Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow
This work addresses a fundamental computational bottleneck in statistics and machine learning for researchers and practitioners, though it appears incremental as it builds on existing gradient descent methods with a new geometry.
The authors tackled the computationally hard problem of fitting Gaussian mixture models by proposing a gradient descent algorithm over probability measures using Wasserstein-Fisher-Rao geometry, establishing convergence guarantees and showing effectiveness through numerical experiments compared to benchmarks.
Gaussian mixture models form a flexible and expressive parametric family of distributions that has found applications in a wide variety of applications. Unfortunately, fitting these models to data is a notoriously hard problem from a computational perspective. Currently, only moment-based methods enjoy theoretical guarantees while likelihood-based methods are dominated by heuristics such as Expectation-Maximization that are known to fail in simple examples. In this work, we propose a new algorithm to compute the nonparametric maximum likelihood estimator (NPMLE) in a Gaussian mixture model. Our method is based on gradient descent over the space of probability measures equipped with the Wasserstein-Fisher-Rao geometry for which we establish convergence guarantees. In practice, it can be approximated using an interacting particle system where the weight and location of particles are updated alternately. We conduct extensive numerical experiments to confirm the effectiveness of the proposed algorithm compared not only to classical benchmarks but also to similar gradient descent algorithms with respect to simpler geometries. In particular, these simulations illustrate the benefit of updating both weight and location of the interacting particles.