Sparse least squares solutions of multilinear equations

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 analysed 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 efficiency of our proposed NHTP method.

Keywords:

• Sparse least squares
• multilinear equations
• Newton Hard-Thresholding Pursuit
• completely positive tensor
• symmetric strong M-tensor

Mathematics Subject Classifications:

• 90-08
• 90C46
• 90C26

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Fundamental Research Funds for the Central Universities of China [grant number 2019JBM078].