Geometric Learning of Hidden Markov Models via a Method of Moments Algorithm
This work addresses a domain-specific challenge in geometric machine learning for applications like computer vision or robotics, but it is incremental as it builds on prior method of moments approaches.
The authors tackled the problem of learning hidden Markov model parameters when observations are in Riemannian manifolds, by extending a method of moments algorithm to non-positive curvature symmetric spaces with Riemannian Gaussians, resulting in significantly improved speed and numerical accuracy compared to existing methods.
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.