A new steplength selection for scaled gradient methods with application to image deblurring

TL;DR

This paper introduces a novel steplength selection strategy for scaled gradient methods, improving image deblurring efficiency, especially in constrained problems, by approximating Hessian eigenvalues using a Lanczos-based approach.

Contribution

It extends a Hessian eigenvalue approximation strategy to scaled gradient projection methods for non-negatively constrained problems, enhancing image deblurring performance.

Findings

Improved convergence in image deblurring tasks.

Effective in both regularized and unregularized scenarios.

Enhanced gradient method efficiency for constrained optimization.

Abstract

Gradient methods are frequently used in large scale image deblurring problems since they avoid the onerous computation of the Hessian matrix of the objective function. Second order information is typically sought by a clever choice of the steplength parameter defining the descent direction, as in the case of the well-known Barzilai and Borwein rules. In a recent paper, a strategy for the steplength selection approximating the inverse of some eigenvalues of the Hessian matrix has been proposed for gradient methods applied to unconstrained minimization problems. In the quadratic case, this approach is based on a Lanczos process applied every m iterations to the matrix of the most recent m back gradients but the idea can be extended to a general objective function. In this paper we extend this rule to the case of scaled gradient projection methods applied to non-negatively constrained minimization problems, and we test the effectiveness of the proposed strategy in image deblurring problems in both the presence and the absence of an explicit edge-preserving regularization term.