Sparse least squares solutions of multilinear equations

TL;DR

This paper introduces a sparse least squares model for solving multilinear equations that reduces computational costs, along with a novel algorithm with quadratic convergence, validated by numerical experiments on tensor datasets.

Contribution

It proposes a new sparse least squares approach and the Newton Hard-Threshold Pursuit algorithm for multilinear equations, with convergence analysis and empirical validation.

Findings

NHTP algorithm converges quadratically under certain conditions.

The method effectively solves multilinear equations involving tensors.

Numerical experiments demonstrate the efficiency of the proposed approach.

Abstract

In this paper, we propose a sparse least squares (SLS) optimization model for solving multilinear equations, in which the sparsity constraint on the solutions can effectively reduce storage and computation costs. By employing variational properties of the sparsity set, along with differentiation properties of the objective function in the SLS model, the first-order optimality conditions are analyzed in terms of the stationary points. Based on the equivalent characterization of the stationary points, we propose the Newton Hard-Threshold Pursuit (NHTP) algorithm and establish its locally quadratic convergence under some regularity conditions. Numerical experiments conducted on simulated datasets including cases of Completely Positive(CP)-tensors and symmetric strong M-tensors illustrate the efficacy of our proposed NHTP method.