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.

翻译：我们研究在正交群 $\textbf{O}(\ell, \mathbb{R})\times\textbf{O}(m, \mathbb{R})\times\textbf{O}(n, \mathbb{R})$ 的自然作用下，测试 $\mathbb{R}^\ell\otimes \mathbb{R}^m\otimes \mathbb{R}^n$ 中两个张量是否同构的问题，以及相应的复数域和酉群情形。这些问题自然出现在多个领域，包括图和张量同构（Grochow--Qiao, SIAM J. Comp. '21）、轨线闭包交集的缩放算法（Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18）和量子信息（Liu--Li--Li--Qiao, Phys. Rev. Lett. '12）。我们研究正交和酉张量同构的平均情形算法，其中一个随机张量的每个元素独立同分布于亚高斯分布，另一个张量任意。在算法设计方面，我们将高阶奇异值分解的思想发展为具有严格平均情形分析的多项式时间精确（代数）和近似（数值）算法。继（Allen-Zhu--Garg--Li--Oliveira--Wigderson, STOC '18）之后，我们提出一种用于求解间隙版本的轨线距离近似问题的算法。在平均情形分析中，我们借鉴随机矩阵理论中关于亚高斯Wishart矩阵特征值排斥的最新进展（Christoffersen--Luh--O'Rourke--Shearer and Han, arXiv '25），将其结果从Wishart矩阵边长线性相关的情形推广到多项式相关的情形。