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.
Similar as tree-sparse data, joint-sparse data has the property. It has to be clarified that joint sparsity does not rely on tree sparsity. The former utilizes the structure across different channels, while the later utilizes the structure within each channel. Previous works implies that the minimum measurement bound for such joint sparse data is @math @cite_63 @cite_21 @cite_51 .
@cite_21 @cite_51 @cite_63
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We address the problem of joint sparsity pattern recovery based on low dimensional multiple measurement vectors (MMVs) in resource constrained distributed networks. We assume that distributed nodes observe sparse signals which share the same sparsity pattern and each node obtains measurements via a low dimensional linear operator. When the measurements are collected at distributed nodes in a communication network, it is often required that joint sparse recovery be performed under inherent resource constraints such as communication bandwidth and transmit processing power. We present two approaches to take the communication constraints into account while performing common sparsity pattern recovery. First, we explore the use of a shared multiple access channel (MAC) in forwarding observations residing at each node to a fusion center. With MAC, while the bandwidth requirement does not depend on the number of nodes, the fusion center has access to only a linear combination of the observations. We discuss the conditions under which the common sparsity pattern can be estimated reliably. Second, we develop two collaborative algorithms based on Orthogonal Matching Pursuit (OMP), to jointly estimate the common sparsity pattern in a decentralized manner with a low communication overhead. In the proposed algorithms, each node exploits collaboration among neighboring nodes by sharing a small amount of information for fusion at different stages in estimating the indices of the true support in a greedy manner. Efficiency and effectiveness of the proposed algorithms are demonstrated via simulations along with a comparison with the most related existing algorithms considering the trade-off between the performance gain and the communication overhead. Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonly-used sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper, six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only two of these measures satisfy all six: the pq-mean with p les 1, q > 1 and the Gini index. Correlation-based techniques used for frame synchronization can suffer significant performance degradation over multi-path frequency-selective channels. In this paper, we propose a joint frame synchronization and channel estimation (JFSCE) framework as a remedy to this problem. This framework, however, increases the size of the resulting combined channel vector which should capture both the channel impulse response vector and the frame boundary offset and, therefore, its estimation becomes more challenging. On the other hand, because the combined channel vector is sparse, sparse channel estimation methods can be applied. We propose several JFSCE methods using popular sparse signal recovery algorithms which exploit the sparsity of the combined channel vector. Subsequently, the sparse channel vector estimate is used to design a sparse equalizer. Our simulation results and experimental measurements using software defined radios show that in some scenarios our proposed method improves the overall system performance significantly, in terms of the mean square error between the transmitted and the equalized symbols compared to the conventional method. We consider the problem of recovering clustered sparse signals with no prior knowledge of the sparsity pattern. Beyond simple sparsity, signals of interest often exhibits an underlying sparsity pattern which, if leveraged, can improve the reconstruction performance. However, the sparsity pattern is usually unknown a priori. Inspired by the idea of k-nearest neighbor (k-NN) algorithm, we propose an efficient algorithm termed approximate message passing with nearest neighbor sparsity pattern learning (AMP-NNSPL), which learns the sparsity pattern adaptively. AMP-NNSPL specifies a flexible spike and slab prior on the unknown signal and, after each AMP iteration, sets the sparse ratios as the average of the nearest neighbor estimates via expectation maximization (EM). Experimental results on both synthetic and real data demonstrate the superiority of our proposed algorithm both in terms of reconstruction performance and computational complexity.