Stochastic $p$th root approximation of a stochastic matrix: A Riemannian optimization approach

TL;DR

This paper introduces two Riemannian optimization methods for approximating the $p$th root of a stochastic matrix, improving speed and accuracy over existing constrained optimization techniques, with applications in Markov chain embedding.

Contribution

It presents novel Riemannian manifold approaches for stochastic matrix root approximation, sharing the Perron eigenvector with the original matrix, which is beneficial for Markov chain modeling.

Findings

First approach is faster and more accurate than existing methods.

Second approach yields more accurate approximations despite smaller feasible set.

Methods are validated through extensive numerical experiments.

Abstract

We propose two approaches, based on Riemannian optimization, for computing a stochastic approximation of the $p$th root of a stochastic matrix $A$. In the first approach, the approximation is found in the Riemannian manifold of positive stochastic matrices. In the second approach, we introduce the Riemannian manifold of positive stochastic matrices sharing with $A$ the Perron eigenvector and we compute the approximation of the $p$th root of $A$ in such a manifold. This way, differently from the available methods based on constrained optimization, $A$ and its $p$th root approximation share the Perron eigenvector. Such a property is relevant, from a modelling point of view, in the embedding problem for Markov chains. The extended numerical experimentation shows that, in the first approach, the Riemannian optimization methods are generally faster and more accurate than the available methods based on constrained optimization. In the second approach, even though the stochastic approximation of the $p$th root is found in a smaller set, the approximation is generally more accurate than the one obtained by standard constrained optimization.