P-范数下脉冲噪声下压缩感知的鲁棒稀疏恢复算法

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本文主要探讨了在存在脉冲噪声的条件下，压缩感知（Compressive Sensing, CS）中的鲁棒稀疏恢复问题。传统的CS理论通常假设测量数据是线性的且无噪声，但在实际应用中，诸如电磁干扰、图像传感器故障等常常引入脉冲噪声，这要求恢复算法具有更强的鲁棒性。作者提出了一个基于广义的`P-范数（0<p<2）的稳健模型，该范数作为残差误差在`L1范数正则化下的度量标准。具体而言，他们提出了一种有效的交替方向方法（Alternating Direction Method, ADM），这是一种迭代优化技术，能够有效地求解这种`P-范数约束下的稀疏重构问题。对于非凸的情况，即p值小于1时，他们引入了平滑策略，使得算法能够在保证收敛性的同时处理这类复杂问题。文章详细分析了提出的算法在凸和非凸情况下的一致性和收敛条件，并通过数值模拟展示了新算法在面对脉冲噪声时展现出的卓越鲁棒性能，超越了现有的相关研究成果。该工作的重要性在于，它不仅扩展了CS理论的应用范围，使之能在有噪声环境中保持较高的恢复精度，还提供了实际工程中遇到的噪声挑战的一种有效解决方案。此外，提出的算法具有良好的计算效率和理论支持，为信号处理领域的鲁棒稀疏恢复提供了一个新的优化工具，对于诸如无线通信、图像处理和信号压缩等领域的噪声抑制和信号恢复有着潜在的实际应用价值。

ROBUST SPARSE RECOVERY FOR COMPRESSIVE SENSING IN IMPULSIVE NOISE

USING `

-NORM MODEL FITTING

Fei Wen

, Peilin Liu

, Yipeng Liu

, Robert C. Qiu

, Wenxian Yu

Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China

School of Electronic Engineering, University of Electronic Science and Technology of China, China

{wenfei, liupeilin, wxyu}@sjtu.edu.cn, yipengliu@uestc.edu.cn, rcqiu@sjtu.edu.cn

ABSTRACT

This work considers the robust sparse recovery problem in

compressive sensing (CS) in the presence of impulsive mea-

surement noise. We propose a robust formulation for sparse

recovery using the generalized `

-norm with 0 < p < 2 as

the metric for the residual error under `

-norm regularization.

An alternative direction method (ADM) has been proposed to

solve this formulation efﬁciently. Moreover, a smoothing s-

trategy has been used to derive a convergent method for the

nonconvex case of p < 1. The convergence conditions of the

proposed algorithm for both the convex and nonconvex cas-

es have been provided . Numerical simulations demonstrated

that the new algorithm can achieve state-of-the-art robust per-

formance in highly impulsive noise.

Index Terms— Compressive sensing, robust sparse re-

covery, alternating direction method, `

-norm data-ﬁtting

1. INTRODUCTION

Compressive sensing (CS) allows us to acquire sparse sig-

nals at a signiﬁcantly lower rate than the classical Nyquist

sampling [1]. The CS theory states that if a signal x ∈ R

is sparse, only a small number of linear measurements y =

Ax ∈ R

(m < n) of the signal sufﬁce to accurately re-

construct it, A ∈ R

m×n

is the sensing matrix. Taking the

inevitable measurement noise into consideration, the com-

pressed measurements can be modeled as

y = Ax + n

where n ∈ R

is additive measurement noise.

In the CS setting, the recovery of x from y is generally ill-

posed since m < n. However, provided that x is sparse and

A satisﬁes some stable embedding conditions [2], x can be

reliably recovered with an error upper bounded by the noise

strength. To reconstruct x, the BPDN and LASSO formula-

tions [3], [4] are of the most popular, e.g., LASSO

min



kAx − yk

+ kxk



(1)

This work was supported in part by the National Natural Science Foun-

dation of China (NSFC) under grants 61401501, 61472442 and 61171171.

where µ > 0 is a regularization parameter.

As in BPDN, LASSO and many other variants, the `

norm data-ﬁtting model, which is optimal for Gaussian noise

in the maximum likelihood sense, is the most widely used

one. However, in practical applications, the measurement

noise may be of different kinds or combinations. Impulsive

noise in measurements may come from missing data in the

measurement process, transmission problems [5]–[7], fault-

y memory locations [8], buffer overﬂow [9], and has been

raised in many image and video processing works [10]–[13].

In these cases, the `

-norm model is inefﬁcient as it is highly

sensitive to outliers in the observations.

Recently, various robust formulations have been proposed

for CS to suppress the outliers in measurements. In [14],

the Lorentzian-norm has been employed as the metric for the

residual error. In [15], the `

-norm has been used as the data-

ﬁtting model to obtain a robust formulation as

min



kAx − yk

+ kxk



. (2)

It has been shown in [15] that, when the measurements con-

tain impulsive noise, the `

-loss can result in dramatically bet-

ter performance compared with the `

-one. Subsequently, the

Huber penalty function has been used to design robust formu-

lation for sparse recovery in [16].

In this work, we use the generalized `

-norm, 0 < p <

2, as the loss function for the residual error to propose the

following robust formulation

min



kAx − yk

+ kxk



. (3)

When 0 < p < 1, k·k

is the `

quasi-norm deﬁned in a simi-

lar manner as the case of p ≥ 1, i.e., kvk

i=1

. The

intuition behind utilizing `

-norm loss function is that, com-

pared with the quadratic function, it is a less rapidly increas-

ing function when p < 2, and, accordingly, is less sensitive to

large outliers, especially when p is small.

Except for the special case of p = 1, the problem (3)

has still not been well addressed. When 1 < p < 2, it can

be solved by traditional convex optimization methods such as

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