Doubly iteratively reweighted algorithm for constrained compressed sensing models

TL;DR

This paper introduces a novel iterative reweighted algorithmic framework for constrained compressed sensing models with nonconvex regularizers and loss functions, demonstrating improved recovery accuracy and efficiency.

Contribution

The paper develops a new algorithmic framework employing iteratively reweighted schemes with a novel termination criterion, enabling efficient solutions and convergence guarantees for nonconvex constrained compressed sensing.

Findings

The proposed algorithms outperform existing methods in recovery error.

The algorithms are faster while maintaining or improving solution quality.

Numerical experiments validate the effectiveness of the approach on badly-scaled matrices.

Abstract

We propose a new algorithmic framework for constrained compressed sensing models that admit nonconvex sparsity-inducing regularizers including the log-penalty function as objectives, and nonconvex loss functions such as the Cauchy loss function and the Tukey biweight loss function in the constraint. Our framework employs iteratively reweighted $\ell_1$ and $\ell_2$ schemes to construct subproblems that can be efficiently solved by well-developed solvers for basis pursuit denoising such as SPGL1 [6]. We propose a new termination criterion for the subproblem solvers that allows them to return an infeasible solution, with a suitably constructed feasible point satisfying a descent condition. The feasible point construction step is the key for establishing the well-definedness of our proposed algorithm, and we also prove that any accumulation point of this sequence of feasible points is a stationary point of the constrained compressed sensing model, under suitable assumptions. Finally, we compare numerically our algorithm (with subproblems solved by SPGL1 or the alternating direction method of multipliers) against the SCP$_{\rm ls}$ in [41] on solving constrained compressed sensing models with the log-penalty function as the objective and the Cauchy loss function in the constraint, for badly-scaled measurement matrices. Our computational results show that our approaches return solutions with better recovery errors, and are always faster.