Speaker: Joshua Grochow

Affiliation: University of Colorado

Title: Isomorphism of tensors, algebras, and polynomials, oh my!

Abstract: We consider the problems of testing isomorphism of tensors, p-groups, cubic polynomials, quantum states, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. Despite Graph Isomorphism now being in quasi-polynomial time, and having long had efficient practical software, for the problems we consider no algorithm is known that is asymptotically better than brute force, and state-of-the-art software cannot get beyond small instances. We approach this situation in two ways:

– Complexity-theoretic: we show that all these problems are polynomial-time equivalent, giving rise to a class of problems we call Tensor Isomorphism-complete (TI-complete). Perhaps surprising here is that we show that isomorphism of d-tensors for any fixed d (at least 3) is equivalent to testing isomorphism of 3-tensors. These equivalences let us focus on just one problem rather than dozens, or a whole infinite hierarchy, separately.

– Algorithmic: Adapting the Weisfeiler-Leman method from Graph Isomorphism to tensors, trying to understand tensor isomorphism by taking advantage of isomorphisms of small sub-tensors. This leads to tensor analogues of the Graph Reconstruction conjecture and related questions.

Based on joint works with Vyacheslav V. Futorny and Vladimir V. Sergeichuk (Lin. Alg. Appl., 2019; arXiv:1810.09219), with Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson (arXiv:1905.02518), and with Youming Qiao (arXiv:1907.00309).