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Unformatted text preview: Recovery of Clustered Sparse Signals from Compressive Measurements Volkan Cevher (1) , Piotr Indyk (1 , 2) , Chinmay Hegde (1) , and Richard G. Baraniuk (1) (1) Electrical and Computer Engineering, Rice University, Houston, TX (2) Computer Science and Artificial Intelligence Lab, MIT, Cambridge, MA Abstract: We introduce a new signal model, called ( K, C )-sparse, to capture K-sparse signals in N dimensions whose nonzero coefficients are contained within at most C clusters, with C < K ≪ N . In contrast to the existing work in the sparse approximation and compressive sensing liter- ature on block sparsity, no prior knowledge of the loca- tions and sizes of the clusters is assumed. We prove that O ( K + C log( N/C ))) random projections are sufficient for ( K, C )-model sparse signal recovery based on sub- space enumeration. We also provide a robust polynomial- time recovery algorithm for ( K, C )-model sparse signals with provable estimation guarantees. 1. Introduction Compressive sensing (CS) is an alternative to Shan- non/Nyquist sampling for the acquisition of sparse or compressible signals in an appropriate basis [1, 2]. By sparse, we mean that only K of the N basis coefficients are nonzero, where K ≪ N . By compressible, we mean the basis coefficients, when sorted, decay rapidly enough to zero so that they can be well-approximated as K-sparse. Instead of taking periodic samples of a signal, CS mea- sures inner products with random vectors and then recov- ers the signal via a sparsity-seeking convex optimization or greedy algorithm. The number of compressive mea- surements M necessary to recover a sparse signal under this framework grows as M = O ( K log( N/K )) In many applications, including imaging systems and high-speed analog-to-digital converters, such a saving can be dramatic; however, the dimensionality reduction from N to M is still not on par with state-of-the-art transform coding systems. While many natural and manmade signals can be described to a first-order as sparse or compress- ible, their sparse supports often have an underlying do- main specific structure [3–6]. Exploiting this structure in CS recovery has two immediate benefits. First, the number of compressive measurements required for stable recovery decreases due to the reduction in the degrees of freedom of a sparse or compressible signal. Second, true signal infor- mation can be better differentiated from recovery artifacts during signal recovery, which increases recovery robust- ness. Only by exploiting a priori information on coeffi- cient structure in addition to signal sparsity, can CS hope to be competitive with the state-of-the-art transform cod- ing algorithms for dimensionality reduction....
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