Statistics Papers

Document Type

Journal Article

Date of this Version

9-2012

Publication Source

Journal of Computer and System Sciences

Volume

78

Issue

5

Start Page

1460

Last Page

1480

DOI

10.1016/j.jcss.2011.12.025

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) 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.

Copyright/Permission Statement

© 2012. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.

Keywords

hidden Markov models, latent variable models, observable operator models, time series, spectral algorithm, singular value decomposition, learning probability distributions, unsupervised learning

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Date Posted: 27 November 2017

This document has been peer reviewed.