SIGEST

Abstract

The SIGEST article in this issue is “Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors,” by Antoine Gautier, Francesco Tudisco, and Matthias Hein. Most computational and applied mathematicians will be aware of the results that Perron published in 1907 about the eigensystems of positive matrices, which were then extended by Frobenius in 1912 to the case of nonnegative matrices. This theory has impacted many areas of mathematics, including graph theory, Markov chains, and matrix computation, and it forms a fundamental component in the analysis of a range of models in areas such as demography, economics, wireless networking, and search engine optimization. Our SIGEST article, which first appeared in SIAM Journal on Matrix Analysis and Applications in 2019, extends Perron--Frobenius theory in two directions. First, the authors generalize from matrices to multidimensional arrays. This ties in with one of SIAM Review's most highly cited offerings: •Tensor decompositions and applications, T. G. Kolda and B. W. Bader, SIAM Review, 51 (3) (2009), pp. 455--500. It may also be viewed as extending the theory from graphs to hypergraphs---objects that are currently of much interest, as evidenced by several recent SIAM Review articles, including •Hypergraph cuts with general splitting functions, N. Veldt, A. R. Benson and J. Kleinberg, SIAM Review, 64 (3) (2022), pp. 650--685. By studying this higher-order setting, the authors open up new applications in network science, computer vision, and machine learning. The second major direction of the article is to develop and study nonlinear versions of the underlying spectral problems, and corresponding extensions of the traditional power method. This makes available new classes of iterations for which a comprehensive and satisfactory convergence theory is available. In preparing this SIGEST version, the authors have included new material. The introduction has been extended, and section 2 has been added to provide nontrivial examples of tensor eigenvalue problems in applications, including problems from computer vision and optimal transport. Moreover, subsection 4.1 includes a new nonlinear Perron--Frobenius theorem (Theorem 4.4) that builds on the previously known results in Theorems 4.2. and 4.3.