Weighted quantization using MMD: From mean field to mean shift via gradient flows
This work addresses the problem of probability distribution approximation for machine learning and statistics applications, such as clustering and quantization, providing an incremental improvement over existing methods.
The authors tackled the problem of approximating a probability distribution using a set of particles, achieving more robust results than state-of-the-art methods. Their approach, called mean shift interacting particles (MSIP), yields better performance in high-dimensional and multi-modal numerical experiments.
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.