Probability-One Optimization of Generalized Rayleigh Quotient Sum For Multi-Source Generalized Total Least-Squares

TL;DR

This paper introduces a probability-one homotopy optimization method based on Riemannian Trust Region algorithms to reliably find the global maximum of the sum of Rayleigh quotients, applicable to multi-source generalized total least-squares problems.

Contribution

It extends existing optimization algorithms with a homotopy approach to ensure convergence to the global maximizer for Rayleigh quotient sums.

Findings

Enhanced convergence speed compared to existing methods

High success rate in reaching the global maximizer

Scalability demonstrated with larger problem dimensions

Abstract

This paper addresses the global optimization of the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere. While various methods have been proposed for this problem, they fail to reliably converge to the global maximizer. To overcome this limitation, we propose an extension of the Riemannian Trust Region algorithm based on the probability-one homotopy optimization method, which enhances convergence to a global maximizer and, under certain conditions, ensures convergence to the global maximizer. In addition to the proposed method, existing state-of-the-art approaches are also presented, along with an explanation of their limitations and their connection to the proposed method. The proposed method is evaluated alongside the state-of-the-art approaches through numerical experiments, assessing convergence speed, success in reaching the global maximizer, and scalability with increasing problem dimension. Furthermore, we demonstrate how this ties in with the multi-source Bayesian Generalized Total Least-Squares (B-GTLS) problem, illustrating its applicability.