[Paper Review] A Spectral Algorithm for Learning Hidden Markov Models

Daniel Hsu, Sham M. Kakade+1 more|arXiv (Cornell University)|Nov 26, 2008

Bayesian Methods and Mixture Models137 citations

TL;DR

This paper presents a spectral algorithm for learning Hidden Markov Models (HMMs) under a natural separation condition involving singular values of the observation and transition matrices. The method uses singular value decomposition (SVD) of past-future correlation matrices to recover a low-rank representation of hidden states, achieving provably correct learning with polynomial sample and computational complexity, even in high-observation spaces like natural language processing.

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.

Motivation & Objective

• Address the computational hardness of learning HMMs under general conditions by identifying a tractable setting with provable guarantees.
• Overcome limitations of local search heuristics like EM, which suffer from local optima and lack theoretical guarantees.
• Enable efficient learning in high-dimensional observation spaces—such as word sequences in NLP—where the number of distinct observations is large.
• Develop a method that does not explicitly recover transition and observation matrices but maintains a linearly related hidden state representation.
• Provide theoretical bounds on approximation error for both joint and conditional sequence distributions under spectral separation conditions.

Proposed method

• Use canonical correlation analysis (CCA) via SVD on empirical correlations between past and future observation sequences to estimate a low-dimensional subspace of hidden states.
• Construct a spectral decomposition of the correlation matrix between past and future observations to identify the underlying hidden state structure.
• Employ a two-stage estimation: first estimate the subspace using SVD, then recover the conditional distribution of future observations using matrix operations on the estimated subspace.
• Apply a normalization and renormalization step to ensure the estimated conditional distributions are valid probability vectors.
• Use spectral conditions on the observation matrix (minimum singular value) and transition matrix (correlation between adjacent observations) as separation assumptions.
• Leverage matrix perturbation theory to bound estimation errors and derive sample complexity bounds that depend implicitly on the number of observations through spectral properties.

Experimental results

Research questions

• RQ1 Can we design a provably correct and efficient algorithm for learning HMMs under a natural spectral separation condition?
• RQ2 Does the algorithm maintain good performance in high-dimensional observation spaces, such as those in natural language processing?
• RQ3 Can we achieve bounded error in predicting future observations even as the sequence length increases?
• RQ4 How does the sample complexity scale with respect to the number of distinct observations, and can it be independent of this number?
• RQ5 To what extent can the algorithm recover meaningful hidden state representations without explicitly estimating the full HMM parameters?

Key findings

• The algorithm achieves polynomial sample and computational complexity, making it scalable for large-scale applications.
• The sample complexity depends implicitly on the number of distinct observations through spectral properties of the HMM, not explicitly, which is advantageous in high-observation settings.
• The approximation error for the joint distribution of sequences of length $ t $ degrades polynomially with $ t $, but the error in predicting the next observation is asymptotically bounded.
• The method provides a provable bound on the Kullback-Leibler divergence between the true and estimated conditional distributions, with error terms controlled by spectral conditions and estimation errors.
• The algorithm is robust to estimation errors in the correlation matrix, with error bounds derived using matrix perturbation theory and concentration inequalities.
• Theoretical analysis shows that under appropriate sample size, the estimated model achieves $ O( heta) $ error in predicting the next observation, where $ heta $ depends on spectral gaps and singular values of the HMM parameters.