Spectral dimensionality reduction for HMMs

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method Hsu et al. (2009) in contrast to the usual slow methods like EM or Gibbs sampling. We provide a new spectral method which significantly reduces the number of model parameters that need to be estimated, and generates a sample complexity that does not depend on the size of the observation vocabulary. We present an elementary proof giving bounds on the relative accuracy of probability estimates from our model. (Correlaries show our bounds can be weakened to provide either L1 bounds or KL bounds which provide easier direct comparisons to previous work.) Our theorem uses conditions that are checkable from the data, instead of putting conditions on the unobservable Markov transition matrix.