Tensor-based Dinkelbach method for computing generalized tensor eigenvalues and its applications

TL;DR

This paper introduces a tensor-based Dinkelbach method for efficiently computing extremal tensor generalized eigenvalues, reformulating the problem as a multilinear optimization with proven convergence and practical validation.

Contribution

It presents a novel tensor-based Dinkelbach--Type method, reformulating the eigenvalue problem as a MOP and establishing global convergence and explicit convergence rates.

Findings

Efficient algorithm for extremal tensor eigenvalues.

Global convergence and explicit rate analysis.

Numerical validation on high-order problems.

Abstract

In this paper, we propose a novel tensor-based Dinkelbach--Type method for computing extremal tensor generalized eigenvalues. We show that the extremal tensor generalized eigenvalue can be reformulated as a critical subproblem of the classical Dinkelbach--Type method, which can subsequently be expressed as a multilinear optimization problem (MOP). The MOP is solved under a spherical constraint using an efficient proximal alternative minimization method, in which we rigorously establish the global convergence. Additionally, the equivalent MOP is reformulated as an unconstrained optimization problem, allowing for the analysis of the Kurdyka-Lojasiewicz (KL) exponent and providing an explicit expression for the convergence rate of the proposed algorithm. Preliminary numerical experiments on solving extremal tensor generalized eigenvalues and minimizing high-order trust-region subproblems are provided, validating the efficacy and practical utility of the proposed method.