Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only “approximately sparse”, i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the G-AMP algorithm for the homogeneous measurement matrices is not optimal, a special “seeding” design of a spatially-coupled measurement matrix allows to restore optimality.
For random matrices the evolution of iterations of G-AMP on large system sizes is described by state evolution @cite_1 . The exactness of this description was proven in large generality in @cite_6 . See also @cite_19 @cite_9 @cite_20 for discussions and results on the state evolution.
@cite_9 @cite_1 @cite_6 @cite_19 @cite_20
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We consider a class of approximated message passing (AMP) algorithms and characterize their highdimensional behavior in terms of a suitable state evolution recursion. Our proof applies to Gaussian matrices with independent but not necessarily identically distributed entries. It covers—in particular—the analysis of generalized AMP, introduced by Rangan, and of AMP reconstruction in compressed sensing with spatially coupled sensing matrices. The proof technique builds on that of Bayati & Montanari [2], while simplifying and generalizing several steps. We consider the estimation of a random vector observed through a linear transform followed by a componentwise probabilistic measurement channel. Although such linear mixing estimation problems are generally highly non-convex, Gaussian approximations of belief propagation (BP) have proven to be computationally attractive and highly effective in a range of applications. Recently, Bayati and Montanari have provided a rigorous and extremely general analysis of a large class of approximate message passing (AMP) algorithms that includes many Gaussian approximate BP methods. This paper extends their analysis to a larger class of algorithms to include what we call generalized AMP (G-AMP). G-AMP incorporates general (possibly non-AWGN) measurement channels. Similar to the AWGN output channel case, we show that the asymptotic behavior of the G-AMP algorithm under large i.i.d. Gaussian transform matrices is described by a simple set of state evolution (SE) equations. The general SE equations recover and extend several earlier results, including SE equations for approximate BP on general output channels by Guo and Wang. In this paper, we extend the generalized approximate message passing (G-AMP) approach, originally proposed for high-dimensional generalized-linear regression in the context of compressive sensing, to the generalized-bilinear case, which enables its application to matrix completion, robust PCA, dictionary learning, and related matrix-factorization problems. Here, in Part I of a two-part paper, we derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the sum-product belief propagation algorithm in the high-dimensional limit, where central-limit theorem arguments and Taylor-series approximations apply, and under the assumption of statistically independent matrix entries with known priors. In addition, we propose an adaptive damping mechanism that aids convergence under finite problem sizes, an expectation-maximization (EM)-based method to automatically tune the parameters of the assumed priors, and two rank-selection strategies. In Part II of the paper, we will discuss the specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA, and dictionary learning, and we will present the results of an extensive empirical study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem. “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.