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We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to…
We present a short overview on the strongest variational formulation for gradient flows of geodesically $\lambda$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures.…
Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study…
In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the…
The scarcity of labeled data is a long-standing challenge for many machine learning tasks. We propose our gradient flow method to leverage the existing dataset (i.e., source) to generate new samples that are close to the dataset of interest…
A recurring obstacle in the study of Wasserstein gradient flow is the lack of convexity of the square Wasserstein metric. In this paper, we develop a class of transport metrics that have better convexity properties and use these metrics to…
Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with…
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…
An exciting branch of machine learning research focuses on methods for learning, optimizing, and integrating unknown functions that are difficult or costly to evaluate. A popular Bayesian approach to this problem uses a Gaussian process…
In this work, we analyze two of the most fundamental algorithms in geodesically convex optimization: Riemannian gradient descent and (possibly inexact) Riemannian proximal point. We quantify their rates of convergence and produce different…
The Wasserstein space of probability measures is known for its intricate Riemannian structure, which underpins the Wasserstein geometry and enables gradient flow algorithms. However, the Wasserstein geometry may not be suitable for certain…
We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We…
Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general non-equilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasi-convex…
In this paper, a new probability distribution, referred to as the matrix Fisher-Gaussian (MFG) distribution, is proposed on the nonlinear manifold $\mathrm{SO}(3)\times\mathbb{R}^n$. It is constructed by conditioning a (9+n)-variate…
Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically…
Flow matching has emerged as a powerful generative modeling approach with flexible choices of source distribution. While Gaussian distributions are commonly used, the potential for better alternatives in high-dimensional data generation…
Recently developed particle flow algorithms provide an alternative to importance sampling for drawing particles from a posterior distribution, and a number of particle filters based on this principle have been proposed. Samples are drawn…
We investigate a family of approximate multi-step proximal point methods, framed as implicit linear discretizations of gradient flow. The resulting methods are multi-step proximal point methods, with similar computational cost in each…
The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational…
In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The…