A Spectral Algorithm for Learning Hidden Markov Models

TL;DR

This paper introduces an efficient spectral algorithm for learning Hidden Markov Models under certain conditions, overcoming computational hardness and applicable to large observation spaces like natural language processing.

Contribution

It provides a provably correct, simple spectral method for learning HMMs with guarantees under a natural separation condition, independent of the number of observations.

Findings

Algorithm is computationally efficient and provably correct.

Sample complexity depends on spectral properties, not observation count.

Applicable to large observation spaces such as language words.

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.