SIAM Review, Volume 65, Issue 2, Page 495-536, May 2023.
We present a unifying Perron--Frobenius theory for nonlinear spectral problems defined in terms of nonnegative tensors. By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad set of problems involving tensors (and matrices), including the computation of operator norms, graph and hypergraph matching in computer vision, hypergraph spectral theory, higher-order network analysis, and multimarginal optimal transport. The key to our approach is to recast the eigenvalue problem as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multihomogeneous order-preserving maps to derive new and unifying Perron--Frobenius theorems for nonnegative tensors, which either imply earlier results of this kind or improve them, as weaker assumptions are required. We introduce a general power method for the computation of the dominant tensor eigenpair and provide a detailed convergence analysis. This paper is directly based on our previous work [A. Gautier, F. Tudisco, and M. Hein, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1206--1231] and complements it by providing an extended introduction and several new results.

中文翻译：

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非线性 Perron--非负张量的 Frobenius 定理

SIAM Review，第 65 卷，第 2 期，第 495-536 页，2023 年 5 月。
我们针对根据非负张量定义的非线性谱问题提出了统一的 Perron-Frobenius 理论。通过使用张量形状划分的概念，作为特例，我们的结果包括文献中考虑的各种特定张量谱问题，并且可以应用于涉及张量（和矩阵）的广泛问题集，包括计算算子规范、计算机视觉中的图和超图匹配、超图谱理论、高阶网络分析和多边际最优传输。我们方法的关键是将特征值问题重铸为射影空间的合适乘积上的不动点问题。这使我们能够使用多齐次保序映射的理论来推导出新的和统一的非负张量的 Perron-Frobenius 定理，这要么暗示了这种早期的结果，要么改进了它们，因为需要更弱的假设。我们介绍了一种用于计算主要张量特征对的通用幂方法，并提供了详细的收敛性分析。本文直接基于我们之前的工作 [A. Gautier、F. Tudisco 和 M. Hein，SIAM J. Matrix Anal。Appl., 40 (2019), pp. 1206--1231] 并通过提供扩展介绍和几个新结果对其进行补充。