A Lifted $\ell_1 $ Framework for Sparse Recovery

TL;DR

This paper introduces a generalized lifted  regularization framework for sparse recovery, connecting it to  minimization, and demonstrates its effectiveness through an efficient ADMM-based algorithm and experiments.

Contribution

It proposes a novel lifted  regularization that generalizes existing methods and guarantees equivalence to  minimization under certain conditions.

Findings

Improves sparse recovery performance over state-of-the-art methods.

Provides an efficient ADMM algorithm with proven convergence.

Establishes theoretical connections between lifted  and  minimization.

Abstract

Motivated by re-weighted $\ell_1$ approaches for sparse recovery, we propose a lifted $\ell_1$ (LL1) regularization which is a generalized form of several popular regularizations in the literature. By exploring such connections, we discover there are two types of lifting functions which can guarantee that the proposed approach is equivalent to the $\ell_0$ minimization. Computationally, we design an efficient algorithm via the alternating direction method of multiplier (ADMM) and establish the convergence for an unconstrained formulation. Experimental results are presented to demonstrate how this generalization improves sparse recovery over the state-of-the-art.