A Spectral Algorithm for Learning Hidden Markov Models
Abstract
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 (under cryptographic assumptions), and practitioners typically resort to search heuristics which suffer from the usual local optima issues. We prove that under a natural separation condition (bounds on the smallest singular value of the HMM parameters), 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) observationsit implicitly depends on this quantity 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, employing only a singular value decomposition and matrix multiplications.
 Publication:

arXiv eprints
 Pub Date:
 November 2008
 DOI:
 10.48550/arXiv.0811.4413
 arXiv:
 arXiv:0811.4413
 Bibcode:
 2008arXiv0811.4413H
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence
 EPrint:
 Published in JCSS Special Issue "Learning Theory 2009"