Low-Rank Tensors for Multi-Dimensional Markov Models

TL;DR

This paper introduces a low-rank tensor model for multi-dimensional Markov chains that efficiently captures complex state spaces with fewer parameters, outperforming traditional matrix-based methods.

Contribution

It proposes a novel tensor-based approach for modeling multi-dimensional Markov chains, including an optimization method using ADMM for parameter estimation.

Findings

More efficient estimation with fewer samples

Fewer parameters needed compared to matrix methods

Effective modeling demonstrated on real-world data

Abstract

This work presents a low-rank tensor model for multi-dimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success of these matrix techniques, we present low-rank tensors for representing transition probabilities on multi-dimensional state spaces. Through tensor decomposition, we provide a connection between our method and classical probabilistic models. Moreover, our proposed model yields a parsimonious representation with fewer parameters than matrix-based approaches. Unlike these methods, which impose low-rankness uniformly across all states, our tensor method accounts for the multi-dimensionality of the state space. We also propose an optimization-based approach to estimate a Markov model as a low-rank tensor. Our optimization problem can be solved by the alternating direction method of multipliers (ADMM), which enjoys convergence to a stationary solution. We empirically demonstrate that our tensor model estimates Markov chains more efficiently than conventional techniques, requiring both fewer samples and parameters. We perform numerical simulations for both a synthetic low-rank Markov chain and a real-world example with New York City taxi data, showcasing the advantages of multi-dimensionality for modeling state spaces.