期望传播视角下的广义近似消息传递简洁推导

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本文档主要探讨了"使用期望传播的广义近似消息传递的简洁推导"这一主题，发表在2018年12月的IEEE SIGNAL PROCESSING LETTERS Vol.25, No.12上。作者Qiuyun Zou、Haochuan Zhang、Chao-Kai Wen、Shi Jin和Rong Yu在文中聚焦于一种高效算法——广义近似消息传递（Generalized Approximate Message Passing, GAMP），该算法特别适用于独立同分布随机信号在广义线性模型下的估计问题。 GAMP起源于[1]中提出的近似消息传递方法，旨在从带有加性噪声的线性测量中恢复稀疏信号。它被认为是循环贝叶斯网络中sum-product算法的一个近似实现，然而传统的推导方法往往依赖泰勒展开，这可能较为繁琐且不便于统一处理实数和复数情况下的GAMP。作者们在本文中提出了一种新的视角，即利用期望传播（Expectation Propagation, EP）来理解消息传递过程。与基于泰勒展开的方法相比，这种方法提供了一个更为简洁的推导框架，尤其是在设置高斯密度时，可以将实数GAMP和复数GAMP的推导统一起来。EP作为一种统计学习方法，通过最大化似然函数的下界，能有效地处理复杂概率模型，因此在这个新的视角下，GAMP的理论基础得以更加清晰地展现。文章的关键术语包括：广义近似消息传递、期望传播以及高斯复制性质，这些都是理解和应用GAMP算法时不可或缺的概念。通过这篇文章，读者不仅可以深入理解GAMP的原理，还能了解到如何通过EP简化其推导过程，这对于实际的信号处理和机器学习任务具有重要的理论价值和实践指导意义。

IEEE SIGNAL PROCESSING LETTERS, VOL. 25, NO. 12, DECEMBER 2018 1835

Concise Derivation for Generalized Approximate

Message Passing Using Expectation Propagation

Qiuyun Zou , Haochuan Zhang , Chao-Kai Wen ,ShiJin , and Rong Yu

Abstract—Generalized approximate message passing (GAMP)

is an efﬁcient algorithm for the estimation of independent identi-

cally distributed random signals under generalized linear model.

The sum-product GAMP has long been recognized as an approxi-

mate implementation of the sum-product loopy belief propagation.

In this letter, we propose to view the message passing in a new

perspective of expectation propagation (EP). Comparing with the

previous methods that were based on Taylor expansions, the pro-

posed EP method could unify the derivations for the real and the

complex GAMP, with a difference only in the setup of Gaussian

densities.

Index Terms—Generalized approximate message passing,

expectation propagation, Gaussian reproduction property.

I. INTRODUCTION

PPROXIMATE message passing (AMP) was proposed in

[1] to recover sparse signals from linear measurements

that were corrupted by additive white Gaussian noise. Owing to

its efﬁciency (roughly of quadratic complexity in the problem

size) and effectiveness (asymptotically exact in certain cases),

AMP has since found various applications in engineering. Af-

ter AMP, an extension termed generalized AMP (GAMP) [2]

was developed, which considered a much broader scope, the

generalized linear model (GLM), allowing the componentwise

output mapping to take an arbitrary form. Motivated by GAMP,

the so-called BiGAMP was proposed by [3]. BiGAMP further

extended the linear inverse scope to bi-linear inverse, in which

both the input signal and the measurement matrix are to be

estimated. In these early works, signals were considered to be

Manuscript received September 8, 2018; accepted October 14, 2018. Date

of publication October 18, 2018; date of current version November 1, 2018.

The work of Q. Zou and H. Zhang was supported in part by the National

Natural Science Foundation of China (NSFC) under Grant 61501127; in part

by the NSF of Guangdong under Grant 2016A030313705; and in part by the

Special Fund for Applied Science and Technology of Guangdong under Grant

2015B010129001, Grant 2014B090907010, and Grant 2015B010106010. The

work of C.-K. Wen was supported by the ITRI in Hsinchu, Taiwan. The work of

S. Jin was supported by the NSFC for Distinguished Young Scholars of China

under Grant 61625106. The work of R. Yu was supported by the NSFC under

Grant 61422201, Grant 61370159, and Grant 61503083. The associate editor

coordinating the review of this manuscript and approving it for publication was

Dr. Qing Ling. (Corresponding author: Haochuan Zhang.)

Q. Zou, H. Zhang, and R. Yu are with Guangdong University of Tech-

nology, Guangzhou 510006, China (e-mail:, qiuyunzou@qq.com; haochuan.

zhang@qq.com; yurong@ieee.org).

C.-K. Wen is with the National Sun Yat-sen University, Kaohsiung 80424,

Taiwan (e-mail:,chaokai.wen@mail.nsysu.edu.tw).

S. Jin is with Southeast University, Nanjing 210096, China (e-mail:, jin-

shi@seu.edu.cn).

Color versions of one or more of the ﬁgures in this letter are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/LSP.2018.2876806

real. However, for applications like wireless communications,

magnetic resonance imaging, and radar imaging, a complex

form would be more desired. In this context, complex AMP

and GAMP were proposed in [4]–[7] among others. Although

the sum-product AMP/GAMP has long been recognized as an

approximate implementation of the sum-product loopy belief

propagation (LBP), we ﬁnd that, following this direction, the

derivation for the complex case may face some new challenges

in the expansion and justiﬁcation of Taylor series.

In an attempt to recover complex GAMP without Taylor ex-

pansions, this letter offers a new perspective of viewing the

message passing as in an expectation propagation (EP) [8] man-

ner. To be speciﬁc, each variable and factor node in a factor

graph (FG) updates its message according to scalar EP rules.

The resulting messages are then in a Gaussian form, and after

certain approximation to the means and variances, the complex

sum-product GAMP could be recovered. It is worthy of noting

that the idea of using EP was introduced by [9]–[11] to derive

real and complex AMP. The method there, however, relied on a

key assumption of Gaussian likelihood, which prohibits a direct

extension to complex GAMP. To remove such a limitation, we

borrow an idea from [3], where a key step is to apply central

limit theorem in the Gaussian approximation of one part of the

message. We also note that there exists other ways to recover

the complex GAMP. For instance, GAMP can be treated as a

degenerate Gr-AMP [12], which admits also a concise deriva-

tion (but be aware their factor graph was different from ours as

they required additional nodes δ(·)). As another example, com-

plex GAMP could be approximated by expanding the complex

model into pairs of real variables, t hen applying the HyGAMP

[13] with (real, imaginary) pairs in the prior and likelihood,

leading to a GAMP-like algorithm that passes 2 × 2 covariance

matrices, and ﬁnally taking the SHyGAMP [14] approach to

approximate those covariance matrices by scaled identities.

Comparing with these methods, our derivation contributes

through the following aspects: ﬁrst, offering a new perspective

to view the GAMP message passing in an EP manner; second,

extending prior work to establish a more general link that con-

nects EP and GAMP.

II. S

YSTEM MODEL AND GAMP RECAP

System Model: Consider the generalized linear model [2]

x →

→ p(y|z ) → y (1)

where the input x ∈ C

, randomly drawn from the distribu-

tion p(x)=



p(x

),islinearly mixed by a deterministic mea-

surement matrix H ∈ C

M ×N

to obtain z = Hx. Then z is

passed through a componentwise mapping channel whose transi-

tion probability function is p(y|z)=



a=1

p(y

). The ﬁnal

See http://www.ieee.org/publications

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