In this paper, we investigate a new compressive sensing model for multi-channel sparse data where each channel can be represented as a hierarchical tree and different channels are highly correlated. Therefore, the full data could follow the forest structure and we call this property forest sparsity. It exploits both intra- and inter- channel correlations and enriches the family of existing model-based compressive sensing theories. The proposed theory indicates that only @math measurements are required for multi-channel data with forest sparsity, where @math is the number of channels, @math and @math are the length and sparsity number of each channel, respectively. This result is much better than @math of tree sparsity, @math of joint sparsity, and far better than @math of standard sparsity. In addition, we extend the forest sparsity theory to the multiple measurement vectors problem, where the measurement matrix is a block-diagonal matrix. The result shows that the required measurement bound can be the same as that for dense random measurement matrix, when the data shares equal energy in each channel. A new algorithm is developed and applied on four example applications to validate the benefit of the proposed model. Extensive experiments demonstrate the effectiveness and efficiency of the proposed theory and algorithm.
CS result shows that, if @math , a sub-Gaussian random matrix It includes Gaussian and Bernoulli random matrices etc. @cite_22 . @math can satisfy the RIP with high probability @cite_19 @cite_2 .
@cite_19 @cite_22 @cite_2
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Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary @math . This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the @math -restricted isometry property ( @math -RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in @math . While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the @math -RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary $ ... Hoeffding's U-statistics model combinatorial-type matrix parameters (appearing in CS theory) in a natural way. This paper proposes using these statistics for analyzing random compressed sensing matrices, in the non-asymptotic regime (relevant to practice). The aim is to address certain pessimisms of worst-case restricted isometry analyses, as observed by both Blanchard and Dossal, We show how U-statistics can obtain average-case analyses, by relating to statistical restricted isometry property (StRIP) type recovery guarantees. However unlike standard StRIP, random signal models are not required; the analysis used here holds in the almost sure (probabilistic) sense. For Gaussian bounded entry matrices, we show that both @math -minimization and LASSO essentially require on the order of @math measurements to respectively recover at least @math fraction, and @math fraction, of the signals. Noisy conditions are considered. Empirical evidence suggests our analysis to compare well to Donoho and Tanner's recent large deviation bounds for @math -equivalence, in the regime of block lengths @math with high undersampling ( @math measurements); similar system sizes are found in recent CS implementation. In this work, it is assumed throughout that matrix columns are independently sampled. The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by Candes, Romberg and Tao in 2004; for subgaussian random matrices it was carlier proved by the present authors, as a consequence of a general result based on a generic chaining method of Talagrand. The present proof combines a simple measure concentration and a covering argument, which are standard tools of high-dimensional convexity. Consider a Bernoulli-Gaussian complex n-vector whose components are Vi = XiBi, with Xi C N(0, Px) and binary Bi mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix U, and a randomly chosen subset, of average size n p, p ∈ [0,1], of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically a matrix with iid components, to allow U satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdu, as well as a number of information-theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of n → ∞. We also extend the scope of the large deviation approach of Rangan and characterize the performance of a class of estimators encompassing thresholded linear MMSE and l1 relaxation.