Convergence rates and optimal approximators for Gaussian mixture reduction

Develop rigorous convergence rate guarantees and identify optimal low-order Gaussian mixtures that approximate high-order Gaussian mixtures under standard divergences (e.g., total variation, Hellinger, KL, χ^2), thereby providing theoretical foundations—including minimax rates and optimal constructions—for the Gaussian mixture reduction problem.

Background

The paper highlights a related practical problem: reducing a high-order Gaussian mixture to a low-order one. While many numerical approaches exist (clustering, optimization, greedy methods), there is little theoretical understanding of convergence rates or what constitutes an optimal approximator.

The authors note the absence of rate results and optimal constructions, framing the need for rigorous analysis analogous to the finite mixture approximation theory developed in the main text.