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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 Ksparse 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 wellapproximated as Ksparse. Instead of taking periodic samples of a signal, CS mea sures inner products with random vectors and then recov ers the signal via a sparsityseeking 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 highspeed analogtodigital converters, such a saving can be dramatic; however, the dimensionality reduction from N to M is still not on par with stateoftheart transform coding systems. While many natural and manmade signals can be described to a firstorder 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 stateoftheart transform cod ing algorithms for dimensionality reduction....
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 Spring '10
 RunyiYu

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