Computing the Nonnegative Low-Rank Leading Eigenmatrix and its Applications to Markov Grids and Metzler Operators

TL;DR

This paper introduces a novel algorithm for computing nonnegative low-rank approximations of the dominant eigenpair of real operators, with applications to Markov grids and structured growth-diffusion systems, demonstrating improved effectiveness over existing methods.

Contribution

The paper presents a new differential system-based algorithm for nonnegative low-rank eigenpair approximation, motivated by Perron-Frobenius theorem, with applications to Markov chains and structured population models.

Findings

Algorithm effectively computes nonnegative eigenpairs.

Outperforms standard approaches in experiments.

Applicable to Markov grids and growth-diffusion systems.

Abstract

We consider in this paper the problem of computing a nonnegative low-rank approximation of the rightmost eigenpair of a linear matrix-valued real operator. We propose an algorithm based on the time integration of a suitable differential system, whose solution is parametrized according to a nonnegative factorization. The conservation of the nonnegativity is theoretically motivated by the Perron-Frobenius theorem, while the computation of the rightmost eigenpair is motivated by two applications: (1) a new class of Markov chains, which we called Markov grids, whose transition matrices can be decomposed as the sum of Kronecker products, and (2) spatially structured systems in growth-diffusion operators arising for example in population and epidemic dynamics. Theoretical analysis and computational experiments show the effectiveness of the algorithm compared to standard approaches.