Dominant Z-Eigenpairs of Tensor Kronecker Products are Decoupled and Applications to Higher-Order Graph Matching

TL;DR

This paper introduces a theorem that decouples dominant eigenvectors of tensor Kronecker products, enabling faster algorithms for higher-order graph matching with improved scalability and accuracy.

Contribution

It presents a novel decoupling theorem for tensor eigenvectors of Kronecker products, extending matrix theory to tensors and enhancing graph matching algorithms.

Findings

Decoupling theorem for tensor eigenvectors of Kronecker products

New algorithms that are faster and more scalable

Improved accuracy in higher-order graph matching

Abstract

Tensor Kronecker products, the natural generalization of the matrix Kronecker product, are independently emerging in multiple research communities. Like their matrix counterpart, the tensor generalization gives structure for implicit multiplication and factorization theorems. We present a theorem that decouples the dominant eigenvectors of tensor Kronecker products, which is a rare generalization from matrix theory to tensor eigenvectors. This theorem implies low rank structure ought to be present in the iterates of tensor power methods on Kronecker products. We investigate low rank structure in the network alignment algorithm TAME, a power method heuristic. Using the low rank structure directly or via a new heuristic embedding approach, we produce new algorithms which are faster while improving or maintaining accuracy, and scale to problems that cannot be realistically handled with existing techniques.