A Spectral Algorithm for Learning Hidden Markov Models

Hidden Markov Models (HMMs) are one of the most fundamental and widely used statistical tools for modeling discrete time series. In general, learning HMMs from data is computationally hard; practitioners typically resort to search heuristics (such as the Baum-Welch / EM algorithm) which suffer from the usual local optima issues. We prove that under a natural separation condition (roughly analogous to those considered for learning mixture models), there is an efficient and provably correct algorithm for learning HMMs. The sample complexity of the algorithm does not explicitly depend on the number of distinct (discrete) observations--it implicitly depends on this number through spectral properties of the underlying HMM. This makes the algorithm particularly applicable to settings with a large number of observations, such as those in natural language processing where the space of observation is sometimes the words in a language. The algorithm is also simple: it employs only a singular value decomposition and matrix multiplications.