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We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry.…

We introduce dynamic and static formulations that formally extend unbalanced optimal transport from the space of positive densities to the space of Riemannian metrics. The first construction is based on a dynamic variational formulation in…

This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…

While many Machine Learning methods were developed or transposed on Riemannian manifolds to tackle data with known non Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on…

In this paper, we propose an extended mixed finite element method for elliptic interface problems. By adding some stabilization terms, we present a mixed approximation form based on Brezzi-Douglas-Marini element space and the piecewise…

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with…

Latent-variable energy-based models (LVEBMs) assign a single normalized energy to joint pairs of observed data and latent variables, offering expressive generative modeling while capturing hidden structure. We recast maximum-likelihood…

We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived…

We extend the linear mixed-effects state model to accommodate the correlated individuals and investigate its parameter and state estimation based on disturbance smoothing in this paper. For parameter estimation, EM and score based…

The Expectation Maximisation (EM) algorithm is widely used to optimise non-convex likelihood functions with latent variables. Many authors modified its simple design to fit more specific situations. For instance, the Expectation (E) step…

Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to…

Understanding proper distance measures between distributions is at the core of several learning tasks such as generative models, domain adaptation, clustering, etc. In this work, we focus on mixture distributions that arise naturally in…

A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and…

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate…

Optimization in the Bures-Wasserstein space has been gaining popularity in the machine learning community since it draws connections between variational inference and Wasserstein gradient flows. The variational inference objective function…

Designing experiments that systematically gather data from complex physical systems is central to accelerating scientific discovery. While Bayesian experimental design (BED) provides a principled, information-based framework that integrates…

Empirical divergence maximization (EDM) refers to a recently proposed strategy for estimating f-divergences and likelihood ratio functions. This paper extends the idea to empirical vector quantization where one seeks to empirically derive…

We study first-order optimality conditions for constrained optimization in the Wasserstein space, whereby one seeks to minimize a real-valued function over the space of probability measures endowed with the Wasserstein distance. Our…

This work presents a novel and effective method for fitting multidimensional ellipsoids to scattered data in the contamination of noise and outliers. We approach the problem as a Bayesian parameter estimate process and maximize the…