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Random tensor isomorphism under orthogonal and unitary actions

Jeremy Chizewer, Samuel Everett, Deven Mithal, Youming Qiao

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.

Random tensor isomorphism under orthogonal and unitary actions

Abstract

We study the problem of testing whether two tensors in are isomorphic under the natural action of orthogonal groups , as well as the corresponding question over 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.

Paper Structure

This paper contains 45 sections, 39 theorems, 156 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivations for orthogonal and unitary tensor isomorphism
  3. Local unitary equivalence of quantum states.
  4. Cartesian tensors from data analysis.
  5. Orbit closure intersection problems.
  6. Graph and tensor isomorphism problems.
  7. Average-case complexities of isomorphism problems
  8. Average-case algorithms for graph isomorphism.
  9. Average-case algorithms for GL-tensor isomorphism over finite fields.
  10. Spectral algorithms for random graph isomorphism.
  11. Our results
  12. Our random model of tensors.
  13. Our main results.
  14. Applications of our results.
  15. Overview of techniques
  16. ...and 30 more sections

Key Result

Theorem 1.1

There exists a deterministic algorithm which, given $A=(a_{i,j,k})\in\mathbb{Q}(i)^{n\times n\times n}$ with i.i.d. entries $a_{i,j,k} \sim \xi$ for a fixed---mean zero, unit variance---sub-Gaussian distribution $\xi$, and $B\in\mathbb{Q}(i)^{n\times n\times n}$ arbitrary and deterministic, runs in with implied constants depending only on the distribution of $\xi$.

Theorems & Definitions (64)

  • Theorem 1.1: Global average-case analysis of the unitary tensor isomorphism problem
  • Theorem 1.2: Global average-case analysis of the unitary orbit approximation problem
  • Theorem 1.3: Eigenvalue repulsion with polynomial gap
  • Theorem 1.4: Local analysis of the unitary tensor isomorphism problem
  • Theorem 1.5: Local analysis of the unitary orbit approximation problem
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Proposition : burgisser_et_alCCC.2021.32
  • Proposition : burgisser_et_alCCC.2021.32
  • ...and 54 more