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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":148,"api":176,"body":177,"description":406,"extension":407,"links":176,"meta":408,"navigation":176,"path":149,"seo":409,"stem":150,"__hash__":410},"docs\u002F3.resources\u002F2.assignment-applications.md",null,{"type":178,"value":179,"toc":394},"minimark",[180,184,189,210,213,217,228,244,267,282,286,293,304,314,318,333,360,363,367,370,373,377,384,388,391],[181,182,183],"p",{},"The linear assignment problem is one of the oldest and most broadly applied combinatorial\noptimisation problems. This page traces where it shows up, from its origins in workforce\nscheduling through to the set-prediction losses that train modern vision transformers.",[185,186,188],"h2",{"id":187},"operations-research-origins","Operations research origins",[181,190,191,192,196,197,201,202,205,206,209],{},"The problem was formalised in the context of ",[193,194,195],"strong",{},"personnel assignment",": given a set of\nworkers and a set of jobs, and a productivity rating for each worker-job pair, find the\nassignment that maximises total productivity. Kuhn's 1955 paper ",[198,199],"docyard-cite",{"bib":200},"Kuhn1955"," coined\nthe \"Hungarian method\" motivated precisely by this framing. The same structure appeared\nunder the name ",[193,203,204],{},"transportation problem"," in the linear programming literature of the same\ndecade (Hitchcock 1941, Koopmans 1947) — matching supply nodes to demand nodes at minimum\nshipping cost — and as a ",[193,207,208],{},"bipartite matching"," in graph theory. All three are the same\nproblem with different vocabulary.",[181,211,212],{},"The assignment problem remains a daily tool in operations research: assigning gates to\narriving aircraft, assigning drivers to delivery routes, matching kidney donors to\nrecipients, scheduling nurses across shifts. The combinatorial structure is always the same:\ntwo disjoint sets, a cost or benefit for every pair, and a requirement for a one-to-one\ncorrespondence.",[185,214,216],{"id":215},"multi-object-tracking","Multi-object tracking",[181,218,219,220,223,224,227],{},"The dominant use case in modern computer vision. A tracker maintains a set of ",[193,221,222],{},"tracks","\n(object hypotheses across time) and receives a set of ",[193,225,226],{},"detections"," (bounding-box outputs\nfrom a detector) each frame. The task is to assign each detection to the track it most\nlikely continues, while flagging new tracks and terminating lost ones.",[181,229,230,231,235,236,239,240,243],{},"The cost matrix is typically ",[232,233,234],"code",{},"1 − IoU"," between all track-detection box pairs, optionally\nsupplemented by centroid distance, appearance similarity (re-id embedding distance), or\nlearned affinity scores. Gate entries whose pairs are geometrically infeasible are set to\n",[232,237,238],{},"+inf",". The assignment is solved per-frame; with a batch dimension over frames it maps\ndirectly onto ",[232,241,242],{},"jonker_dense_batch",".",[181,245,246,247,250,251,254,255,258,259,262,263,266],{},"SORT ",[198,248],{"bib":249},"Bewley2016"," established this template in 2016: linear motion prediction via a Kalman\nfilter, IoU cost, Hungarian assignment. DeepSORT ",[198,252],{"bib":253},"Wojke2017"," added appearance features to\nthe cost. ByteTrack ",[198,256],{"bib":257},"Zhang2022"," extended the idea to also process low-confidence\ndetections through a second assignment pass. BoT-SORT ",[198,260],{"bib":261},"Aharon2022"," and OC-SORT\n",[198,264],{"bib":265},"Cao2023"," refined the motion model and re-identification respectively, but the assignment\nstep remains structurally unchanged across the entire family.",[181,268,269,270,273,274,277,278,281],{},"The ",[232,271,272],{},"_unpacked"," op variants — ",[232,275,276],{},"jonker_dense_batch_unpacked",",\n",[232,279,280],{},"jonker_compact_batch_unpacked"," — return matched pairs, unmatched track indices, and\nunmatched detection indices in one pass, replacing the per-batch-element Python loop that\nmost tracker implementations use to recover these three sets from a raw assignment.",[185,283,285],{"id":284},"set-prediction-losses-detr-family","Set-prediction losses (DETR family)",[181,287,288,289,292],{},"Transformer-based object detectors output a ",[193,290,291],{},"fixed-size set"," of predictions (bounding\nboxes and class logits), padded with no-object slots to a common count. Computing the\ntraining loss requires matching each ground-truth object to exactly one prediction before\nevaluating the per-pair L1, GIoU, and classification terms. The matching must be globally\noptimal — greedy or random assignment produces a worse training signal.",[181,294,295,296,299,300,303],{},"DETR ",[198,297],{"bib":298},"Carion2020"," introduced this formulation and placed the Hungarian matcher explicitly\non the critical path. Subsequent work in the DETR lineage — Conditional DETR, DN-DETR,\nDINO-DETR, MaskFormer ",[198,301],{"bib":302},"Cheng2021",", Mask2Former, RT-DETR — all preserve the Hungarian\nmatching step, with variations in how the cost components are weighted or whether matching\nis applied across multiple decoder layers. The matcher runs at training time, once per image\nper gradient step; its latency directly affects training throughput at large batch sizes.",[181,305,306,307,309,310,313],{},"Batching the per-image cost matrices into a single 3D tensor and solving with\n",[232,308,242],{}," eliminates the per-image Python loop and exposes the problem to\n",[232,311,312],{},"at::parallel_for"," on CPU or the tiled CUDA kernel for small square problems.",[185,315,317],{"id":316},"cluster-evaluation","Cluster evaluation",[181,319,320,321,324,325,328,329,332],{},"Metrics for unsupervised learning and semi-supervised segmentation require an ",[193,322,323],{},"optimal\nrelabelling"," between predicted cluster IDs and ground-truth class IDs before counting\nagreements. Because cluster labels are arbitrary integers — the model could call its cat\ncluster ",[232,326,327],{},"3"," and the ground truth calls it ",[232,330,331],{},"7"," — a naive comparison gives a meaningless\nscore. The correct score uses the permutation that maximises agreements.",[181,334,335,336,339,340,343,344,347,348,351,352,355,356,359],{},"The confusion matrix yields an assignment problem: build a ",[232,337,338],{},"K × K"," cost matrix\nas the negative confusion-matrix entry ",[232,341,342],{},"−C[i,j]"," (number of samples with predicted label\n",[232,345,346],{},"i"," and true label ",[232,349,350],{},"j","), then find the permutation of predicted labels that maximises\ntotal agreement. The resulting ",[193,353,354],{},"assignment accuracy (ACC)"," and the ",[193,357,358],{},"adjusted Rand index\n(ARI)"," (a metric that measures agreement between two clusterings, corrected for chance)\nboth depend on this optimal permutation.",[181,361,362],{},"Domains where this matters include unsupervised semantic segmentation, clustering\nbenchmarks (ImageNet-1K linear evaluation, STL-10, CIFAR-100 supercategory), online\nquantization (codebook alignment in VQ-VAE variants), and re-identification evaluation.",[185,364,366],{"id":365},"pose-estimation-and-part-matching","Pose estimation and part matching",[181,368,369],{},"Multi-person pose estimators that use bottom-up detection (methods that detect individual\nbody parts across the image and group them into person instances) must group detected\nkeypoints into person instances. Each detected keypoint is a candidate node; a bipartite\nassignment on a joint confidence matrix groups keypoints across body-part types into\ncoherent skeletons. Top-down methods instead assign detected bounding boxes to tracked\npersons across frames — again LAP.",[181,371,372],{},"In 6-DoF pose estimation, the assignment problem arises when matching predicted object\ninstances to ground-truth annotations where multiple overlapping instances are present.",[185,374,376],{"id":375},"graph-matching-and-molecular-structure","Graph matching and molecular structure",[181,378,379,380,383],{},"Computing the similarity between two graphs requires a ",[193,381,382],{},"bijection between nodes"," that\nmaximises the number of matched edges — the graph isomorphism problem in its weighted\nvariant. For small graphs or graphs with good node features, a LAP over a node-similarity\ncost matrix gives a tractable approximation. Applications include molecular graph\nmatching (drug-candidate similarity), knowledge-graph entity alignment, and circuit\nnetlist comparison.",[185,385,387],{"id":386},"sequence-alignment-discrete","Sequence alignment (discrete)",[181,389,390],{},"Discrete sequence alignment with known length — matching tokens in a predicted sequence to\ntokens in a reference — appears in evaluation metrics for structured prediction tasks:\nmatching predicted named entities to gold entities, matching predicted spans to gold spans\nin reading comprehension, or aligning predicted parses to gold parses. When the two\nsequences are already segmented into labelled spans, a LAP on a span-overlap cost matrix\ngives the optimal alignment.",[392,393],"bibliography",{},{"title":395,"searchDepth":396,"depth":396,"links":397},"",3,[398,400,401,402,403,404,405],{"id":187,"depth":399,"text":188},2,{"id":215,"depth":399,"text":216},{"id":284,"depth":399,"text":285},{"id":316,"depth":399,"text":317},{"id":365,"depth":399,"text":366},{"id":375,"depth":399,"text":376},{"id":386,"depth":399,"text":387},"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.","md",{},{"title":148,"description":406},"aT62CooML5CS9TdQ2RrMOf7f9au9nKGIHUFLrKb9oGA",[412,414],{"title":144,"path":145,"stem":146,"description":413,"icon":121,"children":-1},"What you will learn",{"title":152,"path":153,"stem":154,"description":415,"children":-1},"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.",[417,427,433,443,454,463,473,481,490,500,509,519,528,537,547,555,565,575,584,593,600,610,620,630,638,648,655,664,673,681,688,694,703,708,716,724,731,741,747,752,757,767,775,785,792,801],{"key":418,"type":419,"author":420,"title":421,"journal":422,"booktitle":-1,"publisher":-1,"year":423,"volume":424,"number":-1,"pages":425,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":426},"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":428,"type":419,"author":429,"title":430,"journal":422,"booktitle":-1,"publisher":-1,"year":423,"volume":424,"number":-1,"pages":431,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":432},"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":200,"type":419,"author":434,"title":435,"journal":436,"booktitle":-1,"publisher":-1,"year":437,"volume":438,"number":439,"pages":440,"doi":441,"url":-1,"institution":-1,"edition":-1,"note":442},"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":444,"type":419,"author":445,"title":446,"journal":447,"booktitle":-1,"publisher":-1,"year":448,"volume":449,"number":450,"pages":451,"doi":452,"url":-1,"institution":-1,"edition":-1,"note":453},"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":455,"type":419,"author":456,"title":457,"journal":458,"booktitle":-1,"publisher":-1,"year":459,"volume":450,"number":438,"pages":460,"doi":461,"url":-1,"institution":-1,"edition":-1,"note":462},"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":464,"type":419,"author":465,"title":466,"journal":467,"booktitle":-1,"publisher":-1,"year":468,"volume":469,"number":438,"pages":470,"doi":471,"url":-1,"institution":-1,"edition":-1,"note":472},"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":474,"type":475,"author":476,"title":477,"journal":-1,"booktitle":-1,"publisher":478,"year":479,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":480},"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":482,"type":483,"author":484,"title":485,"journal":-1,"booktitle":-1,"publisher":-1,"year":486,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":487,"institution":488,"edition":-1,"note":489},"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":491,"type":419,"author":492,"title":493,"journal":494,"booktitle":-1,"publisher":-1,"year":495,"volume":424,"number":496,"pages":497,"doi":498,"url":-1,"institution":-1,"edition":-1,"note":499},"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":501,"type":419,"author":484,"title":502,"journal":503,"booktitle":-1,"publisher":-1,"year":504,"volume":505,"number":450,"pages":506,"doi":507,"url":-1,"institution":-1,"edition":-1,"note":508},"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":510,"type":419,"author":511,"title":512,"journal":513,"booktitle":-1,"publisher":-1,"year":514,"volume":515,"number":438,"pages":516,"doi":517,"url":-1,"institution":-1,"edition":-1,"note":518},"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":520,"type":521,"author":434,"title":435,"journal":-1,"booktitle":522,"publisher":523,"year":524,"volume":-1,"number":-1,"pages":525,"doi":526,"url":-1,"institution":-1,"edition":-1,"note":527},"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":529,"type":475,"author":530,"title":531,"journal":-1,"booktitle":-1,"publisher":532,"year":533,"volume":-1,"number":-1,"pages":-1,"doi":534,"url":-1,"institution":-1,"edition":535,"note":536},"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":538,"type":419,"author":539,"title":540,"journal":541,"booktitle":-1,"publisher":-1,"year":542,"volume":543,"number":496,"pages":544,"doi":545,"url":-1,"institution":-1,"edition":-1,"note":546},"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":548,"type":419,"author":549,"title":550,"journal":551,"booktitle":-1,"publisher":-1,"year":552,"volume":-1,"number":-1,"pages":553,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":554},"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":556,"type":419,"author":557,"title":558,"journal":559,"booktitle":-1,"publisher":-1,"year":560,"volume":561,"number":562,"pages":563,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":564},"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":566,"type":419,"author":567,"title":568,"journal":569,"booktitle":-1,"publisher":-1,"year":570,"volume":571,"number":327,"pages":572,"doi":573,"url":-1,"institution":-1,"edition":-1,"note":574},"BenamouBrenier2000","Benamou, Jean-David and Brenier, Yann","A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem","Numerische Mathematik","2000","84","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":576,"type":419,"author":577,"title":578,"journal":579,"booktitle":-1,"publisher":-1,"year":570,"volume":580,"number":438,"pages":581,"doi":582,"url":-1,"institution":-1,"edition":-1,"note":583},"Rubner2000","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":585,"type":475,"author":586,"title":587,"journal":-1,"booktitle":-1,"publisher":588,"year":589,"volume":590,"number":-1,"pages":-1,"doi":591,"url":-1,"institution":-1,"edition":-1,"note":592},"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":594,"type":475,"author":586,"title":595,"journal":-1,"booktitle":-1,"publisher":523,"year":596,"volume":597,"number":-1,"pages":-1,"doi":598,"url":-1,"institution":-1,"edition":-1,"note":599},"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":601,"type":419,"author":602,"title":603,"journal":604,"booktitle":-1,"publisher":-1,"year":605,"volume":606,"number":496,"pages":607,"doi":608,"url":-1,"institution":-1,"edition":-1,"note":609},"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":611,"type":612,"author":613,"title":614,"journal":-1,"booktitle":615,"publisher":616,"year":617,"volume":618,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":619},"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":621,"type":612,"author":622,"title":623,"journal":-1,"booktitle":624,"publisher":625,"year":626,"volume":627,"number":-1,"pages":628,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":629},"Arjovsky2017","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":631,"type":612,"author":632,"title":633,"journal":-1,"booktitle":634,"publisher":625,"year":635,"volume":571,"number":-1,"pages":636,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":637},"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":639,"type":419,"author":640,"title":641,"journal":642,"booktitle":-1,"publisher":-1,"year":635,"volume":643,"number":644,"pages":645,"doi":646,"url":-1,"institution":-1,"edition":-1,"note":647},"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":649,"type":419,"author":650,"title":651,"journal":652,"booktitle":-1,"publisher":-1,"year":653,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":654},"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":656,"type":419,"author":657,"title":658,"journal":659,"booktitle":-1,"publisher":-1,"year":653,"volume":660,"number":327,"pages":661,"doi":662,"url":-1,"institution":-1,"edition":-1,"note":663},"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":665,"type":419,"author":666,"title":667,"journal":513,"booktitle":-1,"publisher":-1,"year":668,"volume":669,"number":438,"pages":670,"doi":671,"url":-1,"institution":-1,"edition":-1,"note":672},"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":674,"type":419,"author":675,"title":676,"journal":677,"booktitle":-1,"publisher":-1,"year":617,"volume":660,"number":449,"pages":678,"doi":679,"url":-1,"institution":-1,"edition":-1,"note":680},"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":249,"type":612,"author":682,"title":683,"journal":-1,"booktitle":684,"publisher":-1,"year":542,"volume":-1,"number":-1,"pages":685,"doi":686,"url":-1,"institution":-1,"edition":-1,"note":687},"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":253,"type":612,"author":689,"title":690,"journal":-1,"booktitle":684,"publisher":-1,"year":626,"volume":-1,"number":-1,"pages":691,"doi":692,"url":-1,"institution":-1,"edition":-1,"note":693},"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":257,"type":612,"author":695,"title":696,"journal":-1,"booktitle":697,"publisher":-1,"year":698,"volume":699,"number":-1,"pages":700,"doi":701,"url":-1,"institution":-1,"edition":-1,"note":702},"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":261,"type":419,"author":704,"title":705,"journal":706,"booktitle":-1,"publisher":-1,"year":698,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":707},"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":265,"type":612,"author":709,"title":710,"journal":-1,"booktitle":711,"publisher":-1,"year":712,"volume":-1,"number":-1,"pages":713,"doi":714,"url":-1,"institution":-1,"edition":-1,"note":715},"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":298,"type":612,"author":717,"title":718,"journal":-1,"booktitle":697,"publisher":-1,"year":719,"volume":720,"number":-1,"pages":721,"doi":722,"url":-1,"institution":-1,"edition":-1,"note":723},"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":302,"type":612,"author":725,"title":726,"journal":-1,"booktitle":615,"publisher":-1,"year":727,"volume":728,"number":-1,"pages":729,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":730},"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":732,"type":419,"author":733,"title":734,"journal":735,"booktitle":-1,"publisher":-1,"year":736,"volume":737,"number":439,"pages":738,"doi":739,"url":-1,"institution":-1,"edition":-1,"note":740},"Pitie2007","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":742,"type":612,"author":743,"title":744,"journal":-1,"booktitle":745,"publisher":-1,"year":635,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":746},"Tolstikhin2018","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":748,"type":612,"author":749,"title":750,"journal":-1,"booktitle":745,"publisher":-1,"year":712,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":751},"Lipman2022","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":753,"type":612,"author":754,"title":755,"journal":-1,"booktitle":745,"publisher":-1,"year":712,"volume":-1,"number":-1,"pages":-1,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":756},"Liu2022","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":758,"type":419,"author":759,"title":760,"journal":761,"booktitle":-1,"publisher":-1,"year":626,"volume":762,"number":763,"pages":764,"doi":765,"url":-1,"institution":-1,"edition":-1,"note":766},"Courty2017","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":768,"type":612,"author":769,"title":770,"journal":-1,"booktitle":697,"publisher":-1,"year":635,"volume":771,"number":-1,"pages":772,"doi":773,"url":-1,"institution":-1,"edition":-1,"note":774},"Damodaran2018","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":776,"type":419,"author":777,"title":778,"journal":779,"booktitle":-1,"publisher":-1,"year":780,"volume":781,"number":438,"pages":782,"doi":783,"url":-1,"institution":-1,"edition":-1,"note":784},"Agueh2011","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":786,"type":612,"author":787,"title":788,"journal":-1,"booktitle":624,"publisher":625,"year":789,"volume":561,"number":-1,"pages":790,"doi":-1,"url":-1,"institution":-1,"edition":-1,"note":791},"Kusner2015","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":793,"type":419,"author":794,"title":795,"journal":796,"booktitle":-1,"publisher":-1,"year":653,"volume":797,"number":496,"pages":798,"doi":799,"url":-1,"institution":-1,"edition":-1,"note":800},"Schiebinger2019","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":802,"type":419,"author":803,"title":804,"journal":805,"booktitle":-1,"publisher":-1,"year":712,"volume":-1,"number":-1,"pages":-1,"doi":806,"url":-1,"institution":-1,"edition":-1,"note":807},"Klein2023","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).",1785218161842]