Geometric Learning of Hidden Markov Models via a Method of Moments Algorithm

TL;DR

This paper introduces a geometric method of moments algorithm for learning hidden Markov models with observations on Riemannian manifolds, improving speed and accuracy over existing methods.

Contribution

It extends second-order moments algorithms to Riemannian symmetric spaces, combining Gaussian mixture estimation with convex optimization for HMM parameter learning.

Findings

Improved speed over existing learners.

Enhanced numerical accuracy.

Effective in Riemannian manifold settings.

Abstract

We present a novel algorithm for learning the parameters of hidden Markov models (HMMs) in a geometric setting where the observations take values in Riemannian manifolds. In particular, we elevate a recent second-order method of moments algorithm that incorporates non-consecutive correlations to a more general setting where observations take place in a Riemannian symmetric space of non-positive curvature and the observation likelihoods are Riemannian Gaussians. The resulting algorithm decouples into a Riemannian Gaussian mixture model estimation algorithm followed by a sequence of convex optimization procedures. We demonstrate through examples that the learner can result in significantly improved speed and numerical accuracy compared to existing learners.