Addressing Computational Bottlenecks in Higher-Order Graph Matching with Tensor Kronecker Product Structure

Graph matching, also known as network alignment, is the problem of finding a correspondence between the vertices of two separate graphs with strong applications in image correspondence and functional inference in protein networks. One class of successful techniques is based on tensor Kronecker products and tensor eigenvectors. A challenge with these techniques are memory and computational demands that are quadratic (or worse) in terms of problem size. In this manuscript we present and apply a theory of tensor Kronecker products to tensor based graph alignment algorithms to reduce their runtime complexity from quadratic to linear with no appreciable loss of quality. In terms of theory, we show that many matrix Kronecker product identities generalize to straightforward tensor counterparts, which is rare in tensor literature. Improved computation codes for two existing algorithms that utilize this new theory achieve a minimum 10 fold runtime improvement.Comment: 14 pages, 2 pages Supplemental, 5 figure