Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make fewer measurements than were 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 Krzakala et?al (2012 Phys. Rev. X 2 021005) a strategy that allows compressed sensing to be performed at acquisition rates approaching 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 distributions 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.
“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. 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. Compressed sensing aims to undersample certain high-dimensional signals yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity–undersampling tradeoff is achieved when reconstructing by convex optimization, which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity–undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity–undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity–undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity–undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this apparently very different theoretical formalism. Compressed sensing deals with the reconstruction of a high-dimensional signal from far fewer linear measurements, where the signal is known to admit a sparse representation in a certain linear space. The asymptotic scaling of the number of measurements needed for reconstruction as the dimension of the signal increases has been studied extensively. This work takes a fundamental perspective on the problem of inferring about individual elements of the sparse signal given the measurements, where the dimensions of the system become increasingly large. Using the replica method, the outcome of inferring about any fixed collection of signal elements is shown to be asymptotically decoupled, i.e., those elements become independent conditioned on the measurements. Furthermore, the problem of inferring about each signal element admits a single-letter characterization in the sense that the posterior distribution of the element, which is a sufficient statistic, becomes asymptotically identical to the posterior of inferring about the same element in scalar Gaussian noise. The result leads to simple characterization of all other elemental metrics of the compressed sensing problem, such as the mean squared error and the error probability for reconstructing the support set of the sparse signal. Finally, the single-letter characterization is rigorously justified in the special case of sparse measurement matrices where belief propagation becomes asymptotically optimal.