A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices

TL;DR

This paper presents a convergent hierarchy of nonlinear eigenproblems to accurately compute the joint spectral radius of large collections of nonnegative matrices, using a risk-sensitive control interpretation and a scalable iterative method.

Contribution

It introduces a new nonlinear eigenproblem hierarchy and a projective Krasnoselskii-Mann iteration for scalable computation of the joint spectral radius.

Findings

Method efficiently handles large matrices of order 1000 within minutes.

Converges to the joint spectral radius as sequence length increases.

Avoids linear or semidefinite programming, enhancing scalability.

Abstract

We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques.