 130

  • metric: 130

• ground cost: 37, 45, 60, 118, 120, 121, 128, 129, 131

• Gurvits scaling: 135

H¶

• Hausdorff distance: 129, 132

• Hausdorff measure: 15

• heat equation: 144, 145

• Hellinger

  • divergence: 69

  • geometry: 25

• Hellinger-Kantorovich: 141

• Hermitian matrix: 133, 135

• Hilbert

  • embedding: 109

  • metric: 96, 97

  • projective metric: 96, 97

• Hilbertian

  • embedding: 111

  • limit: 90

• histogram: 2, 8, 20, 21, 37, 39, 71, 96, 104, 115, 125, 129

• homogeneous formulation: 105, 106

• homogeneous Sobolev norm: 90

• homogenization: 106

• Hungarian primal-dual method: 1, 6, 7, 50

• hypersurface: 15

I¶

• implicit Euler scheme: 142

• infinite-depth limit: 162

• integer multiplicity: 5

• integral probability metric: 62--64, 69, 70, 153, 155, 160

• interior-point method: 32, 72, 75

• interpolation map: 115

• inverse OT: 120, 121

• inverse square root: 135

• isometric embedding: 132

J¶

• Jacobian determinant: 148

• Jensen inequality: 41, 68, 76, 102, 123, 139, 149

• Jensen-Shannon divergence: 69, 70

• JKO scheme: 143, 164, 169

• joint

  • law: 33, 35, 119

  • measure: 33

• Jordan decomposition: 10

K¶

• Kantorovich

  • duality: ii, 6, 15, 49, 51, 52, 54, 121, 123, 133

  • interpolation: 16

  • potential: 51, 52

  • problem: 5, 28, 31, 34--36, 41, 49, 77, 79, 81, 123, 128

  • relaxation: 1, 4, 12, 15, 26, 28, 31, 32, 35, 104

• Kantorovich-Rubinstein

  • duality: 48, 64, 70

  • formula: 60, 61

• kernel

  • conditionally positive: 65

  • drifting: 161

  • energy: 65

  • Gaussian: 65, 75, 160, 168

  • Laplacian: 65

  • Matern: 65

  • mean embedding: 65, 111

  • norm: 45, 65, 70, 90, 161

  • positive: 65, 88, 96, 160

  • positive definite: 90, 102, 109, 145

  • Riesz: 65

  • translation-invariant: 65, 66

  • universal: 66

• kernel quadrature: 145, 146

• kernelized self-interaction: 145

• kinetic action: 80

• KL

  • projection: 62, 91--93, 95, 96, 118, 135

  • relaxation: 106

• Knothe-Rosenblatt rearrangement: 21, 22

• Kolmogorov-Smirnov distance: 20

• Kullback-Leibler divergence: 68, 76--78

L¶

• Lagrangian

  • description: 137

  • flow: 137

• Laguerre cell: 13, 56, 57, 59

• Langevin dynamics: 80

• large-temperature limit: 90

• latent variable: 153

• Lavenant criterion: 158, 169

• layer normalization: 162, 163

• least-square

  • inversion: 138

  • velocity: 138

• Lebesgue decomposition: 67, 105, 106

• Lebesgue measure: 9, 13, 14, 16, 35, 77

• Legendre transform: 54, 69, 118

• linear

  • convergence: 75, 93, 96, 97

  • functional: 144

  • linearization: 18, 132

  • OT: 110--112

  • tilt: 91

• linear form: 9, 10, 30, 152

• linear objective: 28, 31

• linear programming: i, 30, 31, 72, 115, 170

  • basic feasible solution: 29

  • finite-dimensional: 34, 60, 61

• Lipschitz

  • constant: 60

  • function: 14, 56, 60, 61, 63, 64

  • stability: 54

• Lipschitz stability: 54

• Lloyd algorithm: 59

• local

  • distance distribution: 45

  • matching indicator: 2

• local profile: 45

• localization: 18

• log-domain Sinkhorn: 78

• log-sum-exp: 51, 77, 83--85, 88

• lower semicontinuity: 67, 89, 104, 105, 118, 123, 125, 141

• lower-dimensional set: 15

• Lyapunov equation: 164

M¶

• marginal

  • constraint: 26, 29, 32, 33, 35, 73, 77, 79, 81, 85, 87, 91, 92, 95, 98, 106, 123, 133

  • divergence: 104, 105, 108

  • penalty: 104, 105, 108

  • relaxation: 104, 107

  • residual: 78, 96

• Markov kernel: 21, 45

• masking: 162

• mass

  • balance: 2, 57

  • conservation: 26, 73, 104

  • creation: 104

  • free: 59

  • variation: 108

• matching

  • algorithm: 6

  • non-crossing: 6

  • optimal: 1, 36

  • perfect: 6, 7

  • uniform: 4, 5, 30, 31, 35, 36

• matrix

  • cone: 127

  • incidence: 62

  • nonnegative transport: 4

  • permutation: 28--31, 129

  • polar decomposition: 16

  • scaling: 51, 71--73, 91

  • subproblem: 127

• matrix square root: 99

• matrix-valued measure: 125, 127, 128

• matrix-vector iteration: 72

• maximum mean discrepancy: 65, 66, 70, 101, 102, 111, 145, 160, 166, 167

• McCann interpolation: 17, 21, 40, 41, 114, 115, 140, 147, 149

• McKean-Vlasov limit: 145, 146

• mean-field

  • limit: 91, 150, 162

  • neural network: 142, 150

• measurable function: 9, 63

• measurable selection: 45, 53, 79, 80

• measure

  • atomless: 12

  • evolution: 137, 138, 144

  • isomorphism: 13

• measure-preserving

  • isometry: 129, 130

  • map: 16

• Memoli profile: 45, 131

• metric

  • equivalence: 113

  • learning: 46, 120

• metric space: 9

• metric-measure space: 45, 125, 129--131, 150

• minimax: 112, 113

• minimizing movement: 142, 144, 169

• minimizing movement scheme: 142

• Minkowski inequality: 14, 39, 109, 129, 130

• modulo isometry: 130

• moment matching: 163, 169

• momentum: 125, 126, 139

• Monge

  • distance: 13, 14, 16, 17, 41

  • geodesic: 16, 148

  • interpolation: 16, 20

  • problem: i, 1, 8, 12--14, 16, 19, 22, 23, 26, 34, 35, 40, 41, 115, 139, 147

• Monge-Ampere equation: 17, 18, 111

• Monge-Kantorovich equivalence: 35

• monotone

  • convergence: 93

  • field: 15

  • fixed point: 94

  • matching: 1, 3, 110

  • rearrangement: 2, 3, 19--22, 37

  • spectral gauge: 112

• multi-marginal: 115, 116, 119

  • cost: 119

  • formulation: 119

  • OT: 116, 119, 120

• multi-valued subdifferential: 15

• Muon algorithm: 112, 114

• mutual information: 82

N¶

• neuron: ii, 150--152, 167

• neuron law: 150, 167

• Newton step: 32, 72

• node feature: 132

• noising schedule: 156

• non-atomic probability space: 16

• nonnegative measure: 86, 87, 104, 106

• norm

  • flat: 63

  • RKHS: 65

  • Wasserstein-1: 60

• normal approximation: 47

• normalizing flow: 153

• north-west corner

  • plan: 28, 29

  • rule: 29, 32

O¶

• one-dimensional

  • projection: 13, 104, 108, 109, 167

  • transport: 18, 21, 29

• one-step

  • flow: 160

  • generative model: 160

• operator bound: 113

• operator norm: 10

• operator scaling: 133, 135

• operator-valued coupling: 136

• optimal coupling: 5, 18--20, 28, 31, 35, 36, 40, 49, 54, 59, 115, 118--122, 130, 139, 147, 149

• optimal plan: 15, 27--29, 36, 37, 52, 54, 74, 85, 122, 130, 147

• optimality

  • certificate: 50, 51

  • complementary slackness: 6, 7, 49--51, 122

  • first-order: 57, 92, 135

• order-preserving map: 94

• Ornstein-Uhlenbeck process: 157, 159, 165

P¶

• packing argument: 102

• pairwise distance: 129

• partial transport: 107

• particle ODE: 137, 146, 150

• path

  • measure: 139

  • space: 79--82

• path space: 41

• path-space

  • formulation: 71, 79--82, 140

  • problem: 80

  • transport: 79, 81

• Pearson divergence: 69

• permutation

  • matrix: 29

• Perron-Frobenius theorem: 95, 97

• perspective recession: 125

• perspective transform: 105, 106

• phi-divergence: 63, 66--70, 76, 85--87

• phi-divergence regularized OT: 85--88

• Pinsker inequality: 95, 96

• plan

  • interpolation: 40

  • lifted: 110, 114

  • optimal: 40

  • sparse: 27, 28

  • transport: 5, 26, 35, 40, 41, 49, 110, 114, 121, 125

• polar factorization: 15, 16

• polar formula: 113

• polar set: 112--114

• Polish space: 9, 12, 39, 44, 47, 123

• porous medium equation: 144

• positive

  • cone: 125, 126, 135

  • matrix-valued measure: 127

  • semidefinite matrix: 22--24, 112, 125, 128, 134

  • vector-valued measure: 125

• positive operator: 125, 133

• power diagram: 56, 57

• probabilistic coupling: 10, 12, 153

• probability measure: i, 8--12, 14, 20, 22, 32, 35, 37, 42, 44, 47, 52, 64--66, 68, 87, 108--110, 115, 116, 119, 129, 131, 139, 145, 151, 170

• probability path: 153

• probability simplex: 8, 87

• probability-flow ODE: 153, 155, 157, 159

• product

  • coupling: 26, 33, 35, 41, 88, 149, 157

  • measure: 74, 78, 88, 120

• profile lower bound: 45, 131

• projected formulation: 112

• projective

  • cone: 97

  • diameter: 97

  • geometry: 95

• Prokhorov theorem: 47

• propagation of chaos: 146

• pullback: 11

• push-forward: 8, 10--13, 19, 23, 24, 26, 138, 143, 153--155

• Pythagorean identity: 91, 92, 96

Q¶

• quadratic

  • assignment problem: 129

  • barycenter: 115, 119

  • closure: 99

  • potential: 15, 23, 24, 99, 164, 165

• quantile: 18

  • barycenter: 116

  • formula: 45

  • function: 3, 19--21, 109--111, 116, 117

  • map: 19, 20, 126

  • push-forward: 19

• quantization

  • energy: 59

  • optimal: 58

  • rate: 58

• quantum

  • coupling: 133

  • Kantorovich duality: 133

  • optimal transport: 133

  • OT: 125, 133, 134

  • OT duality: 134

  • Sinkhorn: 135

R¶

• Rademacher bound: 103

• Radon

  • measure: 9, 10, 36, 52

  • transform: 108

• Radon-Nikodym derivative: 155

• random probability measure: 44

• random variable: 10, 12, 35, 42, 48, 164

• rank constraint: 112

• rate

  • parametric: 102, 103

  • sublinear: 58, 95

• rational weights: 4, 5, 44

• recession convention: 104, 105, 125, 139

• rectifiable set: 15

• reference

  • distribution: 153

  • measure: 9, 16, 70, 76, 77, 86, 87, 105, 111, 112, 117, 153

• reflow: 153

• regression loss: 153

• regular conditional distribution: 21

• regularity theory: 17, 18

• reverse flow: 160

• reverse formulation: 105, 106

• reverse KL divergence: 69, 144

• Ricci curvature: 150

• Riemann sum: 42

• Riemannian manifold: 150

• Riesz potential: 65

• Riesz-Markov-Kakutani theorem: 9, 10

• RKHS: 64--66, 70, 102, 111

• robust

  • envelope: 47

  • representation: 113

  • Wasserstein: 112, 113

  • Wasserstein distance: 112, 113

• robustness

  • distributional: 46

  • distributionally robust optimization: 46

  • Wasserstein infinity: 47

• rotational component: 20, 158

• row normalization: 73

S¶

• sample complexity: 59, 91, 101

• scaling

  • algorithm: 71, 120

  • column: 74, 91, 93

  • form: 72, 73, 82, 93, 118, 135

  • iterative proportional fitting: 72, 73

  • potential: 74

  • row: 74, 91, 93

  • vectors: 93, 96

• Schrodinger

  • bridge: 80--82

  • interpolation: 80

  • problem: 71, 79--81, 90

  • system: 93

• score function: 144, 155, 156, 159, 160

• score-based generative modeling: 153

• score-SDE: 153

• second boundary condition: 18

• second moment: 167

• self Sinkhorn: 90

• self-concordance: 72

• self-corrected field: 161

• semi-discrete

  • dual: 57

  • Monge map: 13

  • OT: 13, 34, 53, 56--58, 84, 111

• semi-dual: 56, 57, 118

• semi-relaxed

  • divergence: 161

  • problem: 53

• semidefinite program: 133

• separable space: 9

• Shannon

  • entropy: 67, 68, 71, 134, 144, 145, 148, 149

  • negative entropy: 91, 92, 144

• shearing component: 20, 158

• signed

  • measure: 10, 20, 42, 60, 65, 142, 143, 145

  • negative part: 62

  • positive part: 62

• simplex network: 32

• Sinkhorn

  • algorithm: 51, 55, 62, 71--73, 75, 83, 97, 98

  • barycenter: 117, 141

  • convergence: 91, 93--96

  • coupling: 81

  • divergence: 71, 88--90, 100--103, 147, 160, 166

  • dual: 82, 85, 95, 167

  • half-step: 75, 92, 96

  • iteration: 75, 89, 93, 99, 118, 120, 122, 132

  • scaling: i, 6, 51, 73, 74, 85, 88, 92, 94, 107, 135, 161, 170

  • update: 135

• Slater condition: 133, 134

• sliced Wasserstein

  • distance: 108--111, 167

  • max: 109

  • min transport: 110

• Sobolev norm: 90

• soft

  • c-transform: 82--84, 88, 94, 95, 99, 103

  • minimum: 51, 83, 84

  • transform: 71, 83--85, 91, 93--95, 99, 101, 107

• spanning tree: 32

• spectral

  • gauge: 104, 112--114

  • Wasserstein: 112, 114

  • Wasserstein distance: 112

• spectral dynamics: 114

• splitting obstruction: 13, 15, 18

• static-dynamic equivalence: 141

• stationarity: 55, 132, 134, 151, 152

• stationary condition: 152

• stationary density: 152

• Stein method: 47, 48

• Stiefel manifold: 109

• stochastic

  • gradient: 58, 70, 155

  • interpolant: 153

  • optimization: 58

  • particle: 145

• Strang splitting: 135

• strict

  • convexity: 18, 71, 76, 78, 84, 92

  • positivity: 90, 95

• subdifferential: 15, 18, 35, 120

• subgradient inequality: 152

• subspace

  • projection: 110

  • robust Wasserstein: 113

  • sliced Wasserstein: 109

• superposition principle: 139

• support: 9, 11, 13, 15, 31, 35, 49, 52, 61

• synthetic Ricci curvature: 150

T¶

• teacher distribution: 145, 146

• teacher kernel mean: 145

• tensor product

  • coupling: 33

• tightness: 7, 9, 20, 35, 44, 109

• time-dependent vector field: 137, 153

• token limit: 162

• token measure: 153, 162

• topical map: 94, 95

• topology

  • strong: 43, 68

  • weak: 35, 42--44, 47, 52, 68

• total variation: 8, 10, 37, 42--44, 63, 64, 68, 69, 107, 108

• trace

  • gauge: 114

  • partial: 133, 136

  • partial constraint: 133

• trace entropy: 134

• trace norm: 113, 135

• trace-class operator: 136

• transformer: 162, 163, 166, 168, 170

• transport equation: 163

• transport map: 2, 12, 13, 16, 21, 26, 49, 122, 155, 158

• transportation

  • polytope: 26--29, 32, 33, 71, 72, 77, 92, 96, 129

  • simplex: 29, 32

• triangle inequality: 6, 14, 17, 25, 35, 38--40, 60, 97, 98, 102, 106, 109, 113, 129--131

• triangular

  • map: 21

  • rearrangement: 21, 22

• trivial coupling: 153

• Tweedie identity: 155, 156, 159

• two-layer neural network: 151, 152, 167

U¶

• unbalanced

  • barycenter: 141

  • OT: 76, 104--107, 121, 140, 141

  • OT dual: 105

V¶

• variation seminorm: 94, 95

• variational dual formula: 69

• vector quantile: 110, 111

• vector-valued measure: 125--127

• velocity field: 137, 138, 140, 143, 153, 162, 163, 167

• viscous Benamou-Brenier formulation: 80

• von Neumann entropy: 134

• Voronoi cell: 13, 57, 59

W¶

• Wasserstein

  • barycenter: 112, 115--117, 119, 169

  • coordinate: 111

  • distance: i, 20, 26, 37--43, 45--47, 68, 70, 104, 106, 108, 109, 112, 113, 115, 131, 139, 140, 149, 166, 168

  • flow: 148

  • formula: 20

  • geodesic: 16, 40, 115, 130, 166

  • gradient: 138, 142, 143, 145, 146, 148, 149, 151--153, 160--165, 169

  • gradient flow: 16, 86, 138, 142, 143, 145--149, 152, 153, 160, 163, 169

  • infinity distance: 47

  • over Wasserstein: 44, 45

  • space: 44, 45, 144, 167

• Wasserstein topology: 43, 44

• Wasserstein-Fisher-Rao: 141

• weak

  • convergence: 9, 35, 41--45, 47, 63, 64, 66--68, 108, 109, 113

  • cost: 123, 124

  • Kantorovich duality: 123

  • limit: 13, 22, 42

  • map: 18

  • optimal transport: 122

  • transport: 123

• weighted

  • Poisson equation: 18, 138

  • sweep: 29

Z¶

• zero mass: 22, 57, 65, 72

• zero mean: 138