Identifying Kronecker product factorizations

TL;DR

This paper develops a method to identify all possible Kronecker product factorizations of binary matrices, aiding in revealing latent structures in large networks and matrices for applications in machine learning and graph theory.

Contribution

It introduces a comprehensive approach to determine all Kronecker factorizations of binary matrices and visualizes them via decomposition graphs, enhancing understanding of matrix and network structures.

Findings

Identifies all Kronecker factorizations of binary matrices.

Provides a visualization method through decomposition graphs.

Facilitates approximate factorizations and latent structure discovery.

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.