Related papers: Muon Dynamics as a Spectral Wasserstein Flow
Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment…
Optimization lies at the core of modern deep learning, yet existing methods often face a fundamental trade-off between adapting to problem geometry and leveraging curvature utilization. Steepest descent algorithms adapt to different…
This article establishes the cutoff phenomenon in the Wasserstein distance for systems of nonlinear ordinary differential equations with a unique coercive stable fixed point subject to general additive Markovian noise in the limit of small…
We numerically investigate a mean-field Bayesian approach with the assistance of the Markov chain Monte Carlo method to estimate motion velocity fields and probabilistic models simultaneously in consecutive digital images described by…
Consider a probability measure on a Hilbert space defined via its density with respect to a Gaussian. The purpose of this paper is to demonstrate that an appropriately defined Markov chain, which is reversible with respect to the measure in…
Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of…
In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular,…
We introduce ImitationFlow, a novel Deep generative model that allows learning complex globally stable, stochastic, nonlinear dynamics. Our approach extends the Normalizing Flows framework to learn stable Stochastic Differential Equations.…
Wasserstein gradient flow (WGF) is a common method to perform optimization over the space of probability measures. While WGF is guaranteed to converge to a first-order stationary point, for nonconvex functionals the converged solution does…
In this work, we establish non-asymptotic convergence bounds for the Gauss-Newton method in training neural networks with smooth activations. In the underparameterized regime, the Gauss-Newton gradient flow in parameter space induces a…
We consider $M/Ph/n+M$ queueing systems in steady state. We prove that the Wasserstein distance between the stationary distribution of the normalized system size process and that of a piecewise Ornstein-Uhlenbeck (OU) process is bounded by…
Issued from Optimal Transport, the Wasserstein distance has gained importance in Machine Learning due to its appealing geometrical properties and the increasing availability of efficient approximations. In this work, we consider the problem…
This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the…
Many machine learning problems can be expressed as the optimization of some cost functional over a parametric family of probability distributions. It is often beneficial to solve such optimization problems using natural gradient methods.…
We present a computational framework for efficient learning, sampling, and distribution of general Bayesian posterior distributions. The framework leverages a machine learning approach for the construction of normalizing flows for the…
Optimal Transport has sparked vivid interest in recent years, in particular thanks to the Wasserstein distance, which provides a geometrically sensible and intuitive way of comparing probability measures. For computational reasons, the…
Based on the manifold hypothesis, real-world data often lie on a low-dimensional manifold, while normalizing flows as a likelihood-based generative model are incapable of finding this manifold due to their structural constraints. So, one…
Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine…
Probabilistic smoothing is a standard tool for global optimization, but existing methods rely on Gaussian kernels and specific transforms, often resulting in strong hyperparameter sensitivity and limited robustness. We propose a general…
Recent results have shown that for two-layer fully connected neural networks, gradient flow converges to a global optimum in the infinite width limit, by making a connection between the mean field dynamics and the Wasserstein gradient flow.…