We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math .
More broadly, message passing algorithms for compressed sensing where the object of a number of studies studies, starting with @cite_29 . As mentioned, we will focus on approximate message passing (AMP) as introduced in @cite_19 @cite_40 . As shown in @cite_11 these algorithms can be used in conjunction with a rich class of denoisers @math . A subset of these denoisers arise as posterior mean associated to a prior @math . Several interesting examples were studied by Schniter and collaborators @cite_25 @cite_49 @cite_36 , and by Rangan and collaborators @cite_26 @cite_4 .
@cite_26 @cite_4 @cite_36 @cite_29 @cite_19 @cite_40 @cite_49 @cite_25 @cite_11
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We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math . We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala [30], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Renyi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For “discrete” signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal pX. Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse objects. The formula applies to approximate message passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar soft thresholding denoiser. This paper gives several examples including scalar denoisers not derived from convex penalization-the firm shrinkage nonlinearity and the minimax nonlinearity-and also nonscalar denoisers-block thresholding, monotone regression, and total variation minimization. Let the variables e = k N and δ = n N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N-vector x0 according to y=Ax0. Here, A is an n×N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve δ = δ(e) separating successful from unsuccessful reconstruction of x0 by AMP is given by δ = M(e|Denoiser) where M(e|Denoiser) denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally tuned denoiser in the directly observed problem y = x + z. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem. We prove that this formula follows from state evolution and present numerical results validating it in a wide range of settings. The above formula generates numerous new insights, both in the scalar and in the nonscalar cases. The approximate message passing (AMP) algorithm originally proposed by Donoho, Maleki, and Montanari yields a computationally attractive solution to the usual l 1 -regularized least-squares problem faced in compressed sensing, whose solution is known to be robust to the signal distribution. When the signal is drawn i.i.d from a marginal distribution that is not least-favorable, better performance can be attained using a Bayesian variation of AMP. The latter, however, assumes that the distribution is perfectly known. In this paper, we navigate the space between these two extremes by modeling the signal as i.i.d Bernoulli-Gaussian (BG) with unknown prior sparsity, mean, and variance, and the noise as zero-mean Gaussian with unknown variance, and we simultaneously reconstruct the signal while learning the prior signal and noise parameters. To accomplish this task, we embed the BG-AMP algorithm within an expectation-maximization (EM) framework. Numerical experiments confirm the excellent performance of our proposed EM-BG-AMP on a range of signal types.12