arXiv subjects

Flavien Léger

Publications and source records attributed to Flavien Léger.

15 recordsLinked to original sources

arXiv2025

A synthetic approach to comparison principles for variational problems, with applications to optimal transport

We develop a synthetic, variational framework for deriving comparison principles in infinite-dimensional Banach spaces. Unlike traditional approaches that rely on the regularity of minimizers and Euler--Lagrange equations, our method exploits the order-theoretic structure of the energy. Central to our analysis is the notion of submodularity and its convex dual, substitutability, which we extend here to the infinite-dimensional setting. We prove a duality theorem establishing that a convex functional is submodular if and only if its conjugate is substitutable. We apply these results to problems in optimal transport, and derive comparison principles for Kantorovich potentials in standard, entropic, and unbalanced settings without requiring regularity assumptions on the cost or domain. Finally, we prove that general transport costs are substitutable, yielding comparison principles for JKO schemes driven by internal energies.

math.OC↗

arXiv2025

Nonnegative cross-curvature in infinite dimensions: synthetic definition and spaces of measures

Nonnegative cross-curvature (NNCC) is a geometric property of a cost function defined on a product space that originates in optimal transportation and the Ma-Trudinger-Wang theory. Motivated by applications in optimization, gradient flows and mechanism design, we propose a variational formulation of nonnegative cross-curvature on c-convex domains applicable to infinite dimensions and nonsmooth settings. The resulting class of NNCC spaces is closed under Gromov-Hausdorff convergence and for this class, we extend many properties of classical nonnegative cross-curvature: stability under generalized Riemannian submersions, characterization in terms of the convexity of certain sets of c-concave functions, and in the metric case, it is a subclass of positively curved spaces in the sense of Alexandrov. One of our main results is that Wasserstein spaces of probability measures inherit the NNCC property from their base space. Additional examples of NNCC costs include the Bures-Wasserstein and Fisher-Rao squared distances, the Hellinger-Kantorovich squared distance (in some cases), the relative entropy on probability measures, and the 2-Gromov-Wasserstein squared distance on metric measure spaces.

math.MG↗

arXiv2024

A geometric approach to apriori estimates for optimal transport maps

A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma--Trudinger--Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior $C^1$ estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.

math.DG↗

arXiv2023

Gradient descent with a general cost

We present a new class of gradient-type optimization methods that extends vanilla gradient descent, mirror descent, Riemannian gradient descent, and natural gradient descent. Our approach involves constructing a surrogate for the objective function in a systematic manner, based on a chosen cost function. This surrogate is then minimized using an alternating minimization scheme. Using optimal transport theory we establish convergence rates based on generalized notions of smoothness and convexity. We provide local versions of these two notions when the cost satisfies a condition known as nonnegative cross-curvature. In particular our framework provides the first global rates for natural gradient descent and the standard Newton's method.

math.OC↗

arXiv2022

A geometric Laplace method

A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order Laplace formula is most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order term of the Laplace method. The central tool is the Kim-McCann Riemannian metric which was introduced in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim-McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.

math.DG↗

arXiv2022

Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM

Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.

math.OC↗

arXiv2020

The back-and-forth method for Wasserstein gradient flows

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced by Jacobs and Léger to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large-scale simulations for a large class of internal energies including singular and non-convex energies.

math.NA↗

arXiv2020

Faster Wasserstein Distance Estimation with the Sinkhorn Divergence

The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to $ε$-accuracy by adding an entropic regularization of order $ε$ and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample complexity but allows higher regularization levels, of order $ε^{1/2}$, which leads to improved computational complexity bounds and a strong speedup in practice. Our theoretical analysis covers the case of both randomly sampled densities and deterministic discretizations on uniform grids. 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. We finally demonstrate the efficiency of the proposed estimators with numerical experiments.

math.OC↗

arXiv2020

A gradient descent perspective on Sinkhorn

We present a new perspective on the popular Sinkhorn algorithm, showing that it can be seen as a Bregman gradient descent (mirror descent) of a relative entropy (Kullback-Leibler divergence). This viewpoint implies a new sublinear convergence rate with a robust constant.

math.OC↗

arXiv2020

A fast approach to optimal transport: The back-and-forth method

We present an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs. Given two probability measures supported on a discrete grid with n points, we compute the optimal map using O(n) storage space and O(n log(n)) operations per iteration, with an approximately exponential convergence rate. Our approach allows us to solve optimal transportation problems on spatial grids as large as 4096x4096 and 384x384x384 in a matter of minutes.

math.OC↗

arXiv2019

Hopf-Cole transformation via generalized Schrödinger bridge problem

We study generalized Hopf-Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf-Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf-Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.

math.DS↗

arXiv2018

Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size

We present a primal-dual method to solve L1-type non-smooth optimization problems independently of the grid size. We apply these results to two important problems : the Rudin-Osher-Fatemi image denoising model and the L1 earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 by 4096 in a few minutes without parallelization.

math.OC↗

arXiv2017

A geometric perspective on regularized optimal transport

We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the Schrödinger bridge problem and the Yasue problem. We also propose an analogue of the Hopf--Cole transformation in the geometric setting.

math.OC↗

arXiv2016

A new approach to bounds on mixing

We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an $L^1$ norm of the velocity field. Existing results in the literature use an $L^p$ norm with $p>1$. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perpective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.

math.AP↗

arXiv2010

Efficient Matrix Completion with Gaussian Models

A general framework based on Gaussian models and a MAP-EM algorithm is introduced in this paper for solving matrix/table completion problems. The numerical experiments with the standard and challenging movie ratings data show that the proposed approach, based on probably one of the simplest probabilistic models, leads to the results in the same ballpark as the state-of-the-art, at a lower computational cost.

cs.LG↗