The Kronecker product is an invaluable tool for data-sparse representations of large networks and matrices with countless applications in machine learning, graph theory and numerical linear algebra. In some instances, the sparsity pattern of large matrices may already hide a Kronecker product. Similarly, a large network, represented by its adjacency matrix, may sometimes be factorized as a Kronecker product of smaller adjacency matrices. In this article, we determine all possible Kronecker factorizations of a binary matrix and visualize them through its decomposition graph. Such sparsity-informed factorizations may later enable good (approximate) Kronecker factorizations of real matrices or reveal the latent structure of a network. The latter also suggests a natural visualization of Kronecker graphs.

翻译：Kronecker乘积是大型网络和矩阵数据稀疏表示的重要工具，在机器学习、图论和数值线性代数等领域有广泛应用。在某些情况下，大型矩阵的稀疏模式可能已隐含Kronecker乘积结构。类似地，以邻接矩阵表示的大型网络有时可分解为较小邻接矩阵的Kronecker乘积。本文确定了二元矩阵所有可能的Kronecker分解形式，并通过其分解图进行可视化展示。此类基于稀疏性信息的分解方法，可为实数矩阵提供良好的（近似）Kronecker分解方案，或揭示网络的潜在结构。后者同时为Kronecker图提供了自然的可视化途径。