arXiv subjects

Joaquin Fontbona

Publications and source records attributed to Joaquin Fontbona.

At least 19 recordsLinked to original sources

arXiv2025

An Explicit Euler-type Scheme for Lévy-driven SDEs with Superlinear and Time-Irregular Coefficients

This paper introduces a randomized tamed Euler scheme tailored for Lévy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carathéodory-type drift. Under assumptions that allow for time-irregular drifts while ensuring appropriate time-regularity of the diffusion and jump coefficients, the proposed scheme is shown to achieve the optimal strong $\mathcal{L}^2$-convergence rate, arbitrarily close to $0.5$. A crucial component of our methodology is the incorporation of drift randomization, which overcomes challenges due to low time-regularity, along with a taming technique to handle the superlinear state dependence. Our analysis moreover covers settings where the coefficients are random, providing for instance strong convergence of randomized tamed Euler schemes for Lévy-driven stochastic delay differential equations (SDDEs) with Markovian switching. To our knowledge, this is the first {work} that addresses the case of superlinear coefficients in the numerical analysis of Carathéodory-type SDEs and even for ordinary differential equations.

math.NA↗

arXiv2025

Symmetries in Overparametrized Neural Networks: A Mean-Field View

We develop a Mean-Field (MF) view of the learning dynamics of overparametrized Artificial Neural Networks (NN) under data symmetric in law wrt the action of a general compact group $G$. We consider for this a class of generalized shallow NNs given by an ensemble of $N$ multi-layer units, jointly trained using stochastic gradient descent (SGD) and possibly symmetry-leveraging (SL) techniques, such as Data Augmentation (DA), Feature Averaging (FA) or Equivariant Architectures (EA). We introduce the notions of weakly and strongly invariant laws (WI and SI) on the parameter space of each single unit, corresponding, respectively, to $G$-invariant distributions, and to distributions supported on parameters fixed by the group action (which encode EA). This allows us to define symmetric models compatible with taking $N\to\infty$ and give an interpretation of the asymptotic dynamics of DA, FA and EA in terms of Wasserstein Gradient Flows describing their MF limits. When activations respect the group action, we show that, for symmetric data, DA, FA and freely-trained models obey the exact same MF dynamic, which stays in the space of WI laws and minimizes therein the population risk. We also give a counterexample to the general attainability of an optimum over SI laws. Despite this, quite remarkably, we show that the set of SI laws is also preserved by the MF dynamics even when freely trained. This sharply contrasts the finite-$N$ setting, in which EAs are generally not preserved by unconstrained SGD. We illustrate the validity of our findings as $N$ gets larger in a teacher-student experimental setting, training a student NN to learn from a WI, SI or arbitrary teacher model through various SL schemes. We last deduce a data-driven heuristic to discover the largest subspace of parameters supporting SI distributions for a problem, that could be used for designing EA with minimal generalization error.

stat.ML↗

arXiv2023

Stochastic Gradient Descent for Barycenters in Wasserstein Space

We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters of absolutely continuous probability measures on Euclidean space. The proposed method can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, as well as a manifestation of a Law of Large Numbers therein. The algorithm aims at finding a Karcher mean or critical point in this setting, and can be implemented ``online", sequentially using i.i.d. random measures sampled from the population law. We provide natural sufficient conditions for this algorithm to a.s. converge in the Wasserstein space towards the population barycenter, and we introduce a novel, general condition which ensures uniqueness of Karcher means and moreover allows us to obtain explicit, parametric convergence rates for the expected optimality gap. We furthermore study the mini-batch version of this algorithm, and discuss examples of families of population laws to which our method and results can be applied. This work expands and deepens ideas and results introduced in an early version of \cite{backhoff2018bayesian}, in which a statistical application (and numerical implementation) of this method is developed in the context of Bayesian learning.

math.OC↗

arXiv2022

Bayesian Learning with Wasserstein Barycenters

We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC], and provide a numerical example for experimental validation of the proposed method.

stat.ML↗

arXiv2021

Data Science for Engineers: A Teaching Ecosystem

We describe an ecosystem for teaching data science (DS) to engineers which blends theory, methods, and applications, developed at the Faculty of Physical and Mathematical Sciences, Universidad de Chile, over the last three years. This initiative has been motivated by the increasing demand for DS qualifications both from academic and professional environments. The ecosystem is distributed in a collaborative fashion across three departments in the above Faculty and includes postgraduate programmes, courses, professional diplomas, data repositories, laboratories, trainee programmes, and internships. By sharing our teaching principles and the innovative components of our approach to teaching DS, we hope our experience can be useful to those developing their own DS programmes and ecosystems. The open challenges and future plans for our ecosystem are also discussed at the end of the article.

cs.CY↗

arXiv2020

Skeletal stochastic differential equations for superprocesses

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritial superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is individuals who produce eternal genealogical lines of decent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP). In a previous article, [16], we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever. Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

math.PR↗

arXiv2019

Skeletal stochastic differential equations for continuous-state branching process

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritial CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and super-processes is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article, we revisit these notions for CSBPs and put them in a common framework using the language of (coupled) SDEs. In this way, we are able to deal simultaneously with all types of CSBPs (supercritical, critical and subcritical) as well as understanding how the backbone representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. Our principal motivation is to prepare the way to expand the SDE approach to the spatial setting of super-processes, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes.

math.PR↗

arXiv2018

Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels

In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the number of neurons, the whole system synchronizes exponentially fast as time goes to infinity, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless of the interaction intensities). Combining these results, we deduce that the nonlinear McKean-Vlasov equation describing an infinite network of such neurons concentrates, as time goes to infinity, around the dynamics of a single Hodgkin-Huxley neuron with chemical neurotransmitter channels. Our results are illustrated and complemented with numerical simulations.

math.PR↗

arXiv2017

Quantitative uniform propagation of chaos for Maxwell molecules

We prove propagation of chaos at explicit polynomial rates in Wasserstein distance W_2 for Kac's N-particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules, with and without cutoff. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost N^{-1/3} for W_2^2.

math-ph↗

arXiv2017

Ray-Knight representation of flows of branching processes with competition by pruning of Lévy trees

We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li, we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Lévy-trees, and establish a Ray-Knight representation result for these processes in terms of the local times of suitably pruned forests.

math.PR↗

arXiv2016

Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium

We introduce a stochastic individual model for the spatial behavior of an animal population of dispersive and competitive species, considering various kinds of biological effects, such as heterogeneity of environmental conditions, mutual attractive or repulsive interactions between individuals or competition between them for resources. As a consequence of the study of the large population limit, global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model of competing species is proved. The main new feature of the corresponding integro-differential equation is the nonlocal nonlinearity appearing in the diffusion terms, which may depend on the spatial densities of all population types. Moreover, the diffusion matrix is generally not strictly positive definite and the cross-diffusion effect allows for linearly growing influences of the subpopulations' sizes. We prove uniqueness of the finite measure-valued solution and give conditions under which the solution takes values in a functional space. We then make the competition kernels converge to a Dirac measure and obtain the existence of a solution to a locally competitive version of the previous equation. The large population limit is obtained by means of weak convergence tools for measure valued processes. The techniques employed in the study of the liming equation are novel, and are essentially based on the underlying stochastic flow related to the dispersive part of the dynamics, together with the use of suitable dual distances in the space of finite measures. This is an extended version of the paper "Non local Lotka-Volterra system with cross-diffusion in a heterogeneous medium" appeared in J. Math. Biol (2015) 70:829-854.

math.PR↗

arXiv2016

Quantitative propagation of chaos for generalized Kac particle systems

We study a class of one-dimensional particle systems with true (Bird type) binary interactions, which includes Kac's model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of nonindependent nonlinear processes, as well as on recent sharp estimates for empirical measures.

math.PR↗

arXiv2016

A variational approach to some transport inequalities

We relate transport-entropy inequalities to the study of critical points of functionals defined on the space of probability measures. This approach leads in particular to a new proof of a result by Otto and Villani [43] showing that the logarithmic Sobolev inequality implies Talagrand's transport inequality.

math.PR↗

arXiv2016

Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\textgreater{}1/2$ and multiplicative noise component $σ$. When $σ$ is constant and for every $H\in(0,1)$, it was proved in \cite{hairer} that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-α}$ where $α\in(0,1)$ (depending on $H$). The aim of this paper is to extend such types of results to some multiplicative noise setting. More precisely, we show that we can recover such convergence rates when $H\textgreater{}1/2$ and the inverse of the diffusion coefficient $σ$ is a Jacobian matrix. The main novelty of this work is a type of extension of Foster-Lyapunov like techniques to this non-Markovian setting, which allows us to put in place an asymptotic coupling scheme such as in \cite{hairer} without resorting to deterministic contracting properties.

math.PR↗

arXiv2015

Long time behavior of telegraph processes under convex potentials

We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift toward the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.

math.PR↗

arXiv2015

A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using Itô's calculus and ideas of Arnold, Carlen and Ju \cite{Arnoldcarlenju}, we obtain moreover a new Bakry Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is non-intrisic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide examples where the classic Bakry Emery criterion fails, but our non-intrisic criterion applies without modifying the law of the diffusion process.

math.PR↗

arXiv2015

On the long time behavior of stochastic vortices systems

In this paper, we are interested in the long-time behaviour of stochastic systems of n interacting vortices: the position in R2 of each vortex evolves according to a Brownian motion and a drift summing the influences of the other vortices computed through the Biot and Savart kernel and multiplied by their respective vorticities. For fixed n, we perform the rescalings of time and space used successfully by Gallay and Wayne [5] to study the long-time behaviour of the vorticity formulation of the two dimensional incompressible Navier-Stokes equation, which is the limit as n $\rightarrow$ $\infty$ of the weighted empirical measure of the system under mean-field interaction. When all the vorticities share the same sign, the 2n-dimensional process of the rescaled positions of the vortices is shown to converge exponentially fast as time goes to infinity to some invariant measure which turns out to be Gaussian if all the vorticities are equal. In the particular case n = 2 of two vortices, we prove exponential convergence in law of the 4-dimensional process to an explicit random variable, whatever the choice of the two vorticities. We show that this limit law is not Gaussian when the two vorticities are not equal.

math.PR↗

arXiv2013

Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain

We consider an incompressible kinetic Fokker Planck equation in the flat torus, which is a simplified version of the Lagrangian stochastic models for turbulent flows introduced by S.B. Pope in the context of computational fluid dynamics. The main difficulties in its treatment arise from a pressure type force that couples the Fokker Planck equation with a Poisson equation which strongly depends on the second order moments of the fluid velocity. In this paper we prove short time existence of analytic solutions in the one-dimensional case, for which we are able to use techniques and functional norms that have been recently introduced in the study of a related singular model.

math-ph↗