# Bounds on Lyapunov exponents via entropy accumulation

## TL;DR

This paper introduces entropy accumulation-based bounds for Lyapunov exponents of large random matrix products, providing analytical, tight bounds that are computationally efficient to evaluate.

## Contribution

It establishes a novel link between Lyapunov exponents and entropy accumulation, enabling analytical bounds that are tight and easier to compute.

## Key findings

- Bounds are tight in the commutative case.
- Bounds are expressed as optimization problems involving single matrices.
- Upper bounds can be efficiently computed via convex optimization.

## Abstract

Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively. The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.