Orthogonal Constrained Neural Networks for Solving Structured Inverse Eigenvalue Problems

TL;DR

This paper presents SMLP, a lightweight neural network with orthogonal constraints, designed to efficiently solve Structured Inverse Eigenvalue Problems across various instances with high accuracy.

Contribution

A novel neural network architecture incorporating a Stiefel layer for orthogonality, enabling efficient, unsupervised solutions to diverse SIEPs with a unified framework.

Findings

Demonstrates high accuracy in numerical tests

Efficient unsupervised training process

Applicable to multiple SIEP instances

Abstract

This paper introduces a novel neural network for efficiently solving Structured Inverse Eigenvalue Problems (SIEPs). The main contributions lie in two aspects: firstly, a unified framework is proposed that can handle various SIEPs instances. Particularly, an innovative method for handling nonnegativity constraints is devised using the ReLU function. Secondly, a novel neural network based on multilayer perceptrons, utilizing the Stiefel layer, is designed to efficiently solve SIEP. By incorporating the Stiefel layer through matrix orthogonal decomposition, the orthogonality of similarity transformations is ensured, leading to accurate solutions for SIEPs. Hence, we name this new network Stiefel Multilayer Perceptron (SMLP). Furthermore, SMLP is an unsupervised learning approach with a lightweight structure that is easy to train. Several numerical tests from literature and engineering domains demonstrate the efficiency of SMLP.