A Novel Robust Approach to Least Squares Problems with Bounded Data Uncertainties

TL;DR

This paper presents a robust minimax regret approach to least squares problems with bounded data uncertainties, formulated via semi-definite programming, and demonstrates its effectiveness through simulations.

Contribution

It introduces a novel minimax regret criterion for robust least squares problems and develops semi-definite programming solutions for both unstructured and structured cases.

Findings

Proposed algorithms outperform existing methods in simulations.

Robust formulations handle data uncertainties effectively.

Semi-definite programming enables efficient solutions.

Abstract

In this correspondence, we introduce a minimax regret criteria to the least squares problems with bounded data uncertainties and solve it using semi-definite programming. We investigate a robust minimax least squares approach that minimizes a worst case difference regret. The regret is defined as the difference between a squared data error and the smallest attainable squared data error of a least squares estimator. We then propose a robust regularized least squares approach to the regularized least squares problem under data uncertainties by using a similar framework. We show that both unstructured and structured robust least squares problems and robust regularized least squares problem can be put in certain semi-definite programming forms. Through several simulations, we demonstrate the merits of the proposed algorithms with respect to the the well-known alternatives in the literature.