Optimal transport based theory for latent structured models

TL;DR

This paper explores recent theoretical advances in learning latent structured models, emphasizing the role of optimal transport distances and inverse bounds in connecting unobserved structures to observed data distributions.

Contribution

It introduces the formulation and implications of inverse bounds, a novel structural inequality framework for latent models using optimal transport theory.

Findings

Inverse bounds connect unobserved structures to observed data distributions.

The theory applies to classical mixture models and modern hierarchical Bayesian models.

Optimal transport distances are fundamental in the statistical analysis of latent structured models.

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.