Estimate exponential memory decay in Hidden Markov Model and its applications

TL;DR

This paper introduces a method to estimate the exponential memory decay rate in hidden Markov models, enabling efficient long-sequence inference by leveraging Lyapunov exponents, with theoretical validation and practical applications.

Contribution

It proposes a novel algorithm to numerically estimate the Lyapunov exponent gap, facilitating scalable inference in hidden Markov models.

Findings

The algorithm accurately estimates the Lyapunov exponent gap.

Memory decay rate enables subsequence-based inference for long sequences.

The method is applicable to stochastic gradient algorithms.

Abstract

Inference in hidden Markov model has been challenging in terms of scalability due to dependencies in the observation data. In this paper, we utilize the inherent memory decay in hidden Markov models, such that the forward and backward probabilities can be carried out with subsequences, enabling efficient inference over long sequences of observations. We formulate this forward filtering process in the setting of the random dynamical system and there exist Lyapunov exponents in the i.i.d random matrices production. And the rate of the memory decay is known as $\lambda_2-\lambda_1$, the gap of the top two Lyapunov exponents almost surely. An efficient and accurate algorithm is proposed to numerically estimate the gap after the soft-max parametrization. The length of subsequences $B$ given the controlled error $\epsilon$ is $B=\log(\epsilon)/(\lambda_2-\lambda_1)$. We theoretically prove the validity of the algorithm and demonstrate the effectiveness with numerical examples. The method developed here can be applied to widely used algorithms, such as mini-batch stochastic gradient method. Moreover, the continuity of Lyapunov spectrum ensures the estimated $B$ could be reused for the nearby parameter during the inference.