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ACLR22

Brandon Amos, Samuel Cohen, Giulia Luise, and Ievgen Redko. Meta optimal transport. arXiv preprint arXiv:2206.05262, 2022.

AXK17

Brandon Amos, Lei Xu, and J. Zico Kolter. Input convex neural networks. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, 146–155. PMLR, 06–11 Aug 2017. URL: https://proceedings.mlr.press/v70/amos17b.html.

AV07

David Arthur and Sergei Vassilvitskii. K-means++: the advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '07, 1027–1035. USA, 2007. Society for Industrial and Applied Mathematics.

BCC+15

Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, and Gabriel Peyré. Iterative bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2):A1111–A1138, 2015. doi:10.1137/141000439.

BBV04

Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. URL: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.

Bre91

Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on pure and applied mathematics, 44(4):375–417, 1991.

BKC22

Charlotte Bunne, Andreas Krause, and Marco Cuturi. Supervised training of conditional monge maps. 2022. URL: https://arxiv.org/abs/2206.14262, doi:10.48550/ARXIV.2206.14262.

CLZ19

Song Chen, Blue B. Lake, and Kun Zhang. High-throughput sequencing of the transcriptome and chromatin accessibility in the same cell. Nature Biotechnology, 37(12):1452–1457, Dec 2019. URL: https://doi.org/10.1038/s41587-019-0290-0, doi:10.1038/s41587-019-0290-0.

CDHR08

Yanqing Chen, Timothy A. Davis, William W. Hager, and Sivasankaran Rajamanickam. Algorithm 887: cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Trans. Math. Softw., oct 2008. URL: https://doi.org/10.1145/1391989.1391995, doi:10.1145/1391989.1391995.

CGT19

Yongxin Chen, Tryphon T. Georgiou, and Allen Tannenbaum. Optimal transport for gaussian mixture models. IEEE Access, 7:6269–6278, 2019. doi:10.1109/ACCESS.2018.2889838.

CYL16

Yukun Chen, Jianbo Ye, and Jia Li. A distance for hmms based on aggregated wasserstein metric and state registration. In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, Computer Vision – ECCV 2016, 451–466. Cham, 2016. Springer International Publishing.

CYL20

Yukun Chen, Jianbo Ye, and Jia Li. Aggregated wasserstein distance and state registration for hidden markov models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(9):2133–2147, 2020. doi:10.1109/TPAMI.2019.2908635.

CWW13

Keenan Crane, Clarisse Weischedel, and Max Wardetzky. Geodesics in heat: a new approach to computing distance based on heat flow. ACM Trans. Graph., oct 2013. URL: https://doi.org/10.1145/2516971.2516977, doi:10.1145/2516971.2516977.

Cut13

Marco Cuturi. Sinkhorn distances: lightspeed computation of optimal transport. In C.J. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 26. Curran Associates, Inc., 2013. URL: https://proceedings.neurips.cc/paper/2013/file/af21d0c97db2e27e13572cbf59eb343d-Paper.pdf.

CD14

Marco Cuturi and Arnaud Doucet. Fast computation of wasserstein barycenters. In Eric P. Xing and Tony Jebara, editors, Proceedings of the 31st International Conference on Machine Learning, volume 32 of Proceedings of Machine Learning Research, 685–693. Bejing, China, 22–24 Jun 2014. PMLR. URL: https://proceedings.mlr.press/v32/cuturi14.html.

CTNWV20

Marco Cuturi, Olivier Teboul, Jonathan Niles-Weed, and Jean-Philippe Vert. Supervised quantile normalization for low rank matrix factorization. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, 2269–2279. PMLR, 13–18 Jul 2020. URL: https://proceedings.mlr.press/v119/cuturi20a.html.

CTV19

Marco Cuturi, Olivier Teboul, and Jean-Philippe Vert. Differentiable ranking and sorting using optimal transport. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d\textquotesingle Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL: https://proceedings.neurips.cc/paper/2019/file/d8c24ca8f23c562a5600876ca2a550ce-Paper.pdf.

DD20

Julie Delon and Agnès Desolneux. A wasserstein-type distance in the space of gaussian mixture models. SIAM Journal on Imaging Sciences, 13(2):936–970, 2020. doi:10.1137/19M1301047.

DSS+22

Pinar Demetci, Rebecca Santorella, Björn Sandstede, William Stafford Noble, and Ritambhara Singh. Scot: single-cell multi-omics alignment with optimal transport. Journal of Computational Biology, 29(1):3–18, 2022. PMID: 35050714. doi:10.1089/cmb.2021.0446.

FSejourneV+19

Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouve, and Gabriel Peyré. Interpolating between optimal transport and mmd using sinkhorn divergences. In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, 2681–2690. PMLR, 16–18 Apr 2019. URL: https://proceedings.mlr.press/v89/feydy19a.html.

Gel90

Matthias Gelbrich. On a formula for the l2 wasserstein metric between measures on euclidean and hilbert spaces. Mathematische Nachrichten, 147(1):185–203, 1990. doi:https://doi.org/10.1002/mana.19901470121.

GPC18

Aude Genevay, Gabriel Peyre, and Marco Cuturi. Learning generative models with sinkhorn divergences. In Amos Storkey and Fernando Perez-Cruz, editors, Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, volume 84 of Proceedings of Machine Learning Research, 1608–1617. PMLR, 09–11 Apr 2018. URL: https://proceedings.mlr.press/v84/genevay18a.html.

GPeyreC15

Alexandre Gramfort, Gabriel Peyré, and Marco Cuturi. Fast optimal transport averaging of neuroimaging data. In International Conference on Information Processing in Medical Imaging, 261–272. Springer, 2015.

HBC+21

Matthieu Heitz, Nicolas Bonneel, David Coeurjolly, Marco Cuturi, and Gabriel Peyré. Ground metric learning on graphs. Journal of Mathematical Imaging and Vision, 63(1):89–107, Jan 2021. URL: https://doi.org/10.1007/s10851-020-00996-z, doi:10.1007/s10851-020-00996-z.

Hig97

Nicholas J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15(2):227–242, Sep 1997. URL: https://doi.org/10.1023/A:1019150005407, doi:10.1023/A:1019150005407.

IVWW19

Pitor Indyk, Ali Vakilian, Tal Wagner, and David P Woodruff. Sample-optimal low-rank approximation of distance matrices. In Alina Beygelzimer and Daniel Hsu, editors, Proceedings of the Thirty-Second Conference on Learning Theory, volume 99 of Proceedings of Machine Learning Research, 1723–1751. PMLR, 25–28 Jun 2019. URL: https://proceedings.mlr.press/v99/indyk19a.html.

JCG20

Hicham Janati, Marco Cuturi, and Alexandre Gramfort. Debiased Sinkhorn barycenters. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, 4692–4701. PMLR, 13–18 Jul 2020. URL: https://proceedings.mlr.press/v119/janati20a.html.

JMPeyreC20

Hicham Janati, Boris Muzellec, Gabriel Peyré, and Marco Cuturi. Entropic optimal transport between unbalanced gaussian measures has a closed form. Advances in neural information processing systems, 33:10468–10479, 2020.

KEA+21

Alexander Korotin, Vage Egiazarian, Arip Asadulaev, Alexander Safin, and Evgeny Burnaev. Wasserstein-2 generative networks. In International Conference on Learning Representations. 2021. URL: https://openreview.net/forum?id=bEoxzW_EXsa.

LvRSU21

Tobias Lehmann, Max-K. von Renesse, Alexander Sambale, and André Uschmajew. A note on overrelaxation in the sinkhorn algorithm. Optimization Letters, Dec 2021. URL: https://doi.org/10.1007/s11590-021-01830-0, doi:10.1007/s11590-021-01830-0.

Llo82

S. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28(2):129–137, 1982. doi:10.1109/TIT.1982.1056489.

MTOL20

Ashok Makkuva, Amirhossein Taghvaei, Sewoong Oh, and Jason Lee. Optimal transport mapping via input convex neural networks. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, 6672–6681. PMLR, 13–18 Jul 2020. URL: https://proceedings.mlr.press/v119/makkuva20a.html.

Memoli11

Facundo Mémoli. Gromov–wasserstein distances and the metric approach to object matching. Foundations of Computational Mathematics, 11(4):417–487, Aug 2011. URL: https://doi.org/10.1007/s10208-011-9093-5, doi:10.1007/s10208-011-9093-5.

PC19

Gabriel Peyré and Marco Cuturi. Computational optimal transport: with applications to data science. Foundations and Trends® in Machine Learning, 11(5-6):355–607, 2019. URL: http://dx.doi.org/10.1561/2200000073, doi:10.1561/2200000073.

PCS16

Gabriel Peyré, Marco Cuturi, and Justin Solomon. Gromov-wasserstein averaging of kernel and distance matrices. In Maria Florina Balcan and Kilian Q. Weinberger, editors, Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, 2664–2672. New York, New York, USA, 20–22 Jun 2016. PMLR. URL: https://proceedings.mlr.press/v48/peyre16.html.

PCNW22

Aram-Alexandre Pooladian, Marco Cuturi, and Jonathan Niles-Weed. Debiaser beware: pitfalls of centering regularized transport maps. 2022. URL: https://arxiv.org/abs/2202.08919, doi:10.48550/ARXIV.2202.08919.

PNW21

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic estimation of optimal transport maps. 2021. URL: https://arxiv.org/abs/2109.12004, doi:10.48550/ARXIV.2109.12004.

RPLA21

Jack Richter-Powell, Jonathan Lorraine, and Brandon Amos. Input convex gradient networks. 2021. URL: https://arxiv.org/abs/2111.12187, doi:10.48550/ARXIV.2111.12187.

San15

Filippo Santambrogio. Optimal transport for applied mathematicians. Birkäuser, NY, 55(58-63):94, 2015.

SC22

Meyer Scetbon and Marco Cuturi. Low-rank optimal transport: approximation, statistics and debiasing. 2022. URL: https://arxiv.org/abs/2205.12365, doi:10.48550/ARXIV.2205.12365.

SCPeyre21

Meyer Scetbon, Marco Cuturi, and Gabriel Peyré. Low-rank sinkhorn factorization. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, 9344–9354. PMLR, 18–24 Jul 2021. URL: https://proceedings.mlr.press/v139/scetbon21a.html.

SPeyreC22

Meyer Scetbon, Gabriel Peyré, and Marco Cuturi. Linear-time gromov Wasserstein distances using low rank couplings and costs. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, 19347–19365. PMLR, 17–23 Jul 2022. URL: https://proceedings.mlr.press/v162/scetbon22b.html.

SST+19

Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, Lia Lee, Jenny Chen, Justin Brumbaugh, Philippe Rigollet, Konrad Hochedlinger, Rudolf Jaenisch, Aviv Regev, and Eric S. Lander. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176(4):928–943.e22, Feb 2019. doi:10.1016/j.cell.2019.01.006.

SHB+18

Morgan A. Schmitz, Matthieu Heitz, Nicolas Bonneel, Fred Ngolè, David Coeurjolly, Marco Cuturi, Gabriel Peyré, and Jean-Luc Starck. Wasserstein dictionary learning: optimal transport-based unsupervised nonlinear dictionary learning. SIAM Journal on Imaging Sciences, 11(1):643–678, 2018. doi:10.1137/17M1140431.

SVPeyre22

Thibault Sejourne, Francois-Xavier Vialard, and Gabriel Peyré. Faster unbalanced optimal transport: translation invariant sinkhorn and 1-d frank-wolfe. In Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera, editors, Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, volume 151 of Proceedings of Machine Learning Research, 4995–5021. PMLR, 28–30 Mar 2022. URL: https://proceedings.mlr.press/v151/sejourne22a/sejourne22a.pdf.

SVPeyre21

Thibault Sejourne, Francois-Xavier Vialard, and Gabriel Peyré. The unbalanced gromov wasserstein distance: conic formulation and relaxation. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, 8766–8779. Curran Associates, Inc., 2021. URL: https://proceedings.neurips.cc/paper/2021/file/4990974d150d0de5e6e15a1454fe6b0f-Paper.pdf.

SdGPeyre+15

Justin Solomon, Fernando de Goes, Gabriel Peyré, Marco Cuturi, Adrian Butscher, Andy Nguyen, Tao Du, and Leonidas Guibas. Convolutional wasserstein distances: efficient optimal transportation on geometric domains. ACM Trans. Graph., jul 2015. URL: https://doi.org/10.1145/2766963, doi:10.1145/2766963.

SFV+19

Thibault Séjourné, Jean Feydy, François-Xavier Vialard, Alain Trouvé, and Gabriel Peyré. Sinkhorn divergences for unbalanced optimal transport. 2019. URL: https://arxiv.org/abs/1910.12958, doi:10.48550/ARXIV.1910.12958.

TCDP21

Alexis Thibault, Lénaïc Chizat, Charles Dossal, and Nicolas Papadakis. Overrelaxed sinkhorn–knopp algorithm for regularized optimal transport. Algorithms, 2021. doi:10.3390/a14050143.

TC22

James Thornton and Marco Cuturi. Rethinking initialization of the sinkhorn algorithm. arXiv preprint arXiv:2206.07630, 2022.

TCT+19

Vayer Titouan, Nicolas Courty, Romain Tavenard, Chapel Laetitia, and Rémi Flamary. Optimal transport for structured data with application on graphs. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, 6275–6284. PMLR, 09–15 Jun 2019. URL: https://proceedings.mlr.press/v97/titouan19a.html.

VCF+20

Titouan Vayer, Laetitia Chapel, Remi Flamary, Romain Tavenard, and Nicolas Courty. Fused gromov-wasserstein distance for structured objects. Algorithms, 2020. URL: https://www.mdpi.com/1999-4893/13/9/212, doi:10.3390/a13090212.

AEdelBarrioCAM16

Pedro C. Álvarez-Esteban, E. del Barrio, J.A. Cuesta-Albertos, and C. Matrán. A fixed-point approach to barycenters in wasserstein space. Journal of Mathematical Analysis and Applications, 441(2):744–762, 2016. URL: https://www.sciencedirect.com/science/article/pii/S0022247X16300907, doi:https://doi.org/10.1016/j.jmaa.2016.04.045.