Low-rank tensor methods for Markov chains with applications to tumor progression models

TL;DR

This paper introduces a low-rank tensor approach for efficiently computing time-marginal distributions in high-dimensional continuous-time Markov chains, especially for tumor progression models, overcoming exponential complexity.

Contribution

It develops a convergent low-rank iterative method that guarantees probability normalization, enabling scalable analysis of complex Markov chains with separable transition rates.

Findings

Low-rank tensor methods significantly reduce computational costs.

The proposed method accurately approximates marginal distributions.

Numerical experiments confirm the effectiveness for tumor progression models.

Abstract

Continuous-time Markov chains describing interacting processes exhibit a state space that grows exponentially in the number of processes. This state-space explosion renders the computation or storage of the time-marginal distribution, which is defined as the solution of a certain linear system, infeasible using classical methods. We consider Markov chains whose transition rates are separable functions, which allows for an efficient low-rank tensor representation of the operator of this linear system. Typically, the right-hand side also has low-rank structure, and thus we can reduce the cost for computation and storage from exponential to linear. Previously known iterative methods also allow for low-rank approximations of the solution but are unable to guarantee that its entries sum up to one as required for a probability distribution. We derive a convergent iterative method using low-rank formats satisfying this condition. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank.