The Perron--Frobenius Theorem for Multihomogeneous Mappings

Abstract

The Perron-Frobenius theory for nonnegative matrices has been generalized to orderpreserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings, their associated nonlinear spectral problems and spectral radii. We show several Perron-Frobenius type results for these mappings addressing existence, uniqueness and maximality of nonnegative and positive eigenpairs. We prove a Collatz-Wielandt principle and other characterizations of the spectral radius and analyze the convergence of iterates of these mappings towards their unique positive eigenvectors. On top of providing a remarkable extension of the nonlinear Perron-Frobenius theory to the multi-dimensional case, our contribution poses the basis for several improvements and a deeper understanding of the current spectral theory for nonnegative tensors. In fact, in recent years, important results have been obtained by recasting certain spectral equations for multilinear forms in terms of homogeneous maps, however as our approach is more adapted to such problems, these results can be further refined and improved by employing our new multi-homogeneous setting.

Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs

“…Several notions of spectral radius can be then employed to generalize this concept from the matrix to the higher-order case (see e.g. [27]). Here we adopt the following (4) r p (T) = sup{|λ| : λ is an ℓ p -eigenvalue of T} Note that, when T is symmetric, using Proposition 1 and Remark 2, we have…”

“…Within this range, a distinction must be made: while the case p = d requires assumptions on the irreducibility of T (e.g. strong and weak irreducibility or primitivity, see Definition 4 and [13,24]), it was observed in [27] that p > d is associated with a Lipschitz contractive map and the Perron-Frobenius theorem holds without any special requirement on the non-zero pattern of T. In this work we shall mostly focus on this second case, thus we recall here below the corresponding Perron-Frobenius theorem. We refer to [27] for more details and for a thorough bibliography review on the subject.…”

A framework for second-order eigenvector centralities and clustering coefficients

“…Note that, for α = 1 and M = P in (5) we retrieve exactly the rooted PageRank diffusion (18). Unlike the linear case, the convergence of the second order nonlinear process (20) is not straightforward. However, ideas from the proof of Theorem 4.1 can be used to show that the convergence is guaranteed for any choice of the tensor T , of the matrix M , and of the starting point y 0 ≥ 0, provided that the graph G M is aperiodic.…”

“…In Figure 8 we compare the performance of the link prediction algorithm based on the standard seeded PageRank similarity matrix S P R (18) and the newly introduced similarity matrix S M (20) induced by M with M = P and T = T W , the random walk triangle tensor. The tests were performed on the real-world networks UK faculty and Small World citation.…”

Extrapolation methods for fixed‐point multilinear PageRank computations

“…31 Although much has been done in recent years, much is still unknown for tensor eigenpairs, including the development of fast algorithms for their computation. To this end, higher order and nonlinear versions of the classic power method have been proposed to address the computation of different types of tensors eigenpairs (see References 19,23,[32][33][34][35][36] and, due to the large size of typical problems, enhancing the efficiency and robustness of these methods is of utmost importance.…”