Probabilistic reconstruction in compressed sensing: algorithms, phase diagrams, and threshold achieving matrices

Abstract

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [1] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation maximization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically. arXiv:1206.3953v1 [cond-mat.stat-mech] 18 Jun 2012 36 A. Derivation of the replica analysis for block matrices 37 B. Phase diagram of the 1 reconstruction for seeding matrices 38 C. Equations for a mixture of Gaussians 40 References 41 : unit coupling : no coupling (null elements) : coupling J J J J J J J J : unit coupling : no coupling (null elements) : coupling J J J J J J J J : unit coupling : no coupling (null elements) : coupling J J J J J J J J

Compressive phase retrieval via generalized approximate message passing

“…To address this problem, we now describe how ν w can be learned using an expectation-maximization (EM) [39] procedure. The methodology is similar to that proposed in [40] for the case of a Gaussian p Y |Z (y m |·), but is more involved due to the fact that the p Y |Z (y m |·) used for phaseretrieval (recall (8)) is non-Gaussian.…”

Sparse Estimation Based on a New Random Regularized Matching Pursuit Generalized Approximate Message Passing Algorithm

“…The main advantage is that replica computations give a statistical physical meaning to the linear system. The replica method was used for compressed sensing in recent years [19][20][21][22]; we follow the research of [23].…”

Message Passing Algorithm for Distributed Downlink Regularized Zero-Forcing Beamforming with Cooperative Base Stations

“…The similar methodology was used in [29] in the case of compressed sensing although some modifications are required to reflect the concerned case.…”

“…The used AMP method has recently received considerable interest in the field of compressed sensing [27][28][29][30]. Our form of the message passing algorithm is closely related to the AMP methods in [29] which are a special case of the generalized AMP [30].…”