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.

翻译：本文研究了矩阵的渐近非负秩，该量刻画了固定非负矩阵在Kronecker积下非负秩的渐近增长。由于该量支配着信息论中的若干概念，如所谓的精确Rényi公共信息和分摊通信复杂度，因此具有重要意义。利用V. Strassen（J. Reine Angew. Math. 1988）的渐近谱理论，我们形式化定义了非负矩阵的渐近谱，并给出了渐近非负秩的对偶刻画。作为非负秩的补充，我们引入了非负矩阵子秩的概念，并证明其精确等于矩阵支撑集所定义二分图中最大诱导匹配的大小（因此与元素取值无关）。最后，我们证明秩与分数覆盖数这两个矩阵参数均属于非负矩阵的渐近谱。