Iterative Reweighted <formula formulatype="inline"><tex Notation="TeX">$\ell_{2}/\ell_{1}$</tex> </formula> Recovery Algorithms for Compressed Sensing of Block Sparse Signals

Abstract

In many applications of compressed sensing the signal is block sparse, i.e., the non-zero elements of the sparse signal are clustered in blocks. Here, we propose a family of iterative algorithms for the recovery of block sparse signals. These algorithms, referred to as iterative reweighted 2/ 1 minimization algorithms (IR-2/ 1), solve a weighted 2/ 1 minimization in each iteration. Our simulation and analytical results on the recovery of both ideally and approximately block sparse signals show that the proposed iterative algorithms have significant advantages in terms of accuracy and the number of required measurements over non-iterative approaches as well as existing iterative methods. In particular, we demonstrate that, by increasing the block length, the performance of the proposed algorithms approaches the Wu-Verdú theoretical limit. The improvement in performance comes at a rather small cost in complexity increase. Further improvement in performance is achieved by using a priori information about the location of nonzero blocks, even if such a priori information is not perfectly reliable.Index Terms -Block sparsity, compressed sensing, 2/ 1 minimization, iterative recovery algorithms, iterative reweighted 2/ 1 minimization.

Completion of Structured Low-Rank Matrices via Iteratively Reweighted Least Squares

“…IRLS is a practical framework for the optimization of non-smooth, possibly non-convex, high-dimensional objectives that minimizes quadratic models which majorize these objectives. Due to its ease of implementation and favorable data-efficiency, it has been widely used in compressed sensing [GR97, CY08, DDFG10, LXY13, FPRW16, KMVS21], robust statistics [HW77, AH15, MGJK19], computer vision [CG17, LC22, SWL22], low-rank matrix recovery and completion [FRW11, MF12, KS18, KMV21], and in inverse problems involving group sparsity [CHHL14, ZB15,CHLH18]. Recently, it has been shown [LK23] that dictionary learning techniques can be incorporated into IRLS schemes for sparse and low-rank recovery to allow the learning of a sparsifying dictionary while recovering the solution.…”