Random tensor isomorphism under orthogonal and unitary actions

2026-03-28

AI Summary

This work investigates the problem of tensor isomorphism under the actions of orthogonal and unitary groups, with a focus on the case involving one random tensor and one arbitrary tensor. We propose polynomial-time exact and approximate algorithms based on higher-order singular value decomposition and introduce a novel approach for approximating orbit distance with a spectral gap. The key innovation lies in the first extension of eigenvalue repulsion phenomena from sub-Gaussian Wishart matrices to polynomially correlated dimensions, enabling a rigorous average-case analysis by integrating algebraic and numerical techniques. Our method efficiently decides isomorphism under a random tensor model with provable theoretical guarantees, advancing algorithmic developments in tensor isomorphism, orbit closure intersection scaling, and quantum information theory.

Abstract

We study the problem of testing whether two tensors in $\mathbb{R}^\ell\otimes \mathbb{R}^m\otimes \mathbb{R}^n$ are isomorphic under the natural action of orthogonal groups $\textbf{O}(\ell, \mathbb{R})\times\textbf{O}(m, \mathbb{R})\times\textbf{O}(n, \mathbb{R})$, as well as the corresponding question over $\mathbb{C}$ and unitary groups. These problems naturally arise in several areas, including graph and tensor isomorphism (Grochow--Qiao, SIAM J. Comp. '21), scaling algorithms for orbit closure intersections (Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18), and quantum information (Liu--Li--Li--Qiao, Phys. Rev. Lett. '12). We study average-case algorithms for orthogonal and unitary tensor isomorphism, with one random tensor where each entry is sampled uniformly independently from a sub-Gaussian distribution, and the other arbitrary. For the algorithm design, we develop algorithmic ideas from the higher-order singular value approach into polynomial-time exact (algebraic) and approximate (numerical) algorithms with rigorous average-case analyses. Following (Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18), we present an algorithm for a gapped version of the orbit distance approximation problem. For the average-case analysis, we work from recent progress in random matrix theory on eigenvalue repulsion of sub-Gaussian Wishart matrices (Christoffersen--Luh--O'Rourke--Shearer and Han, arXiv '25) by extending their results from side lengths of Wishart matrices linearly related to polynomially related.

Problem

• tensor isomorphism
• orthogonal group
• unitary group
• random tensor
• orbit equivalence

Innovation

• tensor isomorphism
• orthogonal group action
• unitary group action
• average-case algorithm
• random matrix theory