Reduced-Rank Hidden Markov Models

Hsu et al. (2009) recently proposed an ef- cient, accurate spectral learning algorithm for Hidden Markov Models (HMMs). In this paper we relax their assumptions and prove a tighter nite-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 ecient to work with, this relaxation greatly increases the learning algorithm's scope. In addi- tion, we generalize the algorithm and bounds to models where multiple observations are needed to disambiguate state, and to models that emit multivariate real-valued observa- tions. Finally we prove consistency for learn- ing Predictive State Representations, an even larger class of models. Experiments on syn- thetic data and a toy video, as well as on di- cult robot vision data, yield accurate models that compare favorably with alternatives in simulation quality and prediction accuracy.