变分贝叶斯方法：脉冲噪声下的鲁棒多重测量稀疏信号恢复

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本篇文章《基于变分贝叶斯推理的鲁棒多重测量稀疏信号恢复》主要探讨了在数字信号处理领域中的一个重要问题，即如何设计一种高效且鲁棒的方法来恢复受到脉冲噪声影响的多重测量向量中的稀疏信号。作者们针对这一挑战，提出了一种创新的贝叶斯概率模型，旨在解决普遍存在的脉冲噪声环境下的信号恢复问题。在模型构建中，文章采用了三级分层的贝叶斯估计程序，其中脉冲噪声和联合行稀疏源信号分别被高斯尺度混合分布和多元广义t分布所建模。这种选择能够更准确地捕捉到脉冲噪声的特性，并考虑到信号的稀疏性。变分贝叶斯框架被用于估计隐藏在信号和测量模型中的参数，通过使用多种概率分布来表达这些参数的特征，使得算法能够进行更精确的后验分布估计。不同于传统的方法，作者们提出了一个完整的贝叶斯推理算法，结合了变分贝叶斯估计技术，这使得算法在处理脉冲噪声时展现出强大的鲁棒性。通过估计未知参数，算法能够有效地适应并抵消噪声的影响，从而提高信号恢复的准确性。本文的研究成果通过大量的仿真实验得到了验证，结果显示，在各种类型的脉冲噪声条件下，所提出的算法显著优于其他同类的鲁棒稀疏信号恢复方法。这表明，该方法在实际应用中具有广泛的优势，特别是在需要抵抗突发噪声干扰的场景中，如通信、图像处理和信号压缩等领域。这篇研究论文不仅为稀疏信号恢复问题提供了新的理论框架，还为数字信号处理领域的工程师和研究人员提供了一种实用且有效的工具，对于提升信号质量、提高系统性能具有重要意义。它展示了变分贝叶斯方法在处理复杂噪声环境下的潜力，为未来相关领域的进一步发展开辟了新的路径。

D. Wang, Z. Zhang / Digital Signal Processing 89 (2019) 131–144 133

multiple signal classiﬁcation algorithm (MUSIC) to handle a rank

defective problem in the MMV model. The algorithm provides a

two-stage framework to recover the MMV sparse signals contain-

ing

k nonzero rows. The ﬁrst stage identiﬁes the k − s entries

of the support by using a conventional MMV algorithm, while

the remaining s entries are determined by applying MUSIC on

an augmented subspace. The authors claimed that the compres-

sive

MUSIC could make the l

bound as s approaches the nonzero

support size k. Relatively, MUSIC performs well over the conven-

tional

MMV approaches. In order to fully exploit the advantages

of MUSIC to relax the condition and reduce the iterations, Wen et

al. [32]suggested a semisupervised MUSIC (SS-MUSIC) approach

to recover the support reliably under unfavorable rank-defective

situations in the MMV problem. In SS-MUSIC, both the labeled

MMV and some reliable unlabeled atoms are iteratively exploited

for classiﬁer construction. Hence, the inadequate supervised infor-

mation

in rank defective MMV can be additionally compensated

from those unlabeled data so as to increase the discrimination of

the classiﬁer. As a consequence, SS-MUSIC can successfully classify

all atoms within k − s iterations. In addition, Blanchard et al. [33]

extended

ﬁve known greedy algorithms to solve MMV problems,

e.g., iterative hard thresholding (IHT), normalized IHT (NIHT), etc.

However, the recovery bound acquired by the simultaneous NIHT

algorithm is linearly dependent on the l

-norm of the noise matrix,

which makes that the approach can not ensure to provide accu-

rate

reconstruction when heavy-tailed noise appears. To this end,

Ollila [30]developed a robust simultaneous NIHT (SNIHT) method

to simultaneously compute the estimate of MMV’s sparse signal

matrix and the scale of the related error distribution, depending

on Huber’s criterion and the HUB-SNIHT algorithm. By comparison

against SNIHT, HUB-SNIHT can obtain MMV’s joint sparse signals

well when noise is heavy-tailed non-Gaussian.

Mixed norm optimization

The

recent works have shown that the mixed norm optimiza-

tion

approach is an effective tool for MMV problems with Gaus-

sian

noise. Some valuable achievements [6,13,28,34,35]are concen-

trated

on studying deterministic optimization approaches to han-

dle

p,q

-norm minimization models. For instance, Cotter et al. [6]

acquired

an improved M-FOCUSS, replying upon factored gradient

descent inspirations. They claimed that in the case of q = 1, once

the algorithm did not get stuck into a ﬁxed point, it could converge

to a local or global point. Hyder et al. [13]developed a robust l

2,0

approximation algorithm so-called JLZA for MMV joint-sparse re-

covery

problems. JLZA could get a clear performance improvement

than other convex relaxation algorithms and also could achieve

satisfactory sparse signal recovery with a very high probability.

However, when non-Gaussian noise is included in MMV problems,

less original work can be found in the literature. We notice that He

et al. [28]acquired a smooth block successive minimization algo-

rithm

for an MMV sparse recovery problem with impulsive noise.

In their work, the impulsive signals are expressed by a generalized

-norm divergence function which differs from the conventional

residual l

-norm metric function, while a smooth approximation

function is adopted to replace the non-smoothness l

2,0

joint sparse

measurement function. After that, such an MMV problem is formu-

lated

by a smooth approximate model solved by a block coordinate

descent method with successive quadratic upper-bound minimiza-

tion

(SQUM). The SQUM could achieve outlier-robust sparse recov-

ery.

Robust MMV Bayesian approach

Bayesian

theory-based sparse signal recovery approaches have

recently received much attention in the ﬁeld of CS, as they can

achieve satisfactory recovery performance by comparison against

some popular l

-norm minimization algorithms and FOCUSS ap-

proaches,

and especially can provide posterior distribution for the

sparse signal [16]. On the basis of the Bayesian sparse recovery

method [37], multiple kinds of robust Bayesian algorithms can be

derived to solve the unconstrained l

p,0

-norm (p ≥ 1) minimization

model. Wipf et al. [15] obtained a multiple sparse Bayesian learn-

ing

(MSBL) algorithm for a MMV model, after extending their SBL

approach [38] speciﬁcally designed for SMV problems. Zhang et

al. [16]proposed a fast temporal MSBL for eﬃciently coping with

an MMV model which required all elements in each nonzero row

of the solution matrix be temporally correlated. All these acquired

Bayesian methods belong to the category of two-level hierarchical

estimation capable of being utilized to construct the sparse prior,

in which the expectation-maximization (EM) algorithm performs

the estimates of unknown parameters and random variables. How-

ever,

the EM algorithm has a serious drawback, namely it needs

the knowledge of the posterior of the hidden variables.

Fortunately,

variational methodology, an iterative approach, can

ameliorate EM’s shortcomings and become increasingly popular in

the signal processing community. It can utilize an approximation to

estimate the posterior of the hidden variables given the observa-

tions.

Based on this approximation, Bayesian inference is possible

by maximizing a lower bound of the likelihood function which also

guarantees local convergence. This indicates that sparse recovery

problems can be solved by VB approaches [19,20,22,25,36]. In or-

der

to address the problem of adaptive sparse signal estimation

and system identiﬁcation, Themelis et al. [19]designed a unify-

ing

framework of sparse VB algorithms to facilitate the posterior

inference of the hidden variables by employing heavy-tailed pri-

ors

at a conjugate hierarchical form. Babacan et al. [20]presented

a general class of multivariate priors for group-sparse modeling

under

the Bayesian framework, while gaining some estimation pro-

cedures

used for such priors in terms of variational inference. The

reported experimental results suggested that the proposed formu-

lation

was very powerful and provided better estimation perfor-

mance

than state-of-the-art deterministic approaches. Additionally,

outliers and non-Gaussian noise, presented in MMV problems can

be also handled effectively by the reported VB methods. Shang et

al. [22]claimed that impulsive noise could be modeled by Student-

distributions and then the VB method VB-JSR was designed to

ﬁnd the approximate posterior distribution required by Bayesian

inference. Their experiments indicated that VB-JSR can offer much

better recovery performance than their compared algorithms.

3. Preliminaries and problem statement

3.1. Optimization approaches on MMV sparse signal recov ery

Let  ∈ R

M×N

(M  N) denote a known sensing matrix, y and

e represent a measurement vector and an unknown noise vec-

tor

in R

, respectively. Assume that any two columns in  are

linearly independent. Mathematically, the basic SMV sparse signal

recovery problem is to recover the N-dimensional k-sparse signal

vector x satisfying

y = x +e, (1)

where k-sparse denotes that there are at most k nonzero elements

in x. Correspondingly, once L measurement vectors are available in

some applications, the MMV sparse signals recovery problem [6],

called the MMV problem, is to ﬁnd a k -rowsparse signal matrix

X ∈R

N×L

subjected to

Y = X +E, (2)

where k-rowsparse represents that there exist at most k (k < M)

nonzero row vectors in X, Y ∈ R

M×L

is a measurement matrix

composed of L column measurement vectors and E ∈ R

M×L

is a

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变分贝叶斯方法：脉冲噪声下的鲁棒多重测量稀疏信号恢复

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