Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

TL;DR

This paper introduces an accelerated projected gradient method with convergence guarantees for solving linear inverse problems with sparsity constraints, offering an alternative to traditional soft-thresholding algorithms.

Contribution

It presents a new accelerated gradient algorithm with projection on $ ext{l}_1$-balls for sparse inverse problems, including convergence proofs and variable thresholding.

Findings

Convergence in norm for the proposed projected gradient method.

Effective acceleration techniques improve convergence speed.

The method provides a viable alternative to iterative soft-thresholding.

Abstract

Regularization of ill-posed linear inverse problems via $\ell_1$ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an $\ell_1$ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to $\ell_1$-constraints, using a gradient method, with projection on $\ell_1$-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.