arXiv subjects

Pierre Del Moral

Publications and source records attributed to Pierre Del Moral.

At least 19 recordsLinked to original sources

arXiv2026

On Bridging Mixture Distributions

The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with $d$ the dimension and $N$ the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order $(d\log N/N)^{1/2}+d^2\log N/N$ with probability at least $1-10N^{-1}$. For single Gaussian marginals estimated by their sample moments, it is of order $d^2/N$ in expectation. For the regularized empirical Monge map based on an $ε$-inflation of the sample covariance, the mean squared error is of order $d^2\{(1+ε^{-1})/N+ε^2\}$; as soon as $N$ is a sufficiently large multiple of $d$, it reduces to $d^2/N+d\,ε^2$, up to an exponentially small term. These estimates are illustrated numerically.

math.ST↗

arXiv2026

On the Kantorovich contraction of Markov semigroups

This paper develops a novel operator theoretic framework to study the contraction properties of Markov semigroups with respect to a general class of Kantorovich semi-distances, which notably includes Wasserstein distances. The rather simple contraction cost framework developed in this article, which combines standard Lyapunov techniques with local contraction conditions, helps to unifying and simplifying many arguments in the stability of Markov semigroups, as well as to improve upon some existing results. Our results can be applied to both discrete time and continuous time Markov semigroups, and we illustrate their wide applicability in the context of (i) Markov transitions on models with boundary states, including bounded domains with entrance boundaries, (ii) operator products of a Markov kernel and its adjoint, including two-block-type Gibbs samplers, (iii) iterated random functions and (iv) diffusion models, including overdampted Langevin diffusion with convex at infinity potentials.

math.PR↗

arXiv2026

Fluctuations and Long-Time Stability of Multivariate Ensemble Kalman Filters

We develop a self contained stochastic perturbation theory for discrete generation and multivariate Ensemble Kalman filters. Unlike their continuous-time counterparts, discrete EnKF algorithms are defined through a two steps prediction update mechanism and exhibit non Gaussian fluctuations, even in linear settings. In the multivariate case, these fluctuations take the form of non central Wishart type perturbations, which significantly complicate the mathematical analysis. We establish non asymptotic, time-uniform stability and error estimates for the ensemble covariance matrix processes under minimal structural assumptions on the signal observation model, allowing for possibly unstable dynamics. Our results quantify the impact of ensemble size, dimension, and observation noise, and provide explicit bounds on the propagation of stochastic errors over long time horizons. The analysis relies on a detailed study of stochastic Riccati difference equations driven by matrix-valued noncentral Wishart fluctuations. Beyond their relevance to data assimilation, these results contribute to the probabilistic understanding of ensemble-based filtering methods in high dimension and offer new tools for the analysis of interacting particle systems with matrix-valued dynamics.

math.PR↗

arXiv2026

New Trends in the Stability of Sinkhorn Semigroups

Entropic optimal transport problems play an increasingly important role in machine learning and generative modelling. In contrast with optimal transport maps which often have limited applicability in high dimensions, Schrodinger bridges can be solved using the celebrated Sinkhorn's algorithm, a.k.a. the iterative proportional fitting procedure. The stability properties of Sinkhorn bridges when the number of iterations tends to infinity is a very active research area in applied probability and machine learning. Traditional proofs of convergence are mainly based on nonlinear versions of Perron-Frobenius theory and related Hilbert projective metric techniques, gradient descent, Bregman divergence techniques and Hamilton-Jacobi-Bellman equations, including propagation of convexity profiles based on coupling diffusions by reflection methods. The objective of this review article is to present, in a self-contained manner, recently developed Sinkhorn/Gibbs-type semigroup analysis based upon contraction coefficients and Lyapunov-type operator-theoretic techniques. These powerful, off-the-shelf semigroup methods are based upon transportation cost inequalities (e.g. log-Sobolev, Talagrand quadratic inequality, curvature estimates), $ϕ$-divergences, Kantorovich-type criteria and Dobrushin contraction-type coefficients on weighted Banach spaces as well as Wasserstein distances. This novel semigroup analysis allows one to unify and simplify many arguments in the stability of Sinkhorn algorithm. It also yields new contraction estimates w.r.t. generalized $ϕ$-entropies, as well as weighted total variation norms, Kantorovich criteria and Wasserstein distances.

math.PR↗

arXiv2026

A Contraction Theory for Sinkhorn and Schrodinger Bridges via Log-Sobolev Inequalities

We develop a quantitative contraction framework for Schrodinger and Sinkhorn bridges based on transportation-cost inequalities and Riccati matrix difference equations. Our approach combines logarithmic Sobolev and Talagrand-type inequalities to obtain explicit entropy and Wasserstein contraction bounds for Sinkhorn bridge measures, entropic optimal transport plans, and the associated Markov transport maps. A key feature of the analysis is the interplay between transport-cost inequalities and matrix Riccati difference equations arising in filtering and stochastic control. The results are established under local regularity assumptions on the reference transition, formulated in terms of curvature, Lipschitz continuity, and Fisher-information bounds. Within this general setting, we derive quantitative stability and convergence estimates for Schrodinger bridges and Sinkhorn iterates that are robust with respect to the choice of reference measure. As a main application, we specialize the theory to linear-Gaussian reference transitions, where the Gaussian structure permits sharp constants, refined exponential decay rates, and continuity estimates for Schrodinger bridges, Sinkhorn iterates, barycentric projections, conditional covariances, and proximal sampler semigroups. In this setting, we recover and extend several known contraction results for entropic and Wasserstein distances, and obtain new quantitative bounds that improve previously available rates. Our results provide a unified probabilistic framework for stability, regularity, and convergence of Sinkhorn algorithms.

math.OC↗

arXiv2025

Gaussian entropic optimal transport: Schrödinger bridges and the Sinkhorn algorithm

Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings the Sinkhorn iterations are based on nonlinear conditional/conjugate transformations and exact finite-dimensional solutions cannot be computed. This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrödinger bridges.

stat.ML↗

arXiv2025

Entropic continuity bounds for conditional covariances with applications to Schr\" odinger and Sinkhorn bridges

The article presents new entropic continuity bounds for conditional expectations and conditional covariance matrices. These bounds are expressed in terms of the relative entropy between different coupling distributions. Our approach combines Wasserstein coupling with quadratic transportation cost inequalities. We illustrate the impact of these results in the context of entropic optimal transport problems. The entropic continuity theorem presented in the article allows to estimate the conditional expectations and the conditional covariances of Schr\" odinger and Sinkhorn transitions in terms of the relative entropy between the corresponding bridges. These entropic continuity bounds turns out to be a very useful tool for obtaining remarkably simple proofs of the exponential decays of the gradient and the Hessian of Schrödinger and Sinkhorn bridge potentials.

math.PR↗

arXiv2024

On the Particle Approximation of Lagged Feynman-Kac Formulae

In this paper we examine the numerical approximation of the limiting invariant measure associated with Feynman-Kac formulae. These are expressed in a discrete time formulation and are associated with a Markov chain and a potential function. The typical application considered here is the computation of eigenvalues associated with non-negative operators as found, for example, in physics or particle simulation of rare-events. We focus on a novel \emph{lagged} approximation of this invariant measure, based upon the introduction of a ratio of time-averaged Feynman-Kac marginals associated with a positive operator iterated $l \in\mathbb{N}$ times; a lagged Feynman-Kac formula. This estimator and its approximation using Diffusion Monte Carlo (DMC) have been extensively employed in the physics literature. In short, DMC is an iterative algorithm involving $N\in\mathbb{N}$ particles or walkers simulated in parallel, that undergo sampling and resampling operations. In this work, it is shown that for the DMC approximation of the lagged Feynman-Kac formula, one has an almost sure characterization of the $\mathbb{L}_1$-error as the time parameter (iteration) goes to infinity and this is at most of $\mathcal{O}(\exp\{-κl\}/N)$, for $κ>0$. In addition a non-asymptotic in time, and time uniform $\mathbb{L}_1-$bound is proved which is $\mathcal{O}(l/\sqrt{N})$. We also prove a novel central limit theorem to give a characterization of the exact asymptotic in time variance. This analysis demonstrates that the strategy used in physics, namely, to run DMC with $N$ and $l$ small and, for long time enough, is mathematically justified. Our results also suggest how one should choose $N$ and $l$ in practice. We emphasize that these results are not restricted to physical applications; they have broad relevance to the general problem of particle simulation of the Feynman-Kac formula.

math.PR↗

arXiv2023

On Time Uniform Wong-Zakai Approximation Theorems

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations; e.g. robust filtering problems. In many examples, the mean error estimate bounds that have been derived in the literature can grow exponentially with respect to the time horizon. We show in a simple example that indeed mean error estimates do explode exponentially in the time parameter, i.e. in that case a Wong-Zakai approximation is only useful for extremely short time intervals. Under spectral conditions, we present some quantitative time-uniform convergence theorems, i.e. time-uniform mean error bounds, yielding what seems to be the first results of this type for Wong-Zakai diffusion approximations.

math.PR↗

arXiv2023

Bayesian Parameter Inference for Partially Observed Diffusions using Multilevel Stochastic Runge-Kutta Methods

We consider the problem of Bayesian estimation of static parameters associated to a partially and discretely observed diffusion process. We assume that the exact transition dynamics of the diffusion process are unavailable, even up-to an unbiased estimator and that one must time-discretize the diffusion process. In such scenarios it has been shown how one can introduce the multilevel Monte Carlo method to reduce the cost to compute posterior expected values of the parameters for a pre-specified mean square error (MSE). These afore-mentioned methods rely on upon the Euler-Maruyama discretization scheme which is well-known in numerical analysis to have slow convergence properties. We adapt stochastic Runge-Kutta (SRK) methods for Bayesian parameter estimation of static parameters for diffusions. This can be implemented in high-dimensions of the diffusion and seemingly under-appreciated in the uncertainty quantification and statistics fields. For a class of diffusions and SRK methods, we consider the estimation of the posterior expectation of the parameters. We prove that to achieve a MSE of $\mathcal{O}(ε^2)$, for $ε>0$ given, the associated work is $\mathcal{O}(ε^{-2})$. Whilst the latter is achievable for the Milstein scheme, this method is often not applicable for diffusions in dimension larger than two. We also illustrate our methodology in several numerical examples.

stat.CO↗

arXiv2023

On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter's behaviour, e.g. it's signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman-Bucy filtering.We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

math.ST↗

arXiv2023

A Lyapunov approach to stability of positive semigroups: An overview with illustrations

The stability analysis of possibly time varying positive semigroups on non necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, physics and molecular chemistry. This article presents an overview of some recent Lyapunov-based approaches, focusing principally on practical and powerful tools for designing Lyapunov functions. These techniques include semigroup comparisons as well as conjugacy principles on non necessarily bounded manifolds with locally Lipschitz boundaries. All the Lyapunov methodologies discussed in the article are illustrated in a variety of situations, ranging from conventional Markov semigroups on general state spaces to more sophisticated conditional stochastic processes possibly restricted to some non necessarily bounded domains, including locally Lipschitz and smooth hypersurface boundaries, Langevin diffusions as well as coupled harmonic oscillators.

math.PR↗

arXiv2022

On the exit-problem for self-interacting diffusions

We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $σB_t$ for a constant $σ$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment potential $V$ and a convex interaction potential can be bounded uniformly with respect to $σ$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theory).

math.PR↗

arXiv2021

Log-Normalization Constant Estimation using the Ensemble Kalman-Bucy Filter with Application to High-Dimensional Models

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman-Bucy filter based estimates based upon several nonlinear Kalman-Bucy diffusions. Based upon new conditional bias results for the mean of the afore-mentioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_n-$errors and conditional bias. Depending on the type of nonlinear Kalman-Bucy diffusion, we show that these are of order $(\sqrt{t/N}) + t/N$ or $1/\sqrt{N}$ ($\mathbb{L}_n-$errors) and of order $[t+\sqrt{t}]/N$ or $1/N$ (conditional bias), where $t$ is the time horizon and $N$ is the ensemble size. Finally, we use these results for online static parameter estimation for above filtering models and implement the methodology for both linear and nonlinear models.

stat.CO↗

arXiv2020

Stochastic Epidemic Models inference and diagnosis with Poisson Random Measure Data Augmentation

We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministically from a parameter value and a PRM realisation. This forms the basis of our Data Augmented MCMC, which consists in augmenting parameter space with the unobserved PRM value. The resulting simple Metropolis-Hastings sampler acts as an efficient simulation-based inference method, that can easily be transferred from model to model. Compared with a recent Data Augmentation method based on Gibbs sampling of individual infection histories, PRM-augmented MCMC scales much better with epidemic size and is far more flexible. PRM-augmented MCMC also yields a posteriori estimates of the PRM, that represent process stochasticity, and which can be used to validate the model. If the model is good, the posterior distribution should exhibit no pattern and be close to the PRM prior distribution. We illustrate this by fitting a non-seasonal model to some simulated seasonal case count data. Applied to the Zika epidemic of 2013 in French Polynesia, our approach shows that a simple SEIR model cannot correctly reproduce both the initial sharp increase in the number of cases as well as the final proportion of seropositive. PRM-augmentation thus provides a coherent story for Stochastic Epidemic Model inference, where explicitly inferring process stochasticity helps with model validation.

stat.ME↗

arXiv2020

On the stability of matrix-valued Riccati diffusions

The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. This class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here applies to filtering problems with unstable signals.

math.PR↗

arXiv2019

Backward Nonlinear Smoothing Diffusions

We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow. This solution contrasts with, and complements, the (backward) deterministic flow of probability distributions (viz. a type of Kushner smoothing equation) studied in a number of prior works. A number of corollaries of our main result are given including a derivation of the time-reversal of a stochastic differential equation, and an immediate derivation of the classical Rauch-Tung-Striebel smoothing equations in the linear setting.

math.PR↗

arXiv2019

A perturbation analysis of stochastic matrix Riccati diffusions

Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.

math.PR↗