A variational framework for modal estimation

TL;DR

GERVE is a likelihood-free variational framework that estimates modes of multivariate distributions by approximating Gibbs measures directly from samples, integrating kernel density estimation, mean-shift, and annealing.

Contribution

It introduces GERVE, a novel entropy-regularized variational method for mode estimation that is robust, likelihood-free, and capable of handling unknown numbers of modes.

Findings

Accurate mode recovery demonstrated in simulations and real data.

Robust clustering with reduced sensitivity to component number.

Theoretical guarantees for convergence and asymptotic normality.

Abstract

We approach multivariate mode estimation through Gibbs distributions and introduce GERVE (Gibbs-measure Entropy-Regularised Variational Estimation), a likelihood-free framework that approximates Gibbs measures directly from samples by maximizing an entropy-regularised variational objective with natural-gradient updates. GERVE brings together kernel density estimation, mean-shift, variational inference, and annealing in a single platform for mode estimation. It fits Gaussian mixtures that concentrate on high-density regions and yields cluster assignments from responsibilities, with reduced sensitivity to the chosen number of components. We provide theory in two regimes: as the Gibbs temperature approaches zero, mixture components converge to population modes; at fixed temperature, maximisers of the empirical objective exist, are consistent, and are asymptotically normal. We also propose a bootstrap procedure for per-mode confidence ellipses and stability scores. Simulation and real-data studies show accurate mode recovery and emergent clustering, robust to mixture overspecification. GERVE is a practical likelihood-free approach when the number of modes or groups is unknown and full density estimation is impractical.