Identifying Kronecker product factorizations
Yannis Voet, Leonardo De Novellis
TL;DR
The paper tackles the problem of identifying all Kronecker product factorizations of large sparse binary matrices by exploiting sparsity patterns to guide exact and approximate factorizations. It introduces a matrix-free Cartesian-product criterion for length-2 factorizations, then builds higher-length factorizations by combining these results, and visualizes the entire structure with a decomposition graph. Key contributions include a practical algorithm for detecting factorizability across all compatible pairs, a decomposition-graph visualization, and multiple real-world applications in space-time discretizations, Kronecker-graph visualization, and quantum gate analysis. This framework enables compression, preconditioning, and latent-structure discovery in networks and scientific computing, while also paving the way for broader generalized decompositions.
Abstract
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.