Date of this Version
Journal of Computer and System Sciences
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) observations—it 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.
© 2012. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.
hidden Markov models, latent variable models, observable operator models, time series, spectral algorithm, singular value decomposition, learning probability distributions, unsupervised learning
Hsu, D., Kakade, S. M., & Zhang, T. (2012). A Spectral Algorithm for Learning Hidden Markov Models. Journal of Computer and System Sciences, 78 (5), 1460-1480. http://dx.doi.org/10.1016/j.jcss.2011.12.025
Date Posted: 27 November 2017
This document has been peer reviewed.