A UNITARY TRANSFORM BASED GENERALIZED APPROXIMATE MESSAGE PASSING Jiang ZhuzXiangming MengXupeng LeizQinghua Guoy zOcean College Zhejiang University Zhoushan 316021 China

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A UNITARY TRANSFORM BASED GENERALIZED APPROXIMATE MESSAGE PASSING

Jiang Zhu‡Xiangming Meng?Xupeng Lei‡Qinghua Guo†

‡Ocean College, Zhejiang University, Zhoushan, 316021, China

?Institute for Physics of Intelligence, The University of Tokyo, 7-3-1, Hongo, Tokyo 113-0033, Japan

†SECTE, University of Wollongong, NSW, 2522, Australia

ABSTRACT

We consider the problem of recovering an unknown sig-

nal x∈Rnfrom general nonlinear measurements ob-

tained through a generalized linear model (GLM), i.e., y=

f(Ax +w), where f(·)is a componentwise nonlinear func-

tion. Based on the unitary transform approximate message

passing (UAMP) and expectation propagation, a unitary

transform based generalized approximate message passing

(GUAMP) algorithm is proposed for general measurement

matrices A, in particular highly correlated matrices. Experi-

mental results on quantized compressed sensing demonstrate

that the proposed GUAMP signiﬁcantly outperforms state-of-

the-art GAMP and GVAMP under correlated matrices A.

Index Terms—GLM, AMP, GAMP, message passing,

quantized compressed sensing

1. INTRODUCTION

We consider the general problem of inference on generalized

linear models (GLM) [1], i.e., recovering an unknown signal

x∈Rnfrom a noisy linear transform followed by compo-

nentwise nonlinear measurements

y=f(Ax +w),(1)

where A∈Rm×nis a known linear mixing matrix, w∼

N(w; 0, σ2Im)is an i.i.d. Gaussian noise with known vari-

ance, and f(·) : Rm×1→ Qm×1is an componentwise non-

linear link function. Denote z,Ax ∈Rmas the hidden

linear transform outputs, the componentwise nonlinear func-

tion f(·)can be equivalently described in a probabilistic way

using a fully factorized likelihood function as follows

p(y|z) =

i=1

p(yi|zi),(2)

where p(yi|zi)is determined by the speciﬁc nonlinear func-

tion f(·). The prior distribution p(x)of xis also assumed to

be fully factorized p(x) =

j=1

p(xj)for simplicity. GLM in-

ference has wide applications in science and engineering such

as wireless communications, signal processing, and machine

Corresponding author. E-mail: meng@g.ecc.u-tokyo.ac.jp. This work is

supported by NSFC under grant 61901415.

learning. In the special case of identity function f(·), the non-

linear GLM in (1) will reduce to the popular standard linear

model (SLM) as follows

y=Ax +w.(3)

Thus, GLM is actually an extention of SLM from linear mea-

surements to nonlinear measurements, which are prevalent in

some real-world applications such as quantized compressed

sensing, pattern classiﬁcation, phase retrieval, etc.

A variety of algorithms have been proposed for inference

over GLMs and SLMs. Among them, the past decade has wit-

nessed an advent of one distinguished family of probabilistic

algorithms called message passing algorithm. Among them,

the most famous ones are the approximate message passing

(AMP) algorithm [2, 3] for SLM and generalized approxi-

mate message passing (GAMP) [4] for GLM, which have

been proved to be optimal under i.i.d. Gaussian matrices A.

However, both AMP and GAMP diverge for general A. In [5–

7], the GAMP is incorporated into the sparse Bayesian learn-

ing (SBL) to improve the robustness of GAMP and reduce the

computation complexity of SBL. Vector AMP (VAMP) [8]

(or OAMP [9], one similar algorithm to VAMP) and gener-

alized VAMP (GVAMP) [10] have been proposed to improve

the performance with general Afor SLM and GLM, respec-

tively. The VAMP is derived from expectation propagation

(EP) [11], another powerful message passing algorithm. Re-

markably, it has been demonstrated in [12, 13] that all the

AMP, GAMP, and GVAMP, can be derived concisely as spe-

cial instances of EP under different assumptions and thus be

uniﬁed within a single EP framework. Despite signiﬁcant im-

provement in robustness, in particular for right-orthogonally

invariant matrices, VAMP and GVAMP still suffer poor con-

vergence in some more challenging scenarios, e.g., the mea-

surement matrix is highly correlated [14, 15].

In this work, we focus on the AMP variant: unitary

approximate message passing (UAMP) [16], which was for-

merly called UTAMP [16–18]. The motivation behind UAMP

is this: since AMP has proven to work well when the elements

of Ain (3) are uncorrelated, one can artiﬁcially constructs

such a “good” and equivalent measurement model simply

by performing a singular value decomposition (SVD) on

A=UΣVT, where U∈Rm×r,Σ∈Rr×r,V∈Rn×r

arXiv:2210.08861v1 [cs.IT] 17 Oct 2022

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摘要：

AUNITARYTRANSFORMBASEDGENERALIZEDAPPROXIMATEMESSAGEPASSINGJiangZhuzXiangmingMeng?XupengLeizQinghuaGuoyzOceanCollege,ZhejiangUniversity,Zhoushan,316021,China?InstituteforPhysicsofIntelligence,TheUniversityofTokyo,7-3-1,Hongo,Tokyo113-0033,JapanySECTE,UniversityofWollongong,NSW,2522,AustraliaABSTRACTW...

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A UNITARY TRANSFORM BASED GENERALIZED APPROXIMATE MESSAGE PASSING Jiang ZhuzXiangming MengXupeng LeizQinghua Guoy zOcean College Zhejiang University Zhoushan 316021 China

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