arXiv subjects

Joaquín Fontbona

Publications and source records attributed to Joaquín Fontbona.

8 recordsLinked to original sources

arXiv2026

Quantitative QSD convergence in 1-Wasserstein distance via the Föllmer drift

We develop a novel pathwise approach to study the convergence of the law of killed diffusion processes conditioned on non-absorption, towards a quasi-stationary distribution (QSD) as time goes to infinity. We start from the general observation that the dynamics of an absorbed Markov process conditioned upon survival up to time $T>0$ is the minimizer of the pathwise relative entropy with respect to its unconditioned dynamics, under a simple distributional constraint at that time; in other words, a Föllmer process. We then show how this result applies to a Brownian diffusion process softly-killed at a state-dependent regular rate, and characterize the associated drift change. In the case when the diffusion process is moreover reversible, we leverage this idea and recent results on the propagation of weak log-concavity of HJB semigroups to prove that, under strict asymptotic convexity of the potential, the conditioned dynamics satisfy a contractivity property in $1$-Wasserstein distance, uniformly in $T>0$. Under a general ergodicity condition on the associated Feynman-Kac semigroup, we then establish the existence of a QSD with a large domain of attraction, and the exponentially fast convergence to it of the conditioned semigroup in the $1$-Wasserstein distance as $T$ goes to infinity. Finally, we deduce the exponentially fast convergence, also in $1$-Wasserstein distance, of the law of the corresponding Q-process towards its equilibrium.

math.PR↗

arXiv2025

Towards SFW sampling for diffusion models via external conditioning

Score-based generative models (SBM), also known as diffusion models, are the de facto state of the art for image synthesis. Despite their unparalleled performance, SBMs have recently been in the spotlight for being tricked into creating not-safe-for-work (NSFW) content, such as violent images and non-consensual nudity. Current approaches that prevent unsafe generation are based on the models' own knowledge, and the majority of them require fine-tuning. This article explores the use of external sources for ensuring safe outputs in SBMs. Our safe-for-work (SFW) sampler implements a Conditional Trajectory Correction step that guides the samples away from undesired regions in the ambient space using multimodal models as the source of conditioning. Furthermore, using Contrastive Language Image Pre-training (CLIP), our method admits user-defined NSFW classes, which can vary in different settings. Our experiments on the text-to-image SBM Stable Diffusion validate that the proposed SFW sampler effectively reduces the generation of explicit content while being competitive with other fine-tuning-based approaches, as assessed via independent NSFW detectors. Moreover, we evaluate the impact of the SFW sampler on image quality and show that the proposed correction scheme comes at a minor cost with negligible effect on samples not needing correction. Our study confirms the suitability of the SFW sampler towards aligned SBM models and the potential of using model-agnostic conditioning for the prevention of unwanted images.

cs.CV↗

arXiv2023

A novel notion of barycenter for probability distributions based on optimal weak mass transport

We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlanet al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.

stat.ML↗

arXiv2021

Quantitative mean-field limit for interacting branching diffusions

We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching towards the solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth, shown to describe their large population limits in arXiv:1303.3939. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, which allows us to sharply mimic the trajectory of the interacting binary branching population by certain system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the convergence rate in 2-Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend techniques and ideas on propagation of chaos from kinetic models to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions.

math.PR↗

arXiv2020

Generalized entropy minimization under full marginal constraints

We consider the problem of minimizing a generalized relative entropy, with respect to a reference diffusion law, over the set of path-measures with fully prescribed marginal distributions. When dealing with the actual relative entropy, problems of this kind have appeared in the stochastic mechanics literature, and minimizers go under the name of Nelson Processes. Through convex duality and stochastic control techniques, we obtain in our main result the full characterization of minimizers, containing the related results in the pioneering works of Cattiaux \& Léonard and Mikami as particular cases. We also establish that minimizers need not be Markovian in general, and may depend on the form of the generalized relative entropy if the state space has dimension greater or equal than two. Finally, we illustrate how generalized relative entropy minimization problems of this kind may prove useful beyond stochastic mechanics, by means of two applications: the analysis of certain mean-field games, and the study of scaling limits for a class of backwards SDEs.

math.OC↗

arXiv2018

Dynamics of a planar Coulomb gas

We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincar{é} inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll -- Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.

math-ph↗

arXiv2015

Robust utility maximization without model compactness

We formulate conditions for the solvability of the problem of robust utility maximization from final wealth in continuous time financial markets, without assuming weak compactness of the densities of the uncertainty set, as customary in the literature. Relevant examples of such a situation typically arise when the uncertainty set is determined through moment constraints. Our approach is based on identifying functional spaces naturally associated with the elements of each problem. For general markets these are modular spaces, through which we can prove a minimax equality and the existence of optimal strategies by exploiting the compactness, which we establish, of the image by the utility function of the set of attainable wealths. In complete markets we obtain additionally the existence of a worst-case measure, and combining our ideas with abstract entropy minimization techniques, we moreover provide in that case a novel methodology for the characterization of such measures.

math.OC↗

arXiv2010

The limiting move-to-front search-cost in law of large numbers asymptotic regimes

We explicitly compute the limiting transient distribution of the search-cost in the move-to-front Markov chain when the number of objects tends to infinity, for general families of deterministic or random request rates. Our techniques are based on a "law of large numbers for random partitions," a scaling limit that allows us to exactly compute limiting expectation of empirical functionals of the request probabilities of objects. In particular, we show that the limiting search-cost can be split at an explicit deterministic threshold into one random variable in equilibrium, and a second one related to the initial ordering of the list. Our results ensure the stability of the limiting search-cost under general perturbations of the request probabilities. We provide the description of the limiting transient behavior in several examples where only the stationary regime is known, and discuss the range of validity of our scaling limit.

math.PR↗