Weighted quantization using MMD: From mean field to mean shift via gradient flows

TL;DR

This paper introduces a novel approach for probability distribution approximation using weighted particles optimized under MMD, unifying gradient flows, mean shift, and clustering algorithms.

Contribution

It develops a new fixed-point algorithm called MSIP, extending mean shift and providing a robust method for MMD-based quantization.

Findings

MSIP extends classical mean shift for mode detection.

MSIP acts as a preconditioned gradient descent.

Algorithms outperform state-of-the-art in high-dimensional, multi-modal scenarios.

Abstract

Approximating a probability distribution using a set of particles is a fundamental problem in machine learning and statistics, with applications including clustering and quantization. Formally, we seek a weighted mixture of Dirac measures that best approximates the target distribution. While much existing work relies on the Wasserstein distance to quantify approximation errors, maximum mean discrepancy (MMD) has received comparatively less attention, especially when allowing for variable particle weights. We argue that a Wasserstein-Fisher-Rao gradient flow is well-suited for designing quantizations optimal under MMD. We show that a system of interacting particles satisfying a set of ODEs discretizes this flow. We further derive a new fixed-point algorithm called mean shift interacting particles (MSIP). We show that MSIP extends the classical mean shift algorithm, widely used for identifying modes in kernel density estimators. Moreover, we show that MSIP can be interpreted as preconditioned gradient descent and that it acts as a relaxation of Lloyd's algorithm for clustering. Our unification of gradient flows, mean shift, and MMD-optimal quantization yields algorithms that are more robust than state-of-the-art methods, as demonstrated via high-dimensional and multi-modal numerical experiments.