On Bridging Mixture Distributions

Abstract

In this article we consider bridging between two mixture probability measures. In particular, given access to a Markov kernel between two component distributions, we provide a general mechanism to generate samples from one mixture to the other. Associated to a given reference and extended state space, we prove entropic optimality of this approach. In order to use this idea one needs to know the underlying mixtures and the Markov kernel, which is seldom available, and so we consider the case of Gaussian mixtures and Schr\"odinger Bridges. We prove a general $2-$Wasserstein continuity bound between the exact bridge and one that is approximated, based on $\epsilon-$covariance inflation, and these rely on a novel continuity analysis of perturbed Riccati maps. We apply our results in the context of bridging mixtures of Gaussians, single Gaussians and empirical estimators of the Gaussian parameters and the Monge map. For mixtures of Gaussians, when the parameters are estimated using the Expectation-Maxmization algorithm, the upper-bound on the $2-$Wasserstein distance between the true and approximated bridges is, under assumptions and with probability at least $1-10N^{-1}$, $\mathcal{O}\big(\big[\big(\tfrac{d\log N}{N}\big)^{1/2}\left\{1+\big(\tfrac{d\log N}{N}\right)^{1/2}(\epsilon^{-2}+1)\big\}+\epsilon^2\big]\big) $ and for the other two cases, in expectation, $\mathcal{O}\left(d\left\{\tfrac{1+\epsilon^{-2}}{1+N}+\epsilon^2\right\}\right)$, where $d,N\in\mathbb{N}$ is the dimension of the Gaussian and the number of empirical samples respectively. We also investigate our bounds numerically.