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.
@cite_21: This paper investigates a learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature set, this concept generalizes the group sparsity idea that has become popular in recent years. A general theory is developed for learning with structured sparsity, based on the notion of coding complexity associated with the structure. It is shown that if the coding complexity of the target signal is small, then one can achieve improved performance by using coding complexity regularization methods, which generalize the standard sparse regularization. Moreover, a structured greedy algorithm is proposed to efficiently solve the structured sparsity problem. It is shown that the greedy algorithm approximately solves the coding complexity optimization problem under appropriate conditions. Experiments are included to demonstrate the advantage of structured sparsity over standard sparsity on some real applications.
@cite_51: Compressive sensing (CS) is an alternative to Shannon Nyquist sampling for the acquisition of sparse or compressible signals that can be well approximated by just K ? N elements from an N -dimensional basis. Instead of taking periodic samples, CS measures inner products with M < N random vectors and then recovers the signal via a sparsity-seeking optimization or greedy algorithm. Standard CS dictates that robust signal recovery is possible from M = O(K log(N K)) measurements. It is possible to substantially decrease M without sacrificing robustness by leveraging more realistic signal models that go beyond simple sparsity and compressibility by including structural dependencies between the values and locations of the signal coefficients. This paper introduces a model-based CS theory that parallels the conventional theory and provides concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees. A highlight is the introduction of a new class of structured compressible signals along with a new sufficient condition for robust structured compressible signal recovery that we dub the restricted amplification property, which is the natural counterpart to the restricted isometry property of conventional CS. Two examples integrate two relevant signal models-wavelet trees and block sparsity-into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just M = O(K) measurements. Extensive numerical simulations demonstrate the validity and applicability of our new theory and algorithms.
For both case, we have @math . Similar conclusion has been drawn in previous articles @cite_21 @cite_51 . So far, we have reviewed standard sparsity and tree sparsity on single channel data. For multi-channel data that contains @math channels or vectors (i.e. @math ), each of which is standardly @math -sparse, the bound for the number of measurement should be @math . If each channel is tree-sparse and independently, the measurement bound for a sub-Gaussian random matrix @math is @math .