no code implementations • 23 May 2024 • Ferdinand Genans, Antoine Godichon-Baggioni, François-Xavier Vialard, Olivier Wintenberger

In this work, we answer positively to this question by (i) proving an $\mathcal{O}(t^{-1})$ lower bound rate for the OT map, using the similarity between Laguerre cells estimation and density support estimation, and (ii) proposing a Stochastic Gradient Descent (SGD) algorithm with adaptive entropic regularization and averaging acceleration.

no code implementations • 19 Mar 2024 • Raphaël Barboni, Gabriel Peyré, François-Xavier Vialard

We study the convergence of gradient flow for the training of deep neural networks.

1 code implementation • CVPR 2023 • Lin Tian, Hastings Greer, François-Xavier Vialard, Roland Kwitt, Raúl San José Estépar, Richard Jarrett Rushmore, Nikolaos Makris, Sylvain Bouix, Marc Niethammer

We present an approach to learning regular spatial transformations between image pairs in the context of medical image registration.

no code implementations • 16 Nov 2022 • Thibault Séjourné, Gabriel Peyré, François-Xavier Vialard

Optimal Transport (OT) has recently emerged as a central tool in data sciences to compare in a geometrically faithful way point clouds and more generally probability distributions.

3 code implementations • 13 Jun 2022 • Lin Tian, Hastings Greer, François-Xavier Vialard, Roland Kwitt, Raúl San José Estépar, Richard Jarrett Rushmore, Nikolaos Makris, Sylvain Bouix, Marc Niethammer

We present an approach to learning regular spatial transformations between image pairs in the context of medical image registration.

1 code implementation • 3 Feb 2022 • Bernd Sturmfels, Simon Telen, François-Xavier Vialard, Max von Renesse

Entropic regularization is a method for large-scale linear programming.

no code implementations • 3 Jan 2022 • Thibault Séjourné, François-Xavier Vialard, Gabriel Peyré

In this work, we identify the cause for this deficiency, namely the lack of a global normalization of the iterates, which equivalently corresponds to a translation of the dual OT potentials.

1 code implementation • 10 Dec 2021 • Raphaël Barboni, Gabriel Peyré, François-Xavier Vialard

To bridge the gap between the lazy and mean field regimes, we study Residual Networks (ResNets) in which the residual block has linear parametrization while still being nonlinear.

no code implementations • 3 Dec 2021 • Boris Muzellec, Adrien Vacher, Francis Bach, François-Xavier Vialard, Alessandro Rudi

It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds.

2 code implementations • NeurIPS 2021 • Thibault Séjourné, François-Xavier Vialard, Gabriel Peyré

The GW distance is however limited to the comparison of metric measure spaces endowed with a probability distribution.

no code implementations • NeurIPS 2020 • François-Xavier Vialard, Roland Kwitt, Susan Wei, Marc Niethammer

Continuous-depth neural networks can be viewed as deep limits of discrete neural networks whose dynamics resemble a discretization of an ordinary differential equation (ODE).

no code implementations • NeurIPS 2020 • Lenaic Chizat, Pierre Roussillon, Flavien Léger, François-Xavier Vialard, Gabriel Peyré

We also propose and analyze an estimator based on Richardson extrapolation of the Sinkhorn divergence which enjoys improved statistical and computational efficiency guarantees, under a condition on the regularity of the approximation error, which is in particular satisfied for Gaussian densities.

4 code implementations • 28 Oct 2019 • Thibault Séjourné, Jean Feydy, François-Xavier Vialard, Alain Trouvé, Gabriel Peyré

Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems.

1 code implementation • NeurIPS 2019 • Zhengyang Shen, François-Xavier Vialard, Marc Niethammer

We introduce a region-specific diffeomorphic metric mapping (RDMM) registration approach.

Ranked #1 on Image Registration on Osteoarthritis Initiative

1 code implementation • 18 Oct 2018 • Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouvé, Gabriel Peyré

Comparing probability distributions is a fundamental problem in data sciences.

Statistics Theory Statistics Theory 62

1 code implementation • 20 Dec 2016 • Gabriel Peyré, Lenaïc Chizat, François-Xavier Vialard, Justin Solomon

This "quantum" formulation of OT (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy.

Graphics

3 code implementations • 20 Jul 2016 • Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, François-Xavier Vialard

This article introduces a new class of fast algorithms to approximate variational problems involving unbalanced optimal transport.

Optimization and Control 65K10

1 code implementation • 21 Aug 2015 • Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, François-Xavier Vialard

These distances are defined by two equivalent alternative formulations: (i) a "fluid dynamic" formulation defining the distance as a geodesic distance over the space of measures (ii) a static "Kantorovich" formulation where the distance is the minimum of an optimization program over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures.

Optimization and Control

1 code implementation • 22 Jun 2015 • Lenaic Chizat, Bernhard Schmitzer, Gabriel Peyré, François-Xavier Vialard

This metric interpolates between the quadratic Wasserstein and the Fisher-Rao metrics and generalizes optimal transport to measures with different masses.

Analysis of PDEs

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