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

Approximating a probability distribution using a set of particles is a fundamental problem in machine learning and statistics, with applications including clustering and quantization. Formally, we seek a weighted mixture of Dirac measures…

Wasserstein Gradient Flows (WGF) with respect to specific functionals have been widely used in the machine learning literature. Recently, neural networks have been adopted to approximate certain intractable parts of the underlying…

Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as a powerful discrepancy measure for probability distributions. The recent developments on…

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in…

For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such…

The $E$-optimality criterion for a regression model maximizes the smallest eigenvalue of the information matrix and becomes non-differentiable when this eigenvalue has multiplicity greater than one. Working in the $2$-Wasserstein space, we…

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because…

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal…

We propose a novel supervised learning method to optimize the kernel in the maximum mean discrepancy generative adversarial networks (MMD GANs), and the kernel support vector machines (SVMs). Specifically, we characterize a distributionally…

We present a computationally efficient framework, called $\texttt{FlowDRO}$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets while aiming to find continuous worst-case…

We study the convergence of gradient flow for the training of deep neural networks. If Residual Neural Networks are a popular example of very deep architectures, their training constitutes a challenging optimization problem due notably to…

We propose a learning framework for graph kernels, which is theoretically grounded on regularizing optimal transport. This framework provides a novel optimal transport distance metric, namely Regularized Wasserstein (RW) discrepancy, which…

Many machine learning problems can be seen as approximating a \textit{target} distribution using a \textit{particle} distribution by minimizing their statistical discrepancy. Wasserstein Gradient Flow can move particles along a path that…

Most graph kernels are an instance of the class of $\mathcal{R}$-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of…

We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces…

Gradient flow in the 2-Wasserstein space is widely used to optimize functionals over probability distributions and is typically implemented using an interacting particle system with $n$ particles. Analyzing these algorithms requires showing…

Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood…

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the…

Flow Matching has become a cornerstone of modern generative models like Stable Diffusion 3, largely due to the efficiency of its Rectified Flow (RF) variant. The success of RF hinges on iteratively learning straight trajectories, pushing…

Stein variational gradient descent (SVGD) and its variants have shown promising successes in approximate inference for complex distributions. In practice, we notice that the kernel used in SVGD-based methods has a decisive effect on the…