Splitting-based randomized iterative methods for solving indefinite least squares problem

TL;DR

This paper introduces a simple splitting-based iterative method for indefinite least squares problems, along with two randomized variants that significantly improve computational efficiency and convergence speed.

Contribution

The paper proposes a novel splitting (SP) method for ILS problems and develops two randomized iterative algorithms based on it, enhancing efficiency and convergence.

Findings

The SP method converges unconditionally for any initial value.

The randomized methods outperform existing iterative methods in speed and iteration count.

Numerical experiments confirm the effectiveness and acceleration of the proposed methods.

Abstract

The indefinite least squares (ILS) problem is a generalization of the famous linear least squares problem. It minimizes an indefinite quadratic form with respect to a signature matrix. For this problem, we first propose an impressively simple and effective splitting (SP) method according to its own structure and prove that it converges 'unconditionally' for any initial value. Further, to avoid implementing some matrix multiplications and calculating the inverse of large matrix and considering the acceleration and efficiency of the randomized strategy, we develop two randomized iterative methods on the basis of the SP method as well as the randomized Kaczmarz, Gauss-Seidel and coordinate descent methods, and describe their convergence properties. Numerical results show that our three methods all have quite decent performance in both computing time and iteration numbers compared with the latest iterative method of the ILS problem, and also demonstrate that the two randomized methods indeed yield significant acceleration in term of computing time.