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We introduce and study the class of totally dissipative multivalued probability vector fields (MPVF) $\boldsymbol{\mathrm F}$ on the Wasserstein space $(\mathcal{P}_2(\mathsf{X}),W_2)$ of Euclidean or Hilbertian probability measures. We…
We study the convergence of stochastic time-discretization schemes for evolution equations driven by random velocity fields, including examples like stochastic gradient descent and interacting particle systems. Using a unified framework…
We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which…
The mean field variational inference (MFVI) formulation restricts the general Bayesian inference problem to the subspace of product measures. We present a framework to analyze MFVI algorithms, which is inspired by a similar development for…
The mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional on the Wasserstein space over $\mathbb{R}^d$, and has gained attention recently as a model for the gradient descent dynamics of interacting…
We present a simple approach to study the one-dimensional pressureless Euler system via adhesion dynamics in the Wasserstein space of probability measures with finite quadratic moments. Starting from a discrete system of a finite number of…
We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus,…
We develop a foundational framework for inverse problems governed by evolutionary partial differential equations (PDEs) on the Wasserstein space of probability measures. While the forward problems for such transport-type PDEs have been…
Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. This paper studies the stability properties of the mean-field approximation when the target…
We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure…
We present the first minimax risk bounds for estimators of the spectral measure in multivariate linear factor models, where observations are linear combinations of regularly varying latent factors. Non-asymptotic convergence rates are…
The max-stable process is an asymptotically justified model for spatial extremes. In particular, we focus on the hierarchical extreme-value process (HEVP), which is a particular max-stable process that is conducive to Bayesian computing.…
We study distribution-on-distribution regression problems in which a response distribution depends on multiple distributional predictors. Such settings arise naturally in applications where the outcome distribution is driven by several…
An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete Riemannian metrics, parametrized by $I=(-\infty,T)$. Given an additional $C^{1,1}$ family…
In this article, we propose a general framework for the study of differential inclusions in the Wasserstein space of probability measures. Based on earlier geometric insights on the structure of continuity equations, we define solutions of…
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact…
In this work we investigate the long-time behavior, that is the existence and characterization of invariant measures as well as convergence of transition probabilities, for Markov processes obtained as the unique mild solution to stochastic…
The Mean Field Variational Bayes (MFVB) method is one of the most computationally efficient techniques for Bayesian inference. However, its use has been restricted to models with conjugate priors or those that require analytical…
We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck…
Assume that we observe i.i.d.~points lying close to some unknown $d$-dimensional $\mathcal{C}^k$ submanifold $M$ in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the…