arXiv research

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

Optimizes transport on submanifolds for curvature inequalities.

We extend Sobolev transport to unbalanced measures on graphs.

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

A new metric for comparing probability measures on graphs, scalable and negative definite.

USD algorithm transports distributions with or without mass conservation.

Paper introduces SGA for barycenter optimization in optimal transport.

The paper proves $L^p$-Sobolev inequalities for minimal submanifolds.

Proposes GST for efficient computation of generalized Sobolev transport on graph metrics.

Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.

Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequa…

A new method for transporting unbalanced measures on graphs efficiently.

Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.

We study a simplification of GAN training: the problem of transporting particles from a source to a target distribution. Starting from the Sobolev GAN critic, part of the gradient regularized GAN family, we show a strong relation with Optimal Transport (OT). Specifically with the less popular dynamic formulation of OT …

We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm $||.||_{\dot{H}^{-1}(ν_q)}$, that is known to linearize the Wasserstein $W_2$ distance and plays a fundamental role in the dynamic formulation of…

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\d…

Sharp $L^p$-logarithmic-Sobolev inequalities on submanifolds with applications to hypercontractivity.

New algorithm computes optimal transport barycenter efficiently.

Study nonparametric density estimation via measure transport, achieving optimal rates.

New Langevin dynamics samples from entropy-regularized optimal transport.

Theory of parallel transport on non-collapsed RCD spaces established.

Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its …

Proves error bounds for PGD, extending log-Sobolev and Talagrand inequalities.

Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are $C^1$…

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.

Proves well-posedness for hard phase model in general relativity.

The paper analyzes rates of convergence for optimal transport map estimators using barycentric projections.

Study approximates probability measures using structured classes of functions.

Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform…

Paper explores stability, regularization, and gradient flows for stochastic inverse problems.

We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_μ(M)$ of the diffeomorphism group of a compact manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev $\dot{H}^1$-metric. We construct an explicit isometry f…

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show …

The Sinkhorn flow is a gradient flow in a nonlocal Wasserstein geometry.

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

Sharp inequality found on three-balls for fourth order Sobolev traces.

Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.

New methods improve stability of Sinkhorn algorithm in machine learning.

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $μ$. We show that the Sobolev IPM compares two distributions in hig…

The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding the $L^2$-Wasserstein distance of optimal mass transport, and the Fisher--Rao me…

Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow …

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

Proves Sobolev inequality on manifolds with specific curvature properties.

A new method improves flow matching by dynamically weighting density estimates.