Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complex. We study the popular reconstruction method of @math -regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complex signals and measurements and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP.
Both complex signals and group-sparse signals are special cases of model-based CS @cite_21 . By introducing more structured models for the signal, @cite_21 proves that the number of measurements needed are proportional to the complexity'' of the model rather than the sparsity level @cite_23 . The results in model-based CS also suffer from loose constants in both the number of measurements and the mean square error bounds.
@cite_21 @cite_23
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In traditional framework of compressive sensing (CS), only sparse prior on the property of signals in time or frequency domain is adopted to guarantee the exact inverse recovery. Other than sparse prior, structures on the sparse pattern of the signal have also been used as an additional prior, called model-based compressive sensing, such as clustered structure and tree structure on wavelet coefficients. In this paper, the cluster structured sparse signals are investigated. Under the framework of Bayesian compressive sensing, a hierarchical Bayesian model is employed to model both the sparse prior and cluster prior, then Markov Chain Monte Carlo (MCMC) sampling is implemented for the inference. Unlike the state-of-the-art algorithms which are also taking into account the cluster prior, the proposed algorithm solves the inverse problem automatically-prior information on the number of clusters and the size of each cluster is unknown. The experimental results show that the proposed algorithm outperforms many state-of-the-art algorithms. In this work we derive fundamental limits for many linear and non-linear sparse signal processing models including group testing, quantized compressive sensing, multivariate regression and observations with missing features. In general, sparse signal processing problems can be characterized in terms of the following Markovian property. We are given a set of N variables X1, X2, ..., XN, and there is an unknown subset of variables S ⊂ 1, 2, ..., N that are relevant for predicting outcomes outputs Y. In other words, when Y is conditioned on Xn nϵS it is conditionally independent of the other variables, Xn n∉S. Our goal is to identify the set S from samples of the variables X and the associated outcomes Y. We characterize this problem as a version of the noisy channel coding problem. Using asymptotic information theoretic analyses, we establish mutual information formulas that provide sufficient and necessary conditions on the number of samples required to successfully recover the salient variables. These mutual information expressions unify conditions for both linear and non-linear observations. We then compute sample complexity bounds for the aforementioned models, based on the mutual information expressions. Model-based compressive sensing (CS) for signal-specific applications is of particular interest in the sparse signal approximation. In this paper, we deal with a special class of sparse signals with binary entries. Unlike conventional CS approaches based on l 1 minimization, we model the CS process with a bi-partite graph. We design a novel sampling matrix with unique sum property, which can be universally applied to any binary signal. Moreover, a novel binary CS decoding algorithm (BCS) based on graph and unique sum table, which does not need complex optimization process, is proposed. Proposed method is verified and compared with existing solutions through mathematical analysis and numerical simulations. A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.