no code implementations • ICML 2020 • Marco Cuturi, Olivier Teboul, Jonathan Niles-Weed, Jean-Philippe Vert

Low rank matrix factorization is a fundamental building block in machine learning, used for instance to summarize gene expression profile data or word-document counts.

no code implementations • ICLR 2019 • Jean Alaux, Edouard Grave, Marco Cuturi, Armand Joulin

This paper extends this line of work to the problem of aligning multiple languages to a common space.

no code implementations • 10 Jul 2024 • Théo Uscidda, Luca Eyring, Karsten Roth, Fabian Theis, Zeynep Akata, Marco Cuturi

We propose the Gromov-Monge-Gap (GMG), a regularizer that quantifies the geometry-preservation of an arbitrary push-forward map between two distributions supported on different spaces.

no code implementations • 9 Jul 2024 • Yu-Guan Hsieh, Cheng-Yu Hsieh, Shih-Ying Yeh, Louis Béthune, Hadi Pour Ansari, Pavan Kumar Anasosalu Vasu, Chun-Liang Li, Ranjay Krishna, Oncel Tuzel, Marco Cuturi

The nodes in GBC are created using, in a first stage, object detection and dense captioning tools nested recursively to uncover and describe entity nodes, further linked together in a second stage by highlighting, using new types of nodes, compositions and relations among entities.

no code implementations • 7 Jun 2024 • Parnian Kassraie, Aram-Alexandre Pooladian, Michal Klein, James Thornton, Jonathan Niles-Weed, Marco Cuturi

Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets.

no code implementations • 29 May 2024 • Zoe Piran, Michal Klein, James Thornton, Marco Cuturi

Learning meaningful representations of complex objects that can be seen through multiple ($k\geq 3$) views or modalities is a core task in machine learning.

no code implementations • 14 May 2024 • Antoine Wehenkel, Juan L. Gamella, Ozan Sener, Jens Behrmann, Guillermo Sapiro, Marco Cuturi, Jörn-Henrik Jacobsen

Driven by steady progress in generative modeling, simulation-based inference (SBI) has enabled inference over stochastic simulators.

no code implementations • 13 May 2024 • Ben Blum-Smith, Ningyuan Huang, Marco Cuturi, Soledad Villar

In this work, we present a mathematical formulation for machine learning of (1) functions on symmetric matrices that are invariant with respect to the action of permutations by conjugation, and (2) functions on point clouds that are invariant with respect to rotations, reflections, and permutations of the points.

no code implementations • 5 Mar 2024 • Nina Vesseron, Marco Cuturi

The theorem, known as the polar factorization theorem, states that any field $F$ can be recovered as the composition of the gradient of a convex function $u$ with a measure-preserving map $M$, namely $F=\nabla u \circ M$.

no code implementations • 5 Feb 2024 • Yu-Guan Hsieh, James Thornton, Eugene Ndiaye, Michal Klein, Marco Cuturi, Pierre Ablin

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e. g. performance on another dataset, robustness, agreement with a prior).

no code implementations • 9 Nov 2023 • Othmane Sebbouh, Marco Cuturi, Gabriel Peyré

Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points.

no code implementations • 21 Oct 2023 • Tianyi Lin, Marco Cuturi, Michael I. Jordan

Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples.

no code implementations • 13 Oct 2023 • Dominik Klein, Théo Uscidda, Fabian Theis, Marco Cuturi

Optimal transport (OT) theory has reshaped the field of generative modeling: Combined with neural networks, recent \textit{Neural OT} (N-OT) solvers use OT as an inductive bias, to focus on ``thrifty'' mappings that minimize average displacement costs.

no code implementations • 26 Jul 2023 • Antoine Wehenkel, Jens Behrmann, Andrew C. Miller, Guillermo Sapiro, Ozan Sener, Marco Cuturi, Jörn-Henrik Jacobsen

Over the past decades, hemodynamics simulators have steadily evolved and have become tools of choice for studying cardiovascular systems in-silico.

no code implementations • 20 Jun 2023 • Michal Klein, Aram-Alexandre Pooladian, Pierre Ablin, Eugène Ndiaye, Jonathan Niles-Weed, Marco Cuturi

Given a source and a target probability measure supported on $\mathbb{R}^d$, the Monge problem asks to find the most efficient way to map one distribution to the other.

no code implementations • 9 Feb 2023 • Théo Uscidda, Marco Cuturi

That gap quantifies how far a map $T$ deviates from the ideal properties we expect from a $c$-OT map.

no code implementations • 8 Feb 2023 • Marco Cuturi, Michal Klein, Pierre Ablin

Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure onto another, on those that are the ``thriftiest'', i. e. such that the averaged cost $c(x, T(x))$ between $x$ and its image $T(x)$ be as small as possible.

1 code implementation • 28 Jun 2022 • Charlotte Bunne, Andreas Krause, Marco Cuturi

To account for that context in OT estimation, we introduce CondOT, a multi-task approach to estimate a family of OT maps conditioned on a context variable, using several pairs of measures $\left(\mu_i, \nu_i\right)$ tagged with a context label $c_i$.

no code implementations • 15 Jun 2022 • James Thornton, Marco Cuturi

While the optimal transport (OT) problem was originally formulated as a linear program, the addition of entropic regularization has proven beneficial both computationally and statistically, for many applications.

no code implementations • 24 May 2022 • Meyer Scetbon, Marco Cuturi

The matching principles behind optimal transport (OT) play an increasingly important role in machine learning, a trend which can be observed when OT is used to disambiguate datasets in applications (e. g. single-cell genomics) or used to improve more complex methods (e. g. balanced attention in transformers or self-supervised learning).

no code implementations • 18 Apr 2022 • Yingtao Tian, Marco Cuturi, David Ha

Recent advances in deep learning, such as powerful generative models and joint text-image embeddings, have provided the computational creativity community with new tools, opening new perspectives for artistic pursuits.

1 code implementation • 11 Mar 2022 • Hicham Janati, Marco Cuturi, Alexandre Gramfort

These complex datasets, describing dynamics with both time and spatial components, pose new challenges for data analysis.

no code implementations • 11 Feb 2022 • Charlotte Bunne, Ya-Ping Hsieh, Marco Cuturi, Andreas Krause

The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks to recover an optimal map, or more generally a coupling, to morph a Gaussian into another.

1 code implementation • 28 Jan 2022 • Marco Cuturi, Laetitia Meng-Papaxanthos, Yingtao Tian, Charlotte Bunne, Geoff Davis, Olivier Teboul

Optimal transport tools (OTT-JAX) is a Python toolbox that can solve optimal transport problems between point clouds and histograms.

no code implementations • 25 Nov 2021 • Othmane Sebbouh, Marco Cuturi, Gabriel Peyré

RSGDA can be parameterized using optimal loop sizes that guarantee the best convergence rates known to hold for SGDA.

2 code implementations • 11 Jun 2021 • Charlotte Bunne, Laetitia Meng-Papaxanthos, Andreas Krause, Marco Cuturi

We propose to model these trajectories as collective realizations of a causal Jordan-Kinderlehrer-Otto (JKO) flow of measures: The JKO scheme posits that the new configuration taken by a population at time $t+1$ is one that trades off an improvement, in the sense that it decreases an energy, while remaining close (in Wasserstein distance) to the previous configuration observed at $t$.

1 code implementation • NeurIPS 2021 • Meyer Scetbon, Gabriel Peyré, Marco Cuturi

The ability to align points across two related yet incomparable point clouds (e. g. living in different spaces) plays an important role in machine learning.

1 code implementation • NeurIPS 2021 • Mathieu Blondel, Quentin Berthet, Marco Cuturi, Roy Frostig, Stephan Hoyer, Felipe Llinares-López, Fabian Pedregosa, Jean-Philippe Vert

In this paper, we propose automatic implicit differentiation, an efficient and modular approach for implicit differentiation of optimization problems.

1 code implementation • 8 Mar 2021 • Meyer Scetbon, Marco Cuturi, Gabriel Peyré

Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors.

no code implementations • NeurIPS 2020 • Quentin Berthet, Mathieu Blondel, Olivier Teboul, Marco Cuturi, Jean-Philippe Vert, Francis Bach

Machine learning pipelines often rely on optimizers procedures to make discrete decisions (e. g., sorting, picking closest neighbors, or shortest paths).

no code implementations • NeurIPS 2020 • Hicham Janati, Boris Muzellec, Gabriel Peyré, Marco Cuturi

Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e. g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry.

no code implementations • 22 Jun 2020 • Tianyi Lin, Zeyu Zheng, Elynn Y. Chen, Marco Cuturi, Michael. I. Jordan

Yet, the behavior of minimum Wasserstein estimators is poorly understood, notably in high-dimensional regimes or under model misspecification.

no code implementations • NeurIPS 2020 • Tianyi Lin, Chenyou Fan, Nhat Ho, Marco Cuturi, Michael. I. Jordan

Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance.

no code implementations • 12 Jun 2020 • Meyer Scetbon, Laurent Meunier, Jamal Atif, Marco Cuturi

When there is only one agent, we recover the Optimal Transport problem.

1 code implementation • NeurIPS 2020 • Meyer Scetbon, Marco Cuturi

Although Sinkhorn divergences are now routinely used in data sciences to compare probability distributions, the computational effort required to compute them remains expensive, growing in general quadratically in the size $n$ of the support of these distributions.

1 code implementation • NeurIPS 2020 • Hicham Janati, Boris Muzellec, Gabriel Peyré, Marco Cuturi

Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e. g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry.

Statistics Theory Statistics Theory

3 code implementations • ICML 2020 • Hicham Janati, Marco Cuturi, Alexandre Gramfort

However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters.

1 code implementation • 26 Apr 2020 • Marco Cuturi, Olivier Teboul, Quentin Berthet, Arnaud Doucet, Jean-Philippe Vert

Our goal in this paper is to propose new group testing algorithms that can operate in a noisy setting (tests can be mistaken) to decide adaptively (looking at past results) which groups to test next, with the goal to converge to a good detection, as quickly, and with as few tests as possible.

3 code implementations • 20 Feb 2020 • Quentin Berthet, Mathieu Blondel, Olivier Teboul, Marco Cuturi, Jean-Philippe Vert, Francis Bach

Machine learning pipelines often rely on optimization procedures to make discrete decisions (e. g., sorting, picking closest neighbors, or shortest paths).

no code implementations • NeurIPS 2020 • Tianyi Lin, Nhat Ho, Xi Chen, Marco Cuturi, Michael. I. Jordan

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$.

no code implementations • ICML 2020 • François-Pierre Paty, Marco Cuturi

In this work we depart from this practical perspective and propose a new interpretation of regularization as a robust mechanism, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial.

1 code implementation • ICML 2020 • Boris Muzellec, Julie Josse, Claire Boyer, Marco Cuturi

Missing data is a crucial issue when applying machine learning algorithms to real-world datasets.

no code implementations • 8 Feb 2020 • Marco Cuturi, Olivier Teboul, Jonathan Niles-Weed, Jean-Philippe Vert

Low rank matrix factorization is a fundamental building block in machine learning, used for instance to summarize gene expression profile data or word-document counts.

1 code implementation • 5 Feb 2020 • Ryoma Sato, Marco Cuturi, Makoto Yamada, Hisashi Kashima

Building on \cite{memoli-2011}, who proposed to represent each point in each distribution as the 1D distribution of its distances to all other points, we introduce in this paper the Anchor Energy (AE) and Anchor Wasserstein (AW) distances, which are respectively the energy and Wasserstein distances instantiated on such representations.

1 code implementation • NeurIPS 2019 • Marco Cuturi, Olivier Teboul, Jean-Philippe Vert

From this observation, we propose extended rank and sort operators by considering optimal transport (OT) problems (the natural relaxation for assignments) where the auxiliary measure can be any weighted measure supported on $m$ increasing values, where $m \ne n$.

1 code implementation • 8 Nov 2019 • Matthieu Heitz, Nicolas Bonneel, David Coeurjolly, Marco Cuturi, Gabriel Peyré

Optimal transport (OT) distances between probability distributions are parameterized by the ground metric they use between observations.

2 code implementations • 9 Oct 2019 • Hicham Janati, Marco Cuturi, Alexandre Gramfort

In this paper, we propose Spatio-Temporal Alignments (STA), a new differentiable formulation of DTW, in which spatial differences between time samples are accounted for using regularized optimal transport (OT).

no code implementations • 3 Oct 2019 • Hicham Janati, Thomas Bazeille, Bertrand Thirion, Marco Cuturi, Alexandre Gramfort

Magnetoencephalography and electroencephalography (M/EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity.

1 code implementation • 30 Sep 2019 • Tianyi Lin, Nhat Ho, Marco Cuturi, Michael. I. Jordan

This provides a first \textit{near-linear time} complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when $m = 2$.

no code implementations • 30 May 2019 • Gabriel Dulac-Arnold, Neil Zeghidour, Marco Cuturi, Lucas Beyer, Jean-Philippe Vert

We propose a learning algorithm capable of learning from label proportions instead of direct data labels.

no code implementations • 28 May 2019 • Marco Cuturi, Olivier Teboul, Jean-Philippe Vert

Sorting an array is a fundamental routine in machine learning, one that is used to compute rank-based statistics, cumulative distribution functions (CDFs), quantiles, or to select closest neighbors and labels.

no code implementations • 26 May 2019 • François-Pierre Paty, Alexandre d'Aspremont, Marco Cuturi

On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth.

no code implementations • 26 May 2019 • Josip Djolonga, Mario Lucic, Marco Cuturi, Olivier Bachem, Olivier Bousquet, Sylvain Gelly

Despite the tremendous progress in the estimation of generative models, the development of tools for diagnosing their failures and assessing their performance has advanced at a much slower pace.

1 code implementation • NeurIPS 2019 • Boris Muzellec, Marco Cuturi

A popular approach to avoid this curse is to project input measures on lower-dimensional subspaces (1D lines in the case of sliced Wasserstein distances), solve the OT problem between these reduced measures, and settle for the Wasserstein distance between these reductions, rather than that between the original measures.

no code implementations • 13 Feb 2019 • Hicham Janati, Thomas Bazeille, Bertrand Thirion, Marco Cuturi, Alexandre Gramfort

Inferring the location of the current sources that generated these magnetic fields is an ill-posed inverse problem known as source imaging.

2 code implementations • NeurIPS 2019 • Tam Le, Makoto Yamada, Kenji Fukumizu, Marco Cuturi

Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions.

no code implementations • 25 Jan 2019 • François-Pierre Paty, Marco Cuturi

Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge.

1 code implementation • 19 Nov 2018 • Gwendoline de Bie, Gabriel Peyré, Marco Cuturi

This allows to design discriminative networks (to classify or reduce the dimensionality of input measures), generative architectures (to synthesize measures) and recurrent pipelines (to predict measure dynamics).

no code implementations • 13 Nov 2018 • Marco Cuturi, Gabriel Peyré

Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences.

no code implementations • 2 Nov 2018 • Jean Alaux, Edouard Grave, Marco Cuturi, Armand Joulin

This paper extends this line of work to the problem of aligning multiple languages to a common space.

no code implementations • NeurIPS 2018 • Théo Lacombe, Marco Cuturi, Steve Oudot

Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data.

1 code implementation • 20 May 2018 • Hicham Janati, Marco Cuturi, Alexandre Gramfort

We argue in this paper that these techniques fail to leverage the spatial information associated to regressors.

2 code implementations • NeurIPS 2018 • Boris Muzellec, Marco Cuturi

We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets.

5 code implementations • 1 Mar 2018 • Gabriel Peyré, Marco Cuturi

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site.

2 code implementations • 7 Aug 2017 • Morgan A. Schmitz, Matthieu Heitz, Nicolas Bonneel, Fred Maurice Ngolè Mboula, David Coeurjolly, Marco Cuturi, Gabriel Peyré, Jean-Luc Starck

Wasserstein barycenters) between dictionary atoms; such atoms are themselves synthetic histograms in the probability simplex.

no code implementations • ICML 2017 • Mathieu Carrière, Marco Cuturi, Steve Oudot

To incorporate PDs in a learning pipeline, several kernels have been proposed for PDs with a strong emphasis on the stability of the RKHS distance w. r. t.

Ranked #2 on Graph Classification on NEURON-MULTI

no code implementations • 6 Jun 2017 • Aude Genevay, Gabriel Peyré, Marco Cuturi

This short article revisits some of the ideas introduced in arXiv:1701. 07875 and arXiv:1705. 07642 in a simple setup.

2 code implementations • 1 Jun 2017 • Aude Genevay, Gabriel Peyré, Marco Cuturi

The ability to compare two degenerate probability distributions (i. e. two probability distributions supported on two distinct low-dimensional manifolds living in a much higher-dimensional space) is a crucial problem arising in the estimation of generative models for high-dimensional observations such as those arising in computer vision or natural language.

9 code implementations • ICML 2017 • Marco Cuturi, Mathieu Blondel

We propose in this paper a differentiable learning loss between time series, building upon the celebrated dynamic time warping (DTW) discrepancy.

no code implementations • NeurIPS 2016 • Grégoire Montavon, Klaus-Robert Müller, Marco Cuturi

This metric between observations can then be used to define the Wasserstein distance between the distribution induced by the Boltzmann machine on the one hand, and that given by the training sample on the other hand.

1 code implementation • 29 Aug 2016 • Rémi Flamary, Marco Cuturi, Nicolas Courty, Alain Rakotomamonjy

Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace.

no code implementations • NeurIPS 2016 • Genevay Aude, Marco Cuturi, Gabriel Peyré, Francis Bach

We instantiate these ideas in three different setups: (i) when comparing a discrete distribution to another, we show that incremental stochastic optimization schemes can beat Sinkhorn's algorithm, the current state-of-the-art finite dimensional OT solver; (ii) when comparing a discrete distribution to a continuous density, a semi-discrete reformulation of the dual program is amenable to averaged stochastic gradient descent, leading to better performance than approximately solving the problem by discretization ; (iii) when dealing with two continuous densities, we propose a stochastic gradient descent over a reproducing kernel Hilbert space (RKHS).

1 code implementation • 8 Sep 2015 • Aaditya Ramdas, Nicolas Garcia, Marco Cuturi

In this work, our central object is the Wasserstein distance, as we form a chain of connections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ plots and ROC/ODC curves, to multivariate tests involving energy statistics and kernel based maximum mean discrepancy.

no code implementations • 7 Jul 2015 • Grégoire Montavon, Klaus-Robert Müller, Marco Cuturi

The Boltzmann machine provides a useful framework to learn highly complex, multimodal and multiscale data distributions that occur in the real world.

no code implementations • NeurIPS 2015 • Vivien Seguy, Marco Cuturi

Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family.

no code implementations • 30 Mar 2015 • Alexandre Gramfort, Gabriel Peyré, Marco Cuturi

Data are large, the geometry of the brain is complex and the between subjects variability leads to spatially or temporally non-overlapping effects of interest.

1 code implementation • 9 Mar 2015 • Marco Cuturi, Gabriel Peyré

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures.

1 code implementation • 16 Dec 2014 • Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, Gabriel Peyré

This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport.

Numerical Analysis Analysis of PDEs

no code implementations • NeurIPS 2013 • Marco Cuturi

Optimal transportation distances are a fundamental family of parameterized distances for histograms in the probability simplex.

2 code implementations • 16 Oct 2013 • Marco Cuturi, Arnaud Doucet

We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric.

11 code implementations • NeurIPS 2013 • Marco Cuturi

Optimal transportation distances are a fundamental family of parameterized distances for histograms.

no code implementations • NeurIPS 2009 • Marco Cuturi, Jean-Philippe Vert, Alexandre d'Aspremont

The candidate functionals are estimated in a subset of a reproducing kernel Hilbert space associated with the set where the process takes values.

no code implementations • 6 Oct 2006 • Marco Cuturi, Jean-Philippe Vert, Oystein Birkenes, Tomoko Matsui

We propose in this paper a new family of kernels to handle times series, notably speech data, within the framework of kernel methods which includes popular algorithms such as the Support Vector Machine.

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