The nonnegative rank of nonnegative matrices is an important quantity that
appears in many fields, such as combinatorial optimization, communication
complexity, and information theory. In this paper, we study the asymptotic
growth of the nonnegative rank of a fixed nonnegative matrix under Kronecker
product. This quantity is called the asymptotic nonnegative rank, which is
already studied in information theory. By applying the theory of asymptotic
spectra of V. Strassen (J. Reine Angew. Math. 1988), we introduce the
asymptotic spectrum of nonnegative matrices and give a dual characterization of
the asymptotic nonnegative rank. As the opposite of 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