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clouds","\u002Falgorithms\u002Ftransport\u002Ftutorials\u002Fpoint-clouds","2.algorithms\u002F2.transport\u002F7.tutorials\u002F3.point-clouds",{"title":102,"path":103,"stem":104,"children":105},"Resources","\u002Fresources","3.resources",[106,108,147,151,155],{"title":102,"path":103,"stem":107},"3.resources\u002Findex",{"title":41,"path":109,"stem":110,"children":111},"\u002Fresources\u002Ftutorials","3.resources\u002F1.tutorials\u002Findex",[112,113,131],{"title":41,"path":109,"stem":110},{"title":16,"path":114,"stem":115,"children":116,"page":130},"\u002Fresources\u002Ftutorials\u002Fassignment","3.resources\u002F1.tutorials\u002Fassignment",[117,122,126],{"title":118,"path":119,"stem":120,"icon":121},"Tutorial 1 — The Assignment Problem","\u002Fresources\u002Ftutorials\u002Fassignment\u002F01_the_assignment_problem","3.resources\u002F1.tutorials\u002Fassignment\u002F01_the_assignment_problem","i-lucide-notebook",{"title":123,"path":124,"stem":125,"icon":121},"Tutorial 2 — Backends and Batching","\u002Fresources\u002Ftutorials\u002Fassignment\u002F02_backends_and_batching","3.resources\u002F1.tutorials\u002Fassignment\u002F02_backends_and_batching",{"title":127,"path":128,"stem":129,"icon":121},"Tutorial 3 — Object Tracking with the Assignment Problem","\u002Fresources\u002Ftutorials\u002Fassignment\u002F03_object_tracking","3.resources\u002F1.tutorials\u002Fassignment\u002F03_object_tracking",false,{"title":59,"path":132,"stem":133,"children":134,"page":130},"\u002Fresources\u002Ftutorials\u002Ftransport","3.resources\u002F1.tutorials\u002Ftransport",[135,139,143],{"title":136,"path":137,"stem":138,"icon":121},"Tutorial 1 — What Is Optimal Transport?","\u002Fresources\u002Ftutorials\u002Ftransport\u002F01_optimal_transport","3.resources\u002F1.tutorials\u002Ftransport\u002F01_optimal_transport",{"title":140,"path":141,"stem":142,"icon":121},"Tutorial 2 — The Sinkhorn Algorithm","\u002Fresources\u002Ftutorials\u002Ftransport\u002F02_sinkhorn_algorithm","3.resources\u002F1.tutorials\u002Ftransport\u002F02_sinkhorn_algorithm",{"title":144,"path":145,"stem":146,"icon":121},"Tutorial 3 — Point-Cloud OT and Shape Learning","\u002Fresources\u002Ftutorials\u002Ftransport\u002F03_point_clouds","3.resources\u002F1.tutorials\u002Ftransport\u002F03_point_clouds",{"title":148,"path":149,"stem":150},"Assignment applications","\u002Fresources\u002Fassignment-applications","3.resources\u002F2.assignment-applications",{"title":152,"path":153,"stem":154},"Transport applications","\u002Fresources\u002Ftransport-applications","3.resources\u002F3.transport-applications",{"title":156,"path":157,"stem":158,"children":159},"Benchmarks","\u002Fresources\u002Fbenchmarks","3.resources\u002F4.benchmarks\u002Findex",[160,161],{"title":156,"path":157,"stem":158},{"title":162,"path":163,"stem":164},"Contributing benchmarks","\u002Fresources\u002Fbenchmarks\u002Fcontributing","3.resources\u002F4.benchmarks\u002Fcontributing",{"title":166,"path":167,"stem":168,"icon":169},"API Reference","\u002Fapi","4.api","i-lucide-package",{"title":171,"path":172,"stem":173},"References","\u002Freferences","5.references",{"id":175,"title":152,"api":176,"body":177,"description":455,"extension":456,"links":176,"meta":457,"navigation":176,"path":153,"seo":458,"stem":154,"__hash__":459},"docs\u002F3.resources\u002F3.transport-applications.md",null,{"type":178,"value":179,"toc":442},"minimark",[180,184,189,197,213,217,225,232,236,247,251,261,271,287,305,309,316,344,351,355,365,374,387,391,400,407,411,420,433,439],[181,182,183],"p",{},"Optimal transport has one of the longest application histories in applied mathematics — 240\nyears from Monge's engineering problem to today's generative model training loops. This page\ntraces where the problem arises, why OT is the right formulation, and how torchmatch's\nbackends map onto each use case.",[185,186,188],"h2",{"id":187},"supply-chain-and-resource-allocation","Supply chain and resource allocation",[181,190,191,192,196],{},"The first computational use of the transportation problem (the discrete precursor to general\nOT) was in ",[193,194,195],"strong",{},"centrally-planned resource allocation",": given factories producing a good and\nwarehouses needing it, at what shipping cost should supply be routed to minimise total\nfreight? Kantorovich developed the LP formulation for exactly this purpose in wartime Soviet\nplanning, and the 1975 Nobel Prize citation mentions \"optimal allocation of resources\" as\nthe core contribution. Koopmans, sharing the prize, applied the same structure to shipping\nroute optimisation.",[181,198,199,200,204,205,208,209,212],{},"The transportation LP — assign continuous mass from supply locations to demand locations at\nminimum unit transport cost — is the special case where ",[201,202,203],"code",{},"a"," and ",[201,206,207],{},"b"," are given histograms\nand ",[201,210,211],{},"C"," encodes geographic distance. It remains a daily tool in logistics optimisation and\nsupply chain planning.",[185,214,216],{"id":215},"image-retrieval-and-perceptual-similarity","Image retrieval and perceptual similarity",[181,218,219,220,224],{},"The Earth Mover's Distance entered computer vision through image retrieval\n",[221,222],"docyard-cite",{"bib":223},"Rubner2000",". When comparing two images by their colour histograms, Euclidean distance\nbetween bin counts is insensitive to perceptually small shifts: moving probability mass from\nthe \"red\" bin to the \"orange\" bin registers as a large L2 distance but a small perceptual\ndistance. OT charges the actual cost of moving mass between bins according to the\ncolour-space ground metric (a distance function defined over the colour space, e.g. Euclidean distance in RGB or Lab), so nearby colours are treated as similar.",[181,226,227,228,231],{},"The same idea extends to texture descriptors, shape histograms, and any feature whose\nnatural distance is not Euclidean on the raw representation. The ",[201,229,230],{},"EXACT_EMD"," backend in\ntorchmatch computes the exact earth mover's distance for small histograms where no\nregularisation is desired.",[185,233,235],{"id":234},"colour-transfer-and-style-matching","Colour transfer and style matching",[181,237,238,239,242,243,246],{},"Histogram matching via OT underlies a family of image stylisation techniques. Given a\nsource image and a target whose colour statistics you want to adopt, computing the 1D or 3D\nOT map between their colour histograms and applying it as a pixel-wise transformation\ntransfers the palette of the target to the source with minimal perceptual distortion.\nPitié, Kokaram, and Dahyot ",[221,240],{"bib":241},"Pitie2007"," formalised this as iterated 1D projections\n(sliced Wasserstein transport — an approximation that computes 1D OT on random projections of the distribution, avoiding the full N-dimensional cost), an approximation that scales to full 3D colour histograms\nwithout materialising an ",[201,244,245],{},"n³"," cost.",[185,248,250],{"id":249},"generative-modelling","Generative modelling",[181,252,253,256,257,260],{},[193,254,255],{},"Wasserstein GAN"," ",[221,258],{"bib":259},"Arjovsky2017"," reframed generative adversarial training in terms of\nthe Wasserstein-1 distance between the real and generated distributions. The Wasserstein\ndistance is meaningful even when the two distributions have disjoint support (which\ncommonly occurs during early GAN training), whereas Jensen-Shannon divergence and KL\ndivergence both saturate to a constant in this regime, providing no gradient signal. The\npractical implementation uses the Kantorovich-Rubinstein dual (the equivalent formulation of W1 as a supremum over 1-Lipschitz functions, used to train the discriminator) with gradient-penalised\ndiscriminators rather than directly computing the OT plan, but the connection motivates the\nloss design.",[181,262,263,266,267,270],{},[193,264,265],{},"Wasserstein autoencoders"," (WAE) ",[221,268],{"bib":269},"Tolstikhin2018"," replaced the evidence lower bound\nof the VAE with a Wasserstein distance between the aggregate posterior and the prior,\nproducing sharper reconstructions on image benchmarks.",[181,272,273,256,276,286],{},[193,274,275],{},"Flow matching",[277,278,279,282,283],"span",{},[221,280],{"bib":281},"Lipman2022","; ",[221,284],{"bib":285},"Liu2022"," frames diffusion-model training as learning a\nvector field that transports a source distribution (Gaussian noise) to a target\ndistribution (data). The \"OT-conditioned flow matching\" variant conditions the flow on the\nOT displacement plan between individual noise samples and data samples, producing straighter\ntrajectories and faster inference.",[181,288,289,290,293,294,297,298,293,301,304],{},"For all three model classes, ",[201,291,292],{},"transport.matrix.solve"," with ",[201,295,296],{},"SINKHORN_DIVERGENCE"," or\n",[201,299,300],{},"transport.samples.loss",[201,302,303],{},"debias=True"," provides a differentiable Wasserstein-like\ntraining loss that can be substituted for or combined with the standard objectives.",[185,306,308],{"id":307},"domain-adaptation","Domain adaptation",[181,310,311,312,315],{},"A model trained on a labelled source domain often fails on an unlabelled target domain\nbecause the marginal feature distributions differ. OT provides a principled way to\n",[193,313,314],{},"measure and correct this discrepancy",": it computes a transport plan between source and target feature distributions, then uses that plan to move source features toward the target domain before training a classifier.",[317,318,319,329,338],"ul",{},[320,321,322,256,325,328],"li",{},[193,323,324],{},"OTDA",[221,326],{"bib":327},"Courty2017"," computes the regularised OT plan between labelled source samples\nand unlabelled target samples in the feature space of a pretrained network. The plan then\ntransports source features toward the target, producing pseudo-labelled target\nsamples that train a target-domain classifier.",[320,330,331,256,334,337],{},[193,332,333],{},"DeepJDOT",[221,335],{"bib":336},"Damodaran2018"," integrates OT alignment into an end-to-end deep network:\nthe OT plan between source and target minibatches is recomputed each iteration and used\nas a re-weighting of the classification loss.",[320,339,340,343],{},[193,341,342],{},"Distribution matching for dataset distillation",": the Sinkhorn divergence between\nfeature distributions of real and distilled datasets serves as the distillation\nobjective, requiring a geometry-aware distance that respects the ground metric of feature\nspace.",[181,345,346,347,350],{},"The ",[201,348,349],{},"UNBALANCED_SINKHORN"," backend is particularly useful when source and target have\ngenuinely different class distributions: relaxing the marginal constraints prevents the\ncoupling from being dominated by classes that are abundant in one domain but rare in the\nother.",[185,352,354],{"id":353},"geometric-deep-learning-and-3d-vision","Geometric deep learning and 3D vision",[181,356,357,360,361,364],{},[193,358,359],{},"Point cloud registration and shape matching"," require finding correspondences between two\nunordered point sets. OT provides the soft coupling that minimises transport cost, giving\na probabilistic correspondence matrix as the plan. This soft matching initialises or\nreplaces ICP (iterative closest point) in registration pipelines, is differentiable with\nrespect to the point positions, and scales to large point clouds via the ",[201,362,363],{},"samples.loss","\nstreaming kernel.",[181,366,367,256,370,373],{},[193,368,369],{},"Wasserstein barycenters",[221,371],{"bib":372},"Agueh2011"," — weighted averages of distributions in Wasserstein\nspace — produce interpolations between shapes that respect the geometry of the ground\nmetric. Interpolating between two 3D shapes in Wasserstein space moves each point smoothly\ntoward its corresponding point in the target, unlike Euclidean averaging which collapses\nthe shape when distributions are disjoint.",[181,375,376,379,380,382,383,386],{},[193,377,378],{},"Shape completion and generation",": point-cloud generative models (PointFlow,\nShapeGF, DPM-based point clouds) use Wasserstein or Sinkhorn losses to supervise the\ngenerated shape. ",[201,381,300],{}," is directly applicable; the streaming Triton\nkernel avoids the ",[201,384,385],{},"N × M"," allocation that would dominate memory at generation-scale cloud\nsizes.",[185,388,390],{"id":389},"natural-language-processing","Natural language processing",[181,392,393,256,396,399],{},[193,394,395],{},"Word Mover's Distance",[221,397],{"bib":398},"Kusner2015"," embeds documents as distributions over word\nembedding vectors (weighted by TF-IDF or uniform) and computes the OT cost between them\nusing pretrained embeddings as the ground metric. The resulting distance is insensitive to\nsynonyms and paraphrases — moving mass from \"automobile\" to \"car\" is cheap because the\nembeddings are nearby — and outperforms bag-of-words similarity on several retrieval\nbenchmarks.",[181,401,402,403,406],{},"The same idea applies to ",[193,404,405],{},"sentence-level alignment"," for mining parallel corpora, where\nOT between sentence embedding distributions identifies likely translations without requiring\nexact string matches.",[185,408,410],{"id":409},"computational-biology","Computational biology",[181,412,413,256,416,419],{},[193,414,415],{},"Waddington-OT",[221,417],{"bib":418},"Schiebinger2019"," modelled cellular differentiation as an OT problem:\nsingle-cell RNA-seq profiles from adjacent time points define the source and target\ndistributions over gene-expression space, and the OT plan between them gives the most\nparsimonious account of which early cells give rise to which later cells. This transport\ninterpretation respects the Waddington epigenetic landscape metaphor and produces\nbiologically interpretable developmental trajectories without requiring paired data.",[181,421,422,256,425,428,429,432],{},[193,423,424],{},"Moscot",[221,426],{"bib":427},"Klein2023"," scaled this framework to million-cell datasets and extended it to\nspatial transcriptomics alignment, single-cell multi-omics integration (matching cells\nacross RNA and protein measurements), and lineage tracing. The computational backbone is\nlog-domain Sinkhorn with online batching — structurally identical to ",[201,430,431],{},"LOG_SINKHORN"," —\napplied at dataset scales that require careful memory management.",[181,434,435,438],{},[193,436,437],{},"Protein structure comparison",": OT between residue-coordinate distributions provides a\nrotation-invariant distance between two protein chains that accounts for insertions,\ndeletions, and loop flexibility without requiring global structural alignment.",[440,441],"bibliography",{},{"title":443,"searchDepth":444,"depth":444,"links":445},"",3,[446,448,449,450,451,452,453,454],{"id":187,"depth":447,"text":188},2,{"id":215,"depth":447,"text":216},{"id":234,"depth":447,"text":235},{"id":249,"depth":447,"text":250},{"id":307,"depth":447,"text":308},{"id":353,"depth":447,"text":354},{"id":389,"depth":447,"text":390},{"id":409,"depth":447,"text":410},"Historical and modern applications of optimal transport — from Monge's earth-moving problem through Wasserstein GANs, single-cell genomics, domain adaptation, and geometric deep learning.","md",{},{"title":152,"description":455},"JBIIbxxnPKZ7r_0cIDZtHGy3WO8ec_ntzJcuEtC4WgE",[461,463],{"title":148,"path":149,"stem":150,"description":462,"children":-1},"Historical and modern applications of the linear assignment problem — from 1950s operations research to DETR, multi-object tracking, and cluster evaluation in contemporary deep learning.",{"title":156,"path":157,"stem":158,"description":464,"children":-1},"Per-op benchmark sweep across problem sizes, dtypes, and devices — covering assignment (single-problem, batched) and transport (matrix-face Sinkhorn\u002FEMD, samples-face Triton) ops.",[466,476,482,493,504,513,523,531,540,550,559,569,578,587,597,605,615,626,634,643,650,660,670,679,687,697,704,713,722,730,738,745,755,761,770,779,787,796,801,805,809,818,825,834,840,848],{"key":467,"type":468,"author":469,"title":470,"journal":471,"year":472,"volume":473,"pages":474,"note":475},"Konig1931","article","Kőnig, Dénes","Gráfok és mátrixok","Matematikai és Fizikai Lapok","1931","38","116–119","Hungarian. Origin of Kőnig's theorem on bipartite matchings and covers.",{"key":477,"type":468,"author":478,"title":479,"journal":471,"year":472,"volume":473,"pages":480,"note":481},"Egervary1931","Egerváry, Jenő","Matrixok kombinatorius tulajdonságairól","16–28","Hungarian. Weighted extension of Kőnig's theorem. Translated to English by H. W. Kuhn (1955) as “On Combinatorial Properties of Matrices”, Logistics Papers Issue 11, George Washington University.",{"key":483,"type":468,"author":484,"title":485,"journal":486,"year":487,"volume":488,"number":489,"pages":490,"doi":491,"note":492},"Kuhn1955","Kuhn, Harold W.","The Hungarian method for the assignment problem","Naval Research Logistics Quarterly","1955","2","1--2","83–97","10.1002\u002Fnav.3800020109","Coined the name “Hungarian method” in tribute to Kőnig and Egerváry; first explicit O(n^4) algorithm for the LAP.",{"key":494,"type":468,"author":495,"title":496,"journal":497,"year":498,"volume":499,"number":500,"pages":501,"doi":502,"note":503},"Munkres1957","Munkres, James","Algorithms for the assignment and transportation problems","Journal of the Society for Industrial and Applied Mathematics","1957","5","1","32–38","10.1137\u002F0105003","Six-step state machine on primed and starred zeros under row\u002Fcolumn covers; canonical textbook formulation of the Hungarian method. Implemented as torchmatch.munkres.",{"key":505,"type":468,"author":506,"title":507,"journal":508,"year":509,"volume":500,"number":488,"pages":510,"doi":511,"note":512},"Tomizawa1971","Tomizawa, Nobuaki","On some techniques useful for solution of transportation network problems","Networks","1971","173–194","10.1002\u002Fnet.3230010206","Independent reduction of the Hungarian method to O(n^3) via shortest augmenting paths; predates Edmonds & Karp (1972).",{"key":514,"type":468,"author":515,"title":516,"journal":517,"year":518,"volume":519,"number":488,"pages":520,"doi":521,"note":522},"EdmondsKarp1972","Edmonds, Jack and Karp, Richard M.","Theoretical improvements in algorithmic efficiency for network flow problems","Journal of the ACM","1972","19","248–264","10.1145\u002F321694.321699","Independent O(n^3) improvement to the Hungarian method via shortest augmenting paths in network flow; contemporary with [Tomizawa1971].",{"key":524,"type":525,"author":526,"title":527,"publisher":528,"year":529,"note":530},"Lawler1976","book","Lawler, Eugene L.","Combinatorial Optimization: Networks and Matroids","Holt, Rinehart and Winston","1976","Tree-augmentation reformulation of the Hungarian inner loop; basis for the GPU-parallel BFS structure used in torchmatch.lawler.",{"key":532,"type":533,"author":534,"title":535,"year":536,"url":537,"institution":538,"note":539},"Bertsekas1979","techreport","Bertsekas, Dimitri P.","A distributed algorithm for the assignment problem","1979","https:\u002F\u002Fweb.mit.edu\u002Fdimitrib\u002Fwww\u002FBertsekas_Auction_Distributed_1979.pdf","Laboratory for Information and Decision Systems, MIT","Original distributed-bidding (“auction”) algorithm for the LAP; precursor to the ε-scaling version in [Bertsekas1988].",{"key":541,"type":468,"author":542,"title":543,"journal":544,"year":545,"volume":473,"number":546,"pages":547,"doi":548,"note":549},"JonkerVolgenant1987","Jonker, Roy and Volgenant, Anton","A shortest augmenting path algorithm for dense and sparse linear assignment problems","Computing","1987","4","325–340","10.1007\u002FBF02278710","Dijkstra-based SSP with column-reduction warm start and reduction transfer; basis for torchmatch.jonker\\_* on CPU.",{"key":551,"type":468,"author":534,"title":552,"journal":553,"year":554,"volume":555,"number":500,"pages":556,"doi":557,"note":558},"Bertsekas1988","The auction algorithm: a distributed relaxation method for the assignment problem","Annals of Operations Research","1988","14","105–123","10.1007\u002FBF02186476","Auction with ε-scaling; convergence guarantees. Widely cited follow-up to [Bertsekas1979].",{"key":560,"type":468,"author":561,"title":562,"journal":563,"year":564,"volume":565,"number":488,"pages":566,"doi":567,"note":568},"GoldbergKennedy1995","Goldberg, Andrew V. and Kennedy, Robert","An efficient cost scaling algorithm for the assignment problem","Mathematical Programming","1995","71","153–177","10.1007\u002FBF01585996","Push-relabel cost-scaling for the LAP via min-cost flow reduction.",{"key":570,"type":571,"author":484,"title":485,"booktitle":572,"publisher":573,"year":574,"pages":575,"doi":576,"note":577},"Kuhn2010Variants","incollection","50 Years of Integer Programming 1958–2008","Springer","2010","29–47","10.1007\u002F978-3-540-68279-0_2","Author's retrospective on the 1955 paper; clarifies the attribution chain to Kőnig and Egerváry.",{"key":579,"type":525,"author":580,"title":581,"publisher":582,"year":583,"doi":584,"edition":585,"note":586},"BurkardDellAmicoMartello2012","Burkard, Rainer E. and Dell'Amico, Mauro and Martello, Silvano","Assignment Problems","Society for Industrial and Applied Mathematics","2012","10.1137\u002F1.9781611972238","Revised reprint","Comprehensive treatment of LAP variants, complexity bounds, and empirical comparisons. Original 2009; this is the 2012 revised reprint with corrections.",{"key":588,"type":468,"author":589,"title":590,"journal":591,"year":592,"volume":593,"number":546,"pages":594,"doi":595,"note":596},"Crouse2016","Crouse, David F.","On implementing 2D rectangular assignment algorithms","IEEE Transactions on Aerospace and Electronic Systems","2016","52","1679–1696","10.1109\u002FTAES.2016.140952","Rectangular-native reformulation of Jonker-Volgenant; basis for torchmatch.jonker\\_dense and the tiled CUDA backend of torchmatch.jonker\\_dense\\_batch.",{"key":598,"type":468,"author":599,"title":600,"journal":601,"year":602,"pages":603,"note":604},"Monge1781","Monge, Gaspard","Mémoire sur la théorie des déblais et des remblais","Histoire de l'Académie Royale des Sciences","1781","666–704","Foundational deterministic mass-transport problem; no convex relaxation, making it hard to solve in general.",{"key":606,"type":468,"author":607,"title":608,"journal":609,"year":610,"volume":611,"number":612,"pages":613,"note":614},"Kantorovich1942","Kantorovich, Leonid V.","On the translocation of masses","Doklady Akademii Nauk USSR","1942","37","7--8","199–201","LP relaxation of [Monge1781]: couplings instead of bijections; strong duality yields dual variables interpretable as prices.",{"key":616,"type":468,"author":617,"title":618,"journal":619,"year":620,"volume":621,"number":622,"pages":623,"doi":624,"note":625},"BenamouBrenier2000","Benamou, Jean-David and Brenier, Yann","A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem","Numerische Mathematik","2000","84","3","375–393","10.1007\u002Fs002110050263","Dynamic OT: W_2^2 equals the minimum kinetic energy to deform one distribution into the other; connects OT to PDEs.",{"key":223,"type":468,"author":627,"title":628,"journal":629,"year":620,"volume":630,"number":488,"pages":631,"doi":632,"note":633},"Rubner, Yossi and Tomasi, Carlo and Guibas, Leonidas J.","The earth mover's distance as a metric for image retrieval","International Journal of Computer Vision","40","99–121","10.1023\u002FA:1026543900054","Introduced the Earth Mover's Distance (EMD) as a practical histogram similarity measure; established OT as a tool in computer vision.",{"key":635,"type":525,"author":636,"title":637,"publisher":638,"year":639,"volume":640,"doi":641,"note":642},"Villani2003","Villani, Cédric","Topics in Optimal Transportation","American Mathematical Society","2003","58","10.1090\u002Fgsm\u002F058","First of Villani's two monographs on OT; introduces displacement interpolation and unifies geometry, analysis, and probability.",{"key":644,"type":525,"author":636,"title":645,"publisher":573,"year":646,"volume":647,"doi":648,"note":649},"Villani2008","Optimal Transport: Old and New","2008","338","10.1007\u002F978-3-540-71050-9","Second of Villani's monographs; comprehensive treatment including the Brenier map, regularity, and geometric applications. Fields Medal (2010) awarded partly for this body of work.",{"key":651,"type":468,"author":652,"title":653,"journal":654,"year":655,"volume":656,"number":546,"pages":657,"doi":658,"note":659},"Sinkhorn1967","Sinkhorn, Richard","Diagonal equivalence to matrices with prescribed row and column sums","The American Mathematical Monthly","1967","74","402–405","10.2307\u002F2314570","Proves convergence of the Sinkhorn–Knopp matrix-scaling iteration; foundational for all entropic OT solvers.",{"key":661,"type":662,"author":663,"title":664,"booktitle":665,"publisher":666,"year":667,"volume":668,"note":669},"Cuturi2013","inproceedings","Cuturi, Marco","Sinkhorn distances: lightspeed computation of optimal transportation distances","Advances in Neural Information Processing Systems","Curran Associates","2013","26","Entropic regularisation of OT; reduction to Sinkhorn iteration enables large-scale batched OT on GPU. Direct ancestor of the LOG\\_SINKHORN backend in torchmatch.",{"key":259,"type":662,"author":671,"title":672,"booktitle":673,"publisher":674,"year":675,"volume":676,"pages":677,"note":678},"Arjovsky, Martin and Chintala, Soumith and Bottou, Léon","Wasserstein generative adversarial networks","International Conference on Machine Learning","PMLR","2017","70","214–223","1-Wasserstein GAN objective; meaningful gradient signal even under disjoint support, where Jensen-Shannon divergence saturates.",{"key":680,"type":662,"author":681,"title":682,"booktitle":683,"publisher":674,"year":684,"volume":621,"pages":685,"note":686},"Genevay2018","Genevay, Aude and Peyré, Gabriel and Cuturi, Marco","Learning generative models with Sinkhorn divergences","International Conference on Artificial Intelligence and Statistics","2018","1608–1617","Defines the Sinkhorn divergence by debiasing the entropic OT loss; recovers a proper symmetric divergence zero iff distributions match.",{"key":688,"type":468,"author":689,"title":690,"journal":691,"year":684,"volume":692,"number":693,"pages":694,"doi":695,"note":696},"Chizat2018","Chizat, Lenaïc and Peyré, Gabriel and Vialard, François-Xavier and Schmitzer, Bernhard","Scaling algorithms for unbalanced optimal transport problems","Mathematics of Computation","87","314","2563–2609","10.1090\u002Fmcom\u002F3303","KL-relaxed marginals for unbalanced OT; softened Sinkhorn iteration handles mass differences and is robust to outliers.",{"key":698,"type":468,"author":699,"title":700,"journal":701,"year":702,"note":703},"Sejourne2019","Séjourné, Thibault and Feydy, Jean and Vialard, François-Xavier and Trouvé, Alain and Peyré, Gabriel","Sinkhorn divergences for unbalanced optimal transport","arXiv preprint arXiv:1910.12958","2019","Extends the Sinkhorn divergence to unbalanced marginals; convergence analysis in the unbalanced setting.",{"key":705,"type":468,"author":706,"title":707,"journal":708,"year":702,"volume":709,"number":622,"pages":710,"doi":711,"note":712},"Schmitzer2019","Schmitzer, Bernhard","Stabilized sparse scaling algorithms for entropy regularized transport problems","SIAM Journal on Scientific Computing","41","A1443–A1481","10.1137\u002F16M1106018","Log-domain (Gibbs-potential) Sinkhorn; numerically stable for all regularisation strengths and cost magnitudes.",{"key":714,"type":468,"author":715,"title":716,"journal":563,"year":717,"volume":718,"number":488,"pages":719,"doi":720,"note":721},"Orlin1997","Orlin, James B.","A polynomial time primal network simplex algorithm for minimum cost flows","1997","78","109–129","10.1007\u002FBF02614374","Network simplex with polynomial worst-case guarantee; basis for the EXACT\\_EMD backend in torchmatch.",{"key":723,"type":468,"author":724,"title":725,"journal":726,"year":667,"volume":709,"number":499,"pages":727,"doi":728,"note":729},"Sejdinovic2013","Sejdinovic, Dino and Sriperumbudur, Bharath and Gretton, Arthur and Fukumizu, Kenji","Equivalence of distance-based and RKHS-based statistics in hypothesis testing","The Annals of Statistics","2263–2291","10.1214\u002F13-AOS1140","Proves that MMD with energy-distance kernels equals an OT-based statistic; the connection between Sinkhorn divergence, OT, and MMD.",{"key":731,"type":662,"author":732,"title":733,"booktitle":734,"year":592,"pages":735,"doi":736,"note":737},"Bewley2016","Bewley, Alex and Ge, Zongyuan and Ott, Lionel and Ramos, Fabio and Upcroft, Ben","Simple online and realtime tracking","IEEE International Conference on Image Processing (ICIP)","3464–3468","10.1109\u002FICIP.2016.7533003","SORT; Kalman-filter motion prediction plus IoU cost and Hungarian assignment. Establishes the tracking-by-detection template extended by [Wojke2017], [Zhang2022], [Aharon2022], and [Cao2023].",{"key":739,"type":662,"author":740,"title":741,"booktitle":734,"year":675,"pages":742,"doi":743,"note":744},"Wojke2017","Wojke, Nicolai and Bewley, Alex and Paulus, Dietrich","Simple online and realtime tracking with a deep association metric","3645–3649","10.1109\u002FICIP.2017.8296962","DeepSORT; adds an appearance re-identification embedding distance to the SORT cost matrix in [Bewley2016].",{"key":746,"type":662,"author":747,"title":748,"booktitle":749,"year":750,"volume":751,"pages":752,"doi":753,"note":754},"Zhang2022","Zhang, Yifu and Sun, Peize and Jiang, Yi and Yu, Dongdong and Weng, Fucheng and Yuan, Zehuan and Luo, Ping and Liu, Wenyu and Wang, Xinggang","ByteTrack: Multi-object tracking by associating every detection box","European Conference on Computer Vision (ECCV)","2022","13682","1–21","10.1007\u002F978-3-031-20047-2_1","Associates low-confidence detections through a second Hungarian assignment pass on top of the SORT template.",{"key":756,"type":468,"author":757,"title":758,"journal":759,"year":750,"note":760},"Aharon2022","Aharon, Nir and Orfaig, Roy and Bobrovsky, Ben-Zion","BoT-SORT: Robust associations multi-pedestrian tracking","arXiv preprint arXiv:2206.14651","Camera-motion compensation and appearance embeddings on top of the SORT template; the Hungarian assignment step is unchanged.",{"key":762,"type":662,"author":763,"title":764,"booktitle":765,"year":766,"pages":767,"doi":768,"note":769},"Cao2023","Cao, Jinkun and Pang, Jiangmiao and Weng, Xinshuo and Khirodkar, Rawal and Kitani, Kris","Observation-centric SORT: Rethinking SORT for robust multi-object tracking","IEEE\u002FCVF Conference on Computer Vision and Pattern Recognition (CVPR)","2023","9686–9696","10.1109\u002FCVPR52729.2023.00934","Observation-centric re-update reduces motion-model drift under occlusion; same Hungarian assignment core as [Bewley2016].",{"key":771,"type":662,"author":772,"title":773,"booktitle":749,"year":774,"volume":775,"pages":776,"doi":777,"note":778},"Carion2020","Carion, Nicolas and Massa, Francisco and Synnaeve, Gabriel and Usunier, Nicolas and Kirillov, Alexander and Zagoruyko, Sergey","End-to-end object detection with transformers","2020","12346","213–229","10.1007\u002F978-3-030-58452-8_13","DETR; introduces the Hungarian-matching set-prediction training loss that the entire DETR lineage, including [Cheng2021], preserves.",{"key":780,"type":662,"author":781,"title":782,"booktitle":665,"year":783,"volume":784,"pages":785,"note":786},"Cheng2021","Cheng, Bowen and Schwing, Alexander G. and Kirillov, Alexander","Per-pixel classification is not all you need for semantic segmentation","2021","34","17864–17875","MaskFormer; recasts semantic segmentation as mask classification, preserving the Hungarian matching step from the DETR lineage ([Carion2020]).",{"key":241,"type":468,"author":788,"title":789,"journal":790,"year":791,"volume":792,"number":489,"pages":793,"doi":794,"note":795},"Pitié, François and Kokaram, Anil C. and Dahyot, Rozenn","Automated colour grading using colour distribution transfer","Computer Vision and Image Understanding","2007","107","123–137","10.1016\u002Fj.cviu.2006.11.011","Iterated 1D OT projections (sliced Wasserstein) for colour transfer; scales to full 3D colour histograms.",{"key":269,"type":662,"author":797,"title":798,"booktitle":799,"year":684,"note":800},"Tolstikhin, Ilya and Bousquet, Olivier and Gelly, Sylvain and Schölkopf, Bernhard","Wasserstein auto-encoders","International Conference on Learning Representations","WAE; replaces the VAE evidence lower bound with a Wasserstein distance between the aggregate posterior and the prior.",{"key":281,"type":662,"author":802,"title":803,"booktitle":799,"year":766,"note":804},"Lipman, Yaron and Chen, Ricky T. Q. and Ben-Hamu, Heli and Nickel, Maximilian and Le, Matt","Flow matching for generative modeling","Frames diffusion-model training as learning a vector field transporting a noise source distribution to the data target distribution; the OT-conditioned variant uses the OT displacement plan between individual samples.",{"key":285,"type":662,"author":806,"title":807,"booktitle":799,"year":766,"note":808},"Liu, Xingchao and Gong, Chengyue and Liu, Qiang","Flow straight and fast: Learning to generate and transfer data with rectified flow","Straightens [Lipman2022]-style flow-matching trajectories via an OT-conditioned reflow procedure, reducing inference sampling steps.",{"key":327,"type":468,"author":810,"title":811,"journal":812,"year":675,"volume":813,"number":814,"pages":815,"doi":816,"note":817},"Courty, Nicolas and Flamary, Rémi and Tuia, Devis and Rakotomamonjy, Alain","Optimal transport for domain adaptation","IEEE Transactions on Pattern Analysis and Machine Intelligence","39","9","1853–1865","10.1109\u002FTPAMI.2016.2615921","OTDA; transports labelled source features toward an unlabelled target domain via the regularised OT plan, then trains a target-domain classifier on the result.",{"key":336,"type":662,"author":819,"title":820,"booktitle":749,"year":684,"volume":821,"pages":822,"doi":823,"note":824},"Damodaran, Bharath Bhushan and Kellenberger, Benjamin and Flamary, Rémi and Tuia, Devis and Courty, Nicolas","DeepJDOT: Deep joint distribution optimal transport for unsupervised domain adaptation","11208","467–483","10.1007\u002F978-3-030-01225-0_28","Integrates [Courty2017]-style OT alignment into end-to-end training; the OT plan between minibatches is recomputed every iteration and reweights the classification loss.",{"key":372,"type":468,"author":826,"title":827,"journal":828,"year":829,"volume":830,"number":488,"pages":831,"doi":832,"note":833},"Agueh, Martial and Carlier, Guillaume","Barycenters in the Wasserstein space","SIAM Journal on Mathematical Analysis","2011","43","904–924","10.1137\u002F100805741","Defines Wasserstein barycenters; weighted averages of distributions that respect the geometry of the ground metric, unlike Euclidean averaging.",{"key":398,"type":662,"author":835,"title":836,"booktitle":673,"publisher":674,"year":837,"volume":611,"pages":838,"note":839},"Kusner, Matt J. and Sun, Yu and Kolkin, Nicholas I. and Weinberger, Kilian Q.","From word embeddings to document distances","2015","957–966","Word Mover's Distance; OT between word-embedding distributions of two documents, insensitive to synonymy and paraphrase.",{"key":418,"type":468,"author":841,"title":842,"journal":843,"year":702,"volume":844,"number":546,"pages":845,"doi":846,"note":847},"Schiebinger, Geoffrey and Shu, Jian and Tabaka, Marcin and Cleary, Brian and Subramanian, Vidya and Solomon, Aryeh and Gould, Joshua and Liu, Siyan and Lin, Stacie and Berube, Peter and Lee, Lia and Chen, Jenny and Brumbaugh, Justin and Rigollet, Philippe and Hochedlinger, Konrad and Jaenisch, Rudolf and Regev, Aviv and Lander, Eric S.","Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming","Cell","176","928–943","10.1016\u002Fj.cell.2019.01.006","Waddington-OT; the OT plan between adjacent-timepoint scRNA-seq profiles recovers developmental trajectories without paired data.",{"key":427,"type":468,"author":849,"title":850,"journal":851,"year":766,"doi":852,"note":853},"Klein, Dominik and Palla, Giovanni and Lange, Marius and Klein, Michal and Piran, Zoe and Gander, Manuel and Meng-Papaxanthos, Laetitia and Sterr, Michael and Treutlein, Barbara and Lickert, Heiko and Theis, Fabian J.","Mapping cells through time and space with moscot","bioRxiv","10.1101\u002F2023.05.11.540374","Moscot; scales OT-based single-cell trajectory inference in [Schiebinger2019] to million-cell, multi-omics, and spatial datasets. Later published in Nature (2025).",1785218162860]