On the Asymptotic Nonnegative Rank of Matrices and its Applications in Information Theory
Yeow Meng Chee, Quoc Tung Le, Hoang Ta
TL;DR
The paper develops a Strassen-style asymptotic spectrum theory for nonnegative matrices to characterize the asymptotic nonnegative rank $\widetilde{R}_{+}$ and its dual $\widetilde{Q}_{+}$. It builds a nonnegative matrix semiring $\mathbb{M}_{+}$ with a Strassen preorder and proves a duality formula: $\widetilde{R}_{+}(A)=\max_{\phi\in\mathbf{X}(\mathbb{M}_{+},\le_{+})}\phi(A)$ and $\widetilde{Q}_{+}(A)=\min_{\phi\in\mathbf{X}(\mathbb{M}_{+},\le_{+})}\phi(A)$. A key combinatorial link is proven: the nonnegative subrank $\text{Q}_{+}(A)$ equals the induced matching number $\gamma(A)$ of the support graph, and this drives the asymptotic dual characterization. The work exhibits concrete spectral points, notably the rank and the fractional cover number, and analyzes the spectrum for special cases such as lower triangular nonnegative matrices. These results connect matrix factorization complexity with information-theoretic quantities and combinatorial structures, offering a principled framework to bound exact Rényi common information and amortized communication complexity.
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 Rényi 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.