相关论文: Projected Langevin dynamics and a gradient flow fo…

Given a particle system obeying overdamped Langevin dynamics, we demonstrate that it is always possible to construct a thermodynamically consistent macroscopic model which obeys a gradient flow with respect to its non-equilibrium free…

We present a novel method for drawing samples from Gibbs distributions with densities of the form $\pi(x) \propto \exp(-U(x))$. The method accelerates the unadjusted Langevin algorithm by introducing an inertia term similar to Polyak's…

Our work is motivated by a desire to study the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of neural networks. The key insight, already…

We study a system of drift-diffusion PDEs for a potentially infinite number of incompressible phases, subject to a joint pointwise volume constraint. Our analysis is based on the interpretation as a collection of coupled Wasserstein…

We study the least-energy way to reshape a probability distribution when motion is constrained to a horizontal bundle, that is, optimal transport and distribution steering in sub-Riemannian geometry, motivated by density control over…

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important…

Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…

We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic…

We consider an infinite lattice system of interacting spins living on a smooth compact manifold, with short- but not necessarily finite-range pairwise interactions. We construct the gradient flow of the infinite-volume free energy on the…

We consider the problem of sampling from a target distribution, which is \emph {not necessarily logconcave}, in the context of empirical risk minimization and stochastic optimization as presented in Raginsky et al. (2017). Non-asymptotic…

The sliced-Wasserstein flow is an evolution equation where a probability density evolves in time, advected by a velocity field computed as the average among directions in the unit sphere of the optimal transport displacements from its 1D…

We study sampling as optimization in the space of measures. We focus on gradient flow-based optimization with the Langevin dynamics as a case study. We investigate the source of the bias of the unadjusted Langevin algorithm (ULA) in…

We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost…

We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the $L^2$-Wasserstein metric tensor in the probability density space to a parameter space, equipping the…

The mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional on the Wasserstein space over $\mathbb{R}^d$, and has gained attention recently as a model for the gradient descent dynamics of interacting…

This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport.…

We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density $\nu$ with respect to the natural measure on a manifold with metric $g$. We assume that the target density satisfies a log-Sobolev…

Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach…

The least gradient problem (minimizing the total variation with given boundary data) is equivalent, in the plane, to the Beckmann minimal-flow problem with source and target measures located on the boundary of the domain, which is in turn…

We develop a discrete optimal transport framework for analyzing simulated annealing algorithms on finite state spaces. Building on the discrete Wasserstein metric introduced by Maas (J. Funct. Anal., 2011), we define a generalized discrete…