Related papers: Wasserstein gradient flows for Coulomb discrepancies

Wasserstein gradient flows for Coulomb discrepancies

URL: http://arxiv.org/abs/2607.12579v1
Date: Tue, 14 Jul 2026 09:49:46 GMT
Title: Wasserstein gradient flows for Coulomb discrepancies
Abstract summary: We study the long-time behavior of the gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $$ and a target measure $$, where the underlying kernel is given by a Coulomb potential.<n>First, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove an ultracontractive estimate.<n>We also investigate the regularity of these solutions, showing that the Hlder norm can grow exponentially in time.
Score: 0.0
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
Abstract: We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $ρ$ and a target measure $μ$, where the underlying kernel is given by a Coulomb potential. First, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove an ultracontractive estimate, showing that the density $ρ_t$ becomes instantly bounded in $L^\infty$ for $t>0$. We also investigate the regularity of these solutions, showing that the Hölder norm can grow exponentially in time. Second, on the flat torus $\mathbb T^\mathsf{d}$, we prove exponential decay of the squared MMD along the flow toward a uniformly positive target $μ$, without requiring a lower bound on the initial data. This result is based on a ''defective Polyak-Lojasiewicz (PL) inequality'' whose defect term accounts for possible vacuum regions in the evolving density. We also prove that the usual PL inequality may fail when the target vanishes only at one point, and, in dimensions at least two, that no coercivity constant can depend only on a prescribed positive lower bound for the target. Finally, on $\mathbb R^\mathsf{d}$, we identify an obstruction at spatial infinity. For a compactly supported target, uniformly localized sources initially separated from the target by distance $D$ retain a fixed fraction of their initial squared MMD for times of order $D$. Consequently, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold on the unrestricted whole-space class. By contrast, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence.

Related papers

Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels [1.1593327400723945]
We study the flow of the squared Mean Discrepmath (MMD) generated by the nonsmooth energy kernels.<n>We show that no-data-independent multiplicative regimes exist on $mathRd and that global convergence.<n>in the well-posedness range.
arXiv Detail & Related papers (2026-08-02T11:59:43Z)
Near-Optimal Mode Scaling for Finite-Dimensional Boson Sampling via Lie-Algebraic Leakage Bounds [0.8594140167290097]
Boson sampling on matter-based platforms requires encoding the bosonic modes in finite-dimensional local Hilbert spaces.<n>We develop a unified framework for non-interacting sampling on the irreducible representations of compact Lie groups.<n>We prove, in a Gaussian model of the transition matrix, that its spectral norm concentrates at $tildeO(stn)$ rather than the $O(n)$ worst-case of prior spin-based emulations.
arXiv Detail & Related papers (2026-07-13T15:36:29Z)
Is Spurious Correlation Removal Always Learnable? [56.28155520961125]
Invariant learning can fail even when the structure is statistically identifiable.<n>We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive, there exist emphamplable multi-dimensional environments.<n>Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.
arXiv Detail & Related papers (2026-06-11T05:49:43Z)
Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds [61.14405512940818]
We show that Score-based generative models are trained in high-dimensional ambient spaces.<n>We use a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss Newton.
arXiv Detail & Related papers (2026-05-15T10:20:05Z)
Dimension-Independent Convergence of Underdamped Langevin Monte Carlo in KL Divergence [50.719298242863744]
Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $propto e-V$.<n>We prove the first dimension-free KL divergence bounds for discretized ULD.
arXiv Detail & Related papers (2026-03-02T22:14:38Z)
Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies [10.511277414974613]
We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy functionals.<n>Our setting covers in particular the training dynamics of shallow neural networks in the infinite-width and continuous time limit.
arXiv Detail & Related papers (2026-03-02T15:32:54Z)
Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
arXiv Detail & Related papers (2024-04-29T15:04:07Z)
Estimation of entropy-regularized optimal transport maps between non-compactly supported measures [15.857723276537248]
This paper addresses the problem of estimating entropy-regularized optimal transport maps with squared-Euclidean cost between source and target measures that are subGaussian.
arXiv Detail & Related papers (2023-11-20T17:18:21Z)
Theory of free fermions under random projective measurements [43.04146484262759]
We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers. We derive a non-linear sigma model (NLSM) as an effective field theory of the problem.
arXiv Detail & Related papers (2023-04-06T15:19:33Z)
Estimation and inference for the Wasserstein distance between mixing measures in topic models [18.66039789963639]
The Wasserstein distance between mixing measures has come to occupy a central place in the statistical analysis of mixture models. This work proposes a new canonical interpretation of this distance and provides tools to perform inference on the Wasserstein distance in topic models.
arXiv Detail & Related papers (2022-06-26T02:33:40Z)
Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo [60.785586069299356]
This work provides a general framework for the non-asymotic analysis of sampling error in 2-Wasserstein distance. Our theoretical analysis is further validated by numerical experiments.
arXiv Detail & Related papers (2021-09-08T18:00:05Z)
Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence [8.873449722727026]
We show that the rate estimate $widetildemathcalO(depsilon-1)$ improves the previously known rates in both of these metrics. In particular, for convex and firstorder smooth potentials, we show that LMC algorithm achieves the rate estimate $widetildemathcalO(depsilon-1)$ which improves the previously known rates in both of these metrics.
arXiv Detail & Related papers (2020-07-22T18:18:28Z)
On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
arXiv Detail & Related papers (2020-04-09T17:54:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.