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Macroscopic Analysis of Vector Approximate message passing in a Model-Mismatched Setting

IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第8期68卷 5579-5600页

作者： Takahashi, Takashi Kabashima, Yoshiyuki Univ Tokyo Grad Sch Sci Inst Phys Intelligence Dept Phys Bunkyo City Tokyo 1130033 Japan

In this study, macroscopic properties of the vector approximate message passing (VAMP) algorithm for inference of generalized linear models are investigated using a non-rigorous heuristic method of statistical mechanics when the true posterior cannot be used and the measurement matrix is a sample from rotation-invariant random matrix ensembles. The focus is on the correspondence between the non-rigorous replica analysis of statistical mechanics and the performance assessment of VAMP in the model-mismatched setting. The correspondence of this kind is well-known when the measurement matrix has independent and identically distributed entries. However, when the measurement matrix follows a general rotation-invariant matrix ensemble, the correspondence has been validated only under limited cases, such as the Bayes optimal inference or the convex empirical risk minimization. The result presented in this paper is to extend the scope of such correspondence. Herein, we heuristically derive the explicit formula of state-evolution equations, which macroscopically describe VAMP dynamics for the current model-mismatched case, and show that their fixed point is generally consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, which indicates that message variables continue to move by VAMP while their macroscopically summarized quantities converge to fixed values. The critical condition the for microscopic instability agrees with that for breaking the replica symmetry that is derived within the non-rigorous replica analysis. The results of the numerical experiments cross-check our findings.

关键词： Mathematical models Analytical models Approximation algorithms message passing Heuristic algorithms Inference algorithms Rotation measurement message passing algorithms inference algorithms random matrices statistical mechanics phase transitions compressive sensing

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The Dynamics of message passing on Dense Graphs, with Applications to Compressed Sensing

The Dynamics of Message Passing on Dense Graphs, with Applic...

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IEEE International Symposium on Information Theory

作者： Bayati, Mohsen Montanari, Andrea Stanford Univ Dept Elect Engn Stanford CA 94305 USA Stanford Univ Dept Stat Stanford CA 94305 USA

ISBN: (纸本)9781424469604

"Approximate message passing" (AMP) algorithms have proved to be effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration termed state evolution. In this paper, we provide rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed Gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with a large number of short cycles in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory.

关键词： Compressed sensing density evolution message passing algorithms random matrix theory state evolution

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Bidirectional Approximate message passing for RIS-Assisted Multi-User MISO Communications 94

Bidirectional Approximate Message Passing for RIS-Assisted M...

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94th IEEE Vehicular Technology Conference (VTC-Fall)

作者： Wei, Li Huang, Chongwen Guo, Qinghua Zhang, Zhaoyang Debbah, Merouane Yuen, Chau Singapore Univ Technol & Design Singapore 487372 Singapore Zhejiang Univ Dept Informat & Elect Engn Hangzhou Peoples R China Univ Wollongong Sch Elect Comp & Telecommun Engn Wollongong NSW Australia Lagrange Math & Comp Res Ctr F-75007 Paris France

ISBN: (纸本)9781665413688

Reconfigurable intelligent surfaces (RISs) have been recently considered as a promising candidate for energy-efficient solutions in future wireless networks. Their dynamic and low-power configuration enables coverage extension, massive connectivity, and low-latency communications. Due to a large number of unknown variables referring to the RIS unit elements and the transmitted signals, channel estimation and signal recovery in RIS-based systems are the ones of the most critical technical challenges. To address this problem, we focus on the RIS-assisted multi-user wireless communication system and present a joint channel estimation and signal recovery algorithm in this paper. Specifically, we propose a bidirectional approximate message passing algorithm that applies the Taylor series expansion and Gaussian approximation to simplify the sum-product algorithm in the formulated problem. Our simulation results show that the proposed algorithm shows the superiority over a state-of-art benchmark method. We also provide insights on the impact of different RIS parameter settings on the proposed algorithms.

关键词： Reconfigurable intelligent surfaces message passing algorithms channel estimation signal recovery Gaussian approximation

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An Approximate message passing Approach for DOA Estimation in Phase Noisy Environments 14th

An Approximate Message Passing Approach for DOA Estimation i...

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14th International Conference on Latent Variable Analysis and Signal Separation (LVA/ICA)

作者： Beaumont, Guillaume Fablet, Ronan Dremeau, Angelique ENSTA Bretagne CNRS UMR 6285 Lab STICC F-29200 Brest France UBL UMR 6285 Lab STICC IMT Atlantique F-29200 Brest France

ISBN: (纸本)9783319937649;9783319937632

In underwater acoustics, wave propagation can be greatly disrupted by random fluctuations in the ocean environment. In particular, phase measurements of the complex pressure field can be heavily noisy and can defeat conventional direction-of-arrival (DOA) estimation algorithms. In this paper, we propose a new Bayesian approach to address such phase noise through an informative prior on the measurements. This is combined to a sparse assumption on the directions of arrival to achieve a highly-resolved estimation and integrated into a message-propagation algorithm referred to as the "paSAMP" algorithm (for Phase-Aware Swept Approximate message passing). Our algorithm can be seen as an extension of the recent phase-retrieval algorithm "prSAMP" to phase-aware priors. Experiments on simulated data mimicking real environments demonstrate that paSAMP outperform conventional approaches (e.g. classic beamforming) in terms of DOA estimation. paSAMP also proves to be more robust to additive noise than other variational methods (e.g. based on mean-field approximation).

关键词： DOA estimation Sparse representation Bayesian estimation Variational Bayesian approximations message passing algorithms

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RECONSTRUCTION AND CLUSTERING IN RANDOM CONSTRAINT SATISFACTION PROBLEMS

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2011年第2期25卷 771-808页

作者： Montanari, Andrea Restrepo, Ricardo Tetali, Prasad Stanford Univ Dept Elect Engn Stanford CA 94305 USA Stanford Univ Dept Stat Stanford CA 94305 USA Georgia Inst Technol Sch Math Atlanta GA 30332 USA Univ Antioquia Dept Matemat Medellin Colombia Georgia Inst Technol Sch Comp Sci Atlanta GA 30332 USA

Random instances of constraint satisfaction problems (CSPs) appear to be hard for all known algorithms when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSPs and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSPs. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as random instances of Not-All-Equal SAT, k-XOR formulae, hypergraph 2-coloring, and graph k-coloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of problems with a similar threshold structure.

关键词： random SAT sharp threshold message passing algorithms

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Monte Carlo Optimization of Decentralized Estimation Networks Over Directed Acyclic Graphs Under Communication Constraints

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2011年第11期59卷 5558-5576页

作者： Uney, Murat Cetin, Mujdat Sabanci Univ Fac Engn & Nat Sci TR-34956 Istanbul Turkey

Motivated by the vision of sensor networks, we consider decentralized estimation networks over bandwidth-limited communication links, and are particularly interested in the tradeoff between the estimation accuracy and the cost of communications due to, e. g., energy consumption. We employ a class of in-network processing strategies that admits directed acyclic graph representations and yields a tractable Bayesian risk that comprises the cost of communications and estimation error penalty. This perspective captures a broad range of possibilities for processing under network constraints and enables a rigorous design problem in the form of constrained optimization. A similar scheme and the structures exhibited by the solutions have been previously studied in the context of decentralized detection. Under reasonable assumptions, the optimization can be carried out in a message passing fashion. We adopt this framework for estimation, however, the corresponding optimization scheme involves integral operators that cannot be evaluated exactly in general. We develop an approximation framework using Monte Carlo methods and obtain particle representations and approximate computational schemes for both the in-network processing strategies and their optimization. The proposed Monte Carlo optimization procedure operates in a scalable and efficient fashion and, owing to the nonparametric nature, can produce results for any distributions provided that samples can be produced from the marginals. In addition, this approach exhibits graceful degradation of the estimation accuracy asymptotically as the communication becomes more costly, through a parameterized Bayesian risk.

关键词： Communication constrained inference decentralized estimation graphical models in-network processing message passing algorithms Monte Carlo methods random fields wireless sensor networks

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The LASSO Risk for Gaussian Matrices

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IEEE TRANSACTIONS ON INFORMATION THEORY 2012年第4期58卷 1997-2017页

作者： Bayati, Mohsen Montanari, Andrea Stanford Univ Grad Sch Business Stanford CA 94305 USA Stanford Univ Dept Elect Engn Stanford CA 94305 USA Stanford Univ Dept Stat Stanford CA 94305 USA

We consider the problem of learning a coefficient vector x(o) is an element of R-N from noisy linear observation y = Ax(o) + w is an element of R-n. In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator (x) over cap. In this case, a popular approach consists in solving an l(1)-penalized least-squares problem known as the LASSO or basis pursuit denoising. For sequences of matrices A of increasing dimensions, with independent Gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications.

关键词： Compressed sensing graphical models message passing algorithms random matrix theory state evolution statistical learning

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Joint Channel Estimation and Signal Recovery for RIS-Empowered Multiuser Communications

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IEEE TRANSACTIONS ON COMMUNICATIONS 2022年第7期70卷 4640-4655页

作者： Wei, Li Huang, Chongwen Guo, Qinghua Yang, Zhaohui Zhang, Zhaoyang Alexandropoulos, George C. Debbah, Merouane Yuen, Chau Singapore Univ Technol & Design Engn Prod Dev EPD Pillar Singapore 487372 Singapore Zhejiang Univ Zhejiang Prov Key Lab Informat Proc Commun & Netw Hangzhou 310007 Peoples R China Zhejiang Univ Coll Informat Sci & Elect Engn Hangzhou 310007 Peoples R China Zhejiang Univ Int Joint Innovat Ctr Haining 314400 Peoples R China Zhejiang Singapore Innovat & AI Joint Res Lab Hangzhou 310027 Peoples R China Univ Wollongong Sch Elect Comp & Telecommun Engn Wollongong NSW 2522 Australia Zhejiang Lab Hangzhou 311121 Peoples R China Natl & Kapodistrian Univ Athens Dept Informat & Telecommun Athens 15784 Greece Technol Innovat Inst Abu Dhabi U Arab Emirates Univ Paris Saclay Cent Supelec F-91192 Gif Sur Yvette France

Reconfigurable intelligent surfaces (RISs) have been recently considered as a promising candidate for energy-efficient solutions in future wireless networks. Their dynamic and low-power configuration enables coverage extension, massive connectivity, and low-latency communications. Due to a large number of unknown variables referring to the RIS unit elements and the transmitted signals, channel estimation and signal recovery in RIS-based systems are the ones of the most critical technical challenges. To address this problem, we focus on the RIS-assisted wireless communication system and present two joint channel estimation and signal recovery schemes based on message passing algorithms in this paper. Specifically, the proposed bidirectional scheme applies the Taylor series expansion and Gaussian approximation to simplify the sum-product procedure in the formulated problem. In addition, the inner iteration that adopts two variants of approximate message passing algorithms is incorporated to ensure robustness and convergence. Two ambiguities removal methods are also discussed in this paper. Our simulation results show that the proposed schemes show the superiority over the state-of-art benchmark method. We also provide insights on the impact of different RIS parameter settings on the proposed schemes.

关键词： Channel estimation message passing Approximation algorithms Wireless communication Training Inference algorithms Technological innovation Reconfigurable intelligent surfaces message passing algorithms channel estimation signal recovery Gaussian approximation

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Improving Variational Methods via Pairwise Linear Response Identities

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JOURNAL OF MACHINE LEARNING RESEARCH 2017年第1期18卷 145-180页

作者： Raymond, Jack Ricci-Tersenghi, Federico Univ Roma La Sapienza Dipartimento Fis Piazzale Aldo Moro 5 Rome Italy Univ Roma La Sapienza Dipartimento Fis INFN Sez Romal 1 Piazzale Aldo Moro 5 Rome Italy Univ Roma La Sapienza CNR Nanotec Piazzale Aldo Moro 5 Rome Italy

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks;iterative procedures related to message passing are provided for finding the minima.

关键词： variational inference graphical models message passing algorithms statistical physics linear response

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The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing

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IEEE TRANSACTIONS ON INFORMATION THEORY 2012年第5期58卷 3065-3092页

作者： Reeves, Galen Gastpar, Michael Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA Ecole Polytech Fed Lausanne Sch Comp & Commun Sci CH-1015 Lausanne Switzerland

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models.

关键词： Compressed sensing message passing algorithms model selection random matrix theory sparsity support recovery

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