相关论文: Kernel Approximation of Fisher-Rao Gradient Flows

Accurately reconstructing continuous flow fields from sparse or indirect measurements remains an open challenge, as existing techniques often suffer from oversmoothing artifacts, reliance on heterogeneous architectures, and the…

These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…

Derivative-free Riemannian optimization (DFRO) aims to minimize an objective function using only function evaluations, under the constraint that the decision variables lie on a Riemannian manifold. The rapid increase in problem dimensions…

We propose a general purpose variational inference algorithm that forms a natural counterpart of gradient descent for optimization. Our method iteratively transports a set of particles to match the target distribution, by applying a form of…

We present a novel approach to approximate Gaussian and mixture-of-Gaussians filtering. Our method relies on a variational approximation via a gradient-flow representation. The gradient flow is derived from a Kullback--Leibler discrepancy…

Diffusion models and flow-based methods have shown impressive generative capability, especially for images, but their sampling is expensive because it requires many iterative updates. We introduce W-Flow, a framework for training a…

Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so…

We obtain upper bounds for the estimation error of Kernel Ridge Regression (KRR) for all non-negative regularization parameters, offering a geometric perspective on various phenomena in KRR. As applications: 1. We address the multiple…

We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which…

We develop connections between Stein's approximation method, logarithmic Sobolev and transport inequalities by introducing a new class of functional inequalities involving the relative entropy, the Stein kernel, the relative Fisher…

Directional data consist of observations distributed on a (hyper)sphere, and appear in many applied fields, such as astronomy, ecology, and environmental science. This paper studies both statistical and computational problems of kernel…

This contribution presents substantial computational advancements to compare measures even with varying masses. Specifically, we utilize the nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced…

Generative modeling typically concerns transporting a single source distribution to a target distribution via simple probability flows. However, in fields like computer graphics and single-cell genomics, samples themselves can be viewed as…

In this work, we study the Wasserstein gradient flow of the Riesz energy defined on the space of probability measures. The Riesz kernels define a quadratic functional on the space of measure which is not in general geodesically convex in…

We develop a variant of Stein's method of comparison of generators to bound the Kolmogorov, total variation, and Wasserstein-1 distances between distributions on the real line. Our discrepancy is expressed in terms of the ratio of reverse…

We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no…

We develop and analyze a projected particle Langevin optimization method to learn the distribution in the Sch\"{o}nberg integral representation of the radial basis functions from training samples. More specifically, we characterize a…

Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. Minimizing movement scheme was developed in order to study such curves in metric spaces.…

Wasserstein Policy Optimization (WPO) is a recently proposed reinforcement learning algorithm that leverages Wasserstein gradient flows to optimize stochastic policies in continuous action spaces. Despite its empirical success, the…

Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on…