改进的迭代重权L1方法提升压缩感知稀疏解性能

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本文主要探讨了不确定线性系统中稀疏解的Rewighted L1极小化方法。压缩感知(CS)是一种理论，它指出任何信号，只要其非零元素数量有限，即使在采样远少于信号维度的情况下，仍然可以通过特定算法进行有效恢复。文章的核心贡献是提出了一种简单而高效的迭代加权算法，该算法在每次迭代中考虑信号的支撑集，即包含非零元素的索引集合。传统的L1最小化是CS恢复过程中的常用策略，因为它鼓励稀疏性，即使得解决方案中的大部分元素接近于零。然而，L1优化可能在某些情况下未能充分利用信号特性，尤其是当信号的非零系数分布不均匀时。为了改进这种情况，作者设计了一种迭代重加权的方法，通过动态调整每个系数的权重，使得算法更能适应信号的局部特性。算法的关键在于构建一个merit function（优点函数），该函数能够根据当前估计的信号质量和支撑集信息来调整权重。这种方法的优势在于它能够自适应地对那些非零项进行更精确的支持，从而提高恢复的成功概率。通过数值实验，研究者发现新提出的算法相比于经典L1最小化以及其他迭代重加权的L1算法，在成功率上表现更优。关键词：压缩感知、L1最小化、重加权算法、支撑集、优点函数。这项工作在6th International Congress on Image and Signal Processing (CISP2013)会议上发表，引用号为978-1-4799-2764-7/13/$31.00©2013IEEE。研究者来自南通大学电子与信息学院和图书馆，他们展示了这种技术在实际信号处理问题中的潜力，特别是在信号稀疏表示和高效恢复方面的应用。本文的主要知识点集中在稀疏信号的高效恢复策略上，特别是如何通过迭代重加权方法提升压缩感知的性能，以适应不同类型信号的非零项分布。这种新颖的算法为解决实际问题中的线性系统提供了新的解决方案，具有重要的理论和实际价值。

2013 6th International Congress on Image and Signal Processing (CISP 2013)

Rewighted L1-Minimization for Sparse Solutions to

Underdetermined Linear Systems

Zhengguang Xie

School of Electronics and Information

Nantong University

Nantong City, China

Jianping Hu

Library of Nantong University

Nantong University

Nantong City, China

Abstract—We proposed a simple and efficient iteratively

reweighted algorithm with iterative support set to improve the

recover performance for compressive sensing (CS). The

numerical experiential results demonstrate that the new method

outperforms in successful probabilities, compared with classical

-minimization and other iteratively reweighted

-algorithms.

Keywords- Compressive sensing;

-Minimization; Reweighted

algorithm; Support set; Merit function.

I. INTRODUCTION

Compressive sensing (CS) theory present [1-3] that any

signals

∈x ( N is the number of entries of

x ) which

has a limited number of nonzero entries

(

{: 0}

ix N=≠<<x

), e.g., the coefficient sequence

of the signals in the appropriate frame/basis, can be

reconstructed by solving the combinatorial optimization

problem

（

）

Pmin

subject to =

xAxy (1)

here

∈A ,

∈y , MN<< , M is the number of

measurements or entries of

y . Theoretically, the program

（

）

P can perfectly recover all sparse signals

x obeying

M≤x

. However, this is of little practical use, since

the optimization problem

（

）

P is non-convex and generally

impossible to solve in polynomial time since its solution

usually requires an intractable combinational search.

It is well-known that

-norm is the convex envelope of

x over the region {: 1}

∞

≤xx . One of the main

approaches, and widely used, to attack

（

）

P is through

minimization (

∑

x ), which leads to the more

tractable problem:

（

）

Pmin

subject to =

xAxy (2)

Unlike

（

）

P , this problem is convex—it can actually be

recast as a linear program—and hence can be solved very

efficiently. Various conditions (mutual coherence (MC)[4],

restricted isometric property/Condition (RIP or RIC) [5], and

null space property (NSP) [4]) for the relationship

0 1

arg m in{ : } { } arg min{ : }

x== = =

xAxy x Axy

(3)

have been developed. It has been shown, in terms of sparse

signal recovery, that

-minimization allows recovery of sparse

signals from remarkably few measurements: supposing

A is

chosen randomly from a suitable distribution, then with very

high probability, all sparse signals

x for which

M α≤x

with (log( / ))ONMα = can be perfectly

recovered by using

（

）

P . A comprehensive discussion and

survey of recent results in this field can be found in [1, 6-7].

Inspired by the efficiency of

-minimization, it is natural

to ask whether there are other alternatives to outperform

minimization in finding sparse solutions of linear systems.

Numerical experiments indicate that the iteratively reweighted

-minimization does have an advantage over un-weighted

minimization in many situations [8-22] to find sparsity

solution, which can be formulate weighted

-problems

()WP as follows.

() () ()

()min

ii i

WP subject to

∈x

Wx Ax =y （4）

where

() ()

diag( )

w=w and

() () () () ()

( , ,, ,, ) R

iiiiiTN

wwwww

=∈"" （ means positive

real number ） is the vector of weights determined by the

previous iterate

(1) (1) (1) (1)

(,, ,, )R

iiiiTN

xxx

−−−−

=∈x "" .

The novel contribution of this paper is a new and fast

iteratively reweighted algorithm for CS decoding. The

numerical experiential results show good performance of the

proposed algorithm in successful probabilities, in comparison

to classical

-minimization and other existing algorithms.

In Sect. 2, we summarize the unified framework of

reweighted

-minimization, and give out merit functions for

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