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.

翻译：非负矩阵的非负秩是一个重要量，出现在组合优化、通信复杂性和信息理论等多个领域。本文研究固定非负矩阵在Kronecker积下非负秩的渐近增长。该量被称为渐近非负秩，已在信息理论中有所研究。通过应用V. Strassen的渐近谱理论（J. Reine Angew. Math. 1988），我们引入了非负矩阵的渐近谱，并给出了渐近非负秩的对偶刻画。作为非负秩的对立概念，我们引入了非负矩阵的子秩概念，并证明它恰好等于矩阵支撑集所定义二部图的最大诱导匹配规模（因此与元素值无关）。最后，我们证明两个矩阵参数——秩和分数覆盖数——属于非负矩阵的渐近谱。