Optimal transport based theory for latent structured models
XuanLong Nguyen, Yun Wei
TL;DR
The article surveys theoretical advances in learning latent structured models through optimal transport distances, emphasizing inverse bounds that relate the data-generating distribution $p_G$ to the latent mixing measure $G$. It develops a unified, estimator-agnostic framework using Wasserstein-type distances and test-function (IPM) based metrics to derive pointwise and uniform convergence rates for finite mixtures, including strong and weak identifiability and singular-fisher-information regimes. It then extends these ideas to de Finetti-type mixing measures and hierarchical models, notably the mixture-of-products setting and hierarchical Dirichlet processes, establishing posterior contraction rates via inverse-bounds arguments and transport-based metrics. The work highlights how inverse bounds translate density-estimation rates into latent-structure estimation rates, motivates minimum-IPM and MMD-style estimators, and outlines key open problems, including extensions to networks, weak-identifiability minimax theory, and computational aspects of transport-based inference.
Abstract
This article is an exposition on some recent theoretical advances in learning latent structured models, with a primary focus on the fundamental roles that optimal transport distances play in the statistical theory. We aim at what may be the most critical and novel ingredient in this theory: the motivation, formulation, derivation and ramification of inverse bounds, a rich collection of structural inequalities for latent structured models which connect the space of distributions of unobserved structures of interest to the space of distributions for observed data. This theory is illustrated on classical mixture models, as well as the more modern hierarchical models that have been developed in Bayesian statistics, machine learning and related fields.