Related papers: Mean field approximations via log-concavity
We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight…
We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if…
Mean field variational inference (VI) is the problem of finding the closest product (factorized) measure, in the sense of relative entropy, to a given high-dimensional probability measure $\rho$. The well known Coordinate Ascent Variational…
The estimation of a log-concave density on $\mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet…
As an example of the nonlinear Fokker-Planck equation, the mean field Langevin dynamics recently attracts attention due to its connection to (noisy) gradient descent on infinitely wide neural networks in the mean field regime, and hence the…
In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$.…
Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum…
Mean-Field is an efficient way to approximate a posterior distribution in complex graphical models and constitutes the most popular class of Bayesian variational approximation methods. In most applications, the mean field distribution…
We study the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities. Our main result is the first computationally efficient algorithm for this problem. In more detail, we give an algorithm that, on…
Sampling from Gibbs distributions and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. While efficient algorithms are known for log-concave densities, the worst-case…
In this paper, we study the approximation and estimation of $s$-concave densities via R\'enyi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave densities exists and is unique via the procedure…
We prove sharp estimates for the mean-field limit of weakly interacting diffusions with repulsive logarithmic interaction in arbitrary dimension. More precisely, we show that the associated partition function is uniformly bounded in the…
An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note,…
For algorithms based on interacting particle systems that admit a mean-field description, convergence analysis is often more accessible at the mean-field level. In order to transfer convergence results obtained at the mean-field level to…
The estimation of a log-concave density on $\mathbb{R}^d$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to…
We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the…
The Coordinate Ascent Variational Inference scheme is a popular algorithm used to compute the mean-field approximation of a probability distribution of interest. We analyze its random scan version, under log-concavity assumptions on the…
The recent work arXiv:2407.17373 proposes a derivative-free consensus-based particle method that computes global solutions to nonconvex-nonconcave min-max problems and establishes global exponential convergence in the sense of the…
We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…
This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making novel use of a continuously specified Gaussian random field. We show that for…