Reduced-Rank Hidden Markov Models

Hsu et al. (2009) recently proposed an ef- ficient, accurate spectral learning algorithm for Hidden Markov Models (HMMs). In this paper we relax their assumptions and prove a tighter finite-sample error bound for the case of Reduced-Rank HMMs, i.e., HMMs with low-rank transition matrices. Since rank-k RR-HMMs are a larger class of models than k-state HMMs while being equally efficient to work with, this relaxation greatly increases the learning algorithm’s scope. In addition, we generalize the algorithm and bounds to models where multiple observations are needed to disambiguate state, and to models that emit multivariate real-valued observations. Finally we prove consistency for learning Predictive State Representations, an even larger class of models. Experiments on synthetic data and a toy video, as well as on diffi- cult robot vision data, yield accurate models that compare favorably with alternatives in simulation quality and prediction accuracy