On the Asymptotic Nonnegative Rank of Matrices and its Applications in Information Theory

TL;DR

This paper investigates the asymptotic nonnegative rank of matrices, linking it to information theory concepts and introducing a spectral framework to characterize its properties and related parameters.

Contribution

It formalizes the asymptotic spectrum of nonnegative matrices, providing a dual characterization of the asymptotic nonnegative rank and introducing the subrank concept.

Findings

The asymptotic nonnegative rank governs information-theoretic measures.

The subrank equals the maximum induced matching size of the matrix support.

Rank and fractional cover number are part of the asymptotic spectrum.

Abstract

In this paper, we study the asymptotic nonnegative rank of matrices, which characterizes the asymptotic growth of the nonnegative rank of fixed nonnegative matrices under the Kronecker product. This quantity is important since it governs several notions in information theory such as the so-called exact R\'enyi common information and the amortized communication complexity. By using the theory of asymptotic spectra of V. Strassen (J. Reine Angew. Math. 1988), we define formally the asymptotic spectrum of nonnegative matrices and give a dual characterization of the asymptotic nonnegative rank. As a complementary of the nonnegative rank, we introduce the notion of the subrank of a nonnegative matrix and show that it is exactly equal to the size of the maximum induced matching of the bipartite graph defined on the support of the matrix (therefore, independent of the value of entries). Finally, we show that two matrix parameters, namely rank and fractional cover number, belong to the asymptotic spectrum of nonnegative matrices.