arXiv research

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the $L^2$-space produces the same evolution as the gradient flow of the relative entropy in the $L^2$-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…

Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wassers…

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on …

The curse of dimensionality affects neural network optimization, especially with smooth functions.

The Sinkhorn flow converges to a Wasserstein mirror gradient flow from the Sinkhorn algorithm.

Study shows policy gradient convergence for entropy-regularized MDPs with neural nets in mean-field regime.

New method uses Fokker-Planck equation for sampling and inference.

Improved KL bounds and Wasserstein guarantees for diffusion flow matching under minimal conditions.

Introduces new info-geometric structure for dynamics on graphs and hypergraphs.

This work explores gradient flows and Riemannian structure in Gromov-Wasserstein geometry for data with global structure.

The paper shows how heat flows and Wasserstein distances relate to space rigidity.

New bounds for generative models under weaker assumptions.

Stochastic particle-optimization sampling (SPOS) is a recently-developed scalable Bayesian sampling framework that unifies stochastic gradient MCMC (SG-MCMC) and Stein variational gradient descent (SVGD) algorithms based on Wasserstein gradient flows. With a rigorous non-asymptotic convergence theory developed recently…

Kim-Milman flow map stable under regular target measures

Improved reSGLD accelerates convergence in non-convex learning problems.

New Langevin method achieves third order convergence for strongly log-concave distributions.

A new stochastic algorithm approximates optimal distributions without requiring propagation of chaos.

Upper bound for max-sliced 2-Wasserstein distance between measures.

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 latter with a positive definite metric tensor, under which it becomes a Riemanni…

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manag…

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 observed in the works of Mei, Montanari and Nguyen (2018), Chizat and Bach (2018) as …

New method aligns diffusion models for inference-time properties without retraining.

New algorithms sample from log concave distributions without gradient Lipschitz continuity.

A new robust metric compares distributions more accurately than existing methods.

Paper proposes new Langevin samplers for sampling from log-concave distributions with superlinear gradient growth.

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric meas…

Study entropic regularization of Gaussian measures and processes on Hilbert space.

This work studies Gaussian geometry under entropy-regularized 2-Wasserstein distance.

We propose a fast stochastic Hamilton Monte Carlo (HMC) method, for sampling from a smooth and strongly log-concave distribution. At the core of our proposed method is a variance reduction technique inspired by the recent advance in stochastic optimization. We show that, to achieve $ε$ accuracy in 2-Wasserstein distanc…

New method uses trainable activations to make BNNs behave like GPs.

Paper analyzes convergence of ODE samplers in Wasserstein distances.

This note shows how independent elliptical distributions minimize the Wasserstein distance.

Introduces spectral-domain Wasserstein distance and Gelbrich bound for elliptical processes.

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optim…

Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to g…

As an important Markov Chain Monte Carlo (MCMC) method, stochastic gradient Langevin dynamics (SGLD) algorithm has achieved great success in Bayesian learning and posterior sampling. However, SGLD typically suffers from slow convergence rate due to its large variance caused by the stochastic gradient. In order to allev…

New method estimates covariance in deep heteroscedastic regression without labels.

Study non-asymptotic Langevin Monte Carlo for Gibbs distributions.

A new algorithm computes Wasserstein barycenters without entropic regularization.

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorit…

DE-PSGLD samples from constrained distributions in a decentralized manner.

Improved sampling method using regularized Stein Variational Gradient Flow.

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…

LightSBB-M improves generative diffusion modeling with lower 2-Wasserstein distances.

We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to $e^{-f}$ where $f:\mathbb{R}^d \to \mathbb{R}$ is $μ$-strongly convex and $L$-smooth (the condition number is $κ= L/μ$). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is $O(κ)$, improving…

New HMC framework reduces variance for sampling from log-concave distributions.

Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.