  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.
