On complex power nonnegative matrices

Tudisco, Francesco; Cardinali, Valerio; Fiore, Carmine Di

doi:10.1016/j.laa.2014.12.021

Cited by 12 publications

(15 citation statements)

References 16 publications

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“…In particular, it holds ξ = Hence, there exists α ∈ R d ++ such that z = α ⊗ u and (IV) implies that ω(F , x) = ω(F , α ⊗ u). As A is primitive, we know from Theorem 1.1 [26] Hence, we have ω(F , x) = ω(F , α ⊗ u) = {α B ⊗ u}. So, lim k→∞F k (x) = α B ⊗ u follows from (III).…”

Section: Uniqueness and Simplicity Of Positive Eigenvectorsmentioning

confidence: 87%

The Perron--Frobenius Theorem for Multihomogeneous Mappings

Gautier¹,

Tudisco²,

Hein³

2019

SIAM J. Matrix Anal. Appl.

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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.

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Section: Uniqueness and Simplicity Of Positive Eigenvectorsmentioning

confidence: 87%

The Perron--Frobenius Theorem for Multihomogeneous Mappings

Gautier¹,

Tudisco²,

Hein³

2019

SIAM J. Matrix Anal. Appl.

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“…It follows that A T A is primitive (see e.g. Theorem 1 in [40]). By the same theorem, there exists a positive integer k such that (A T A) k is a matrix with positive entries.…”

Section: Lemma 4 Let a ∈ R M×n Be A Matrix With Nonnegative Entries And Letmentioning

confidence: 93%

The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

Gautier¹,

Hein

Tudisco

2021

J Sci Comput

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We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

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“…More precisely, if A is an irreducible and aperiodic stochastic matrix, then for any positive vector s 0 ∈ Ω + , we are guaranteed that the sequence s ℓ+1 = A s ℓ , ℓ=0,1,… converges to the unique

\overset{true‾}{s} \in Ω_{+}

such that

A \overset{true‾}{s} = \overset{true‾}{s}

. Note that we have, equivalently,

s_{ℓ + 1} = A^{ℓ} s_{0}

and actually, for irreducible and aperiodic chains, the whole matrix sequence A ℓ converges to the rank‐one matrix

\overset{true‾}{s} e^{T}

(see Reference ). The situation is different in the tensor settings.…”

Section: Multilinear Pagerankmentioning

confidence: 99%

“…Such PageRank transition matrix P PR models a new random walk, where one takes a step according to the initial Markov chain with probability α, and with probability 1−α randomly jumps to node i according to the fixed teleportation probability v i >0. Note that, as v ∈ Ω + , for any 0≤α<1, the PageRank matrix P PR is irreducible and thus, by the Perron‐Frobenius theorem, there exists a unique positive eigenvector s ∈ Ω + such that P PR s = s . The analogy with Equation essentially follows by Equation .…”

Section: Multilinear Pagerankmentioning

confidence: 99%

“…Such PageRank transition matrix P PR models a new random walk, where one takes a step according to the initial Markov chain with probability , and with probability 1 − randomly jumps to node i according to the fixed teleportation probability v i > 0. Note that, as v ∈ Ω + , for any 0 ≤ < 1, the PageRank matrix P PR is irreducible and thus, by the Perron-Frobenius theorem, there exists a unique positive eigenvector s ∈ Ω + such that P PR s = s. 4,40,41 The analogy with Equation (3) essentially follows by Equation (2). In fact, it is not difficult to observe that the proposed multilinear PageRank, solution of Equation (3) coincides with a nonnegative Z-eigenvector of the PageRank transition tensor  PR , that is, it solves Equation (2).…”

Section: Introductionmentioning

confidence: 99%

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Extrapolation methods for fixed‐point multilinear PageRank computations

Cipolla

Redivo–Zaglia

Tudisco

2020

Numerical Linear Algebra App

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Summary Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher order power method is one of the most commonly used technique for the computation of positive Z‐eigenvectors of this type of tensors. However, unlike the matrix case, the method may fail to converge even for irreducible tensors. Moreover, when it converges, its convergence rate can be very slow. These two drawbacks often make the computation of the eigenvectors demanding or unfeasible for large problems. In this work, we consider a particular class of nonnegative tensors associated with the multilinear PageRank modification of higher order Markov chains. Based on the simplified topological ε‐algorithm in its restarted form, we introduce an extrapolation‐based acceleration of power method type algorithms, namely, the shifted fixed‐point method and the inner‐outer method. The accelerated methods show remarkably better performance, with faster convergence rates and reduced overall computational time. Extensive numerical experiments on synthetic and real‐world datasets demonstrate the advantages of the introduced extrapolation techniques.

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On complex power nonnegative matrices

Cited by 12 publications

References 16 publications

The Perron--Frobenius Theorem for Multihomogeneous Mappings

The Perron--Frobenius Theorem for Multihomogeneous Mappings

The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

Extrapolation methods for fixed‐point multilinear PageRank computations

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