Minimization of Transformed $L_1$ Penalty: Closed Form Representation and Iterative Thresholding Algorithms

TL;DR

This paper introduces a closed-form fixed point representation and iterative thresholding algorithms for the transformed $L_1$ penalty, demonstrating superior sparse signal recovery performance in compressed sensing.

Contribution

It develops explicit TL1 thresholding functions and iterative algorithms, enhancing sparse recovery methods over existing thresholding techniques.

Findings

TL1 thresholding functions are in closed form for all parameters.

Proposed TL1 iterative thresholding outperforms hard and half thresholding.

Adaptive thresholds improve recovery accuracy in noisy and noiseless scenarios.

Abstract

The transformed $l_1$ penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm ($p \in (0,1)$). In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the $l_p$ thresholding functions ($p \in [0,1]$) are in closed form only for $p=0,1,1/2,2/3$, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds consistently performs the best in sparse signal recovery with and without measurement noise.