compress-acssc-2009 - 1 Compressive Distilled Sensing:...

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Unformatted text preview: 1 Compressive Distilled Sensing: Sparse Recovery Using Adaptivity in Compressive Measurements Jarvis D. Haupt 1 , Richard G. Baraniuk 1 , Rui M. Castro 2 , and Robert D. Nowak 3 1 Dept. of Electrical and Computer Engineering, Rice University, Houston TX 77005 2 Dept. of Electrical Engineering, Columbia University, New York NY 10027 3 Dept. of Electrical and Computer Engineering, University of Wisconsin, Madison WI 53706 Abstract —The recently-proposed theory of distilled sensing establishes that adaptivity in sampling can dramatically improve the performance of sparse recovery in noisy settings. In par- ticular, it is now known that adaptive point sampling enables the detection and/or support recovery of sparse signals that are otherwise too weak to be recovered using any method based on non-adaptive point sampling. In this paper the theory of dis- tilled sensing is extended to highly-undersampled regimes, as in compressive sensing. A simple adaptive sampling-and-refinement procedure called compressive distilled sensing is proposed, where each step of the procedure utilizes information from previous observations to focus subsequent measurements into the proper signal subspace, resulting in a significant improvement in effective measurement SNR on the signal subspace. As a result, for the same budget of sensing resources, compressive distilled sensing can result in significantly improved error bounds compared to those for traditional compressive sensing. I. INTRODUCTION Let x ∈ R n be a sparse vector supported on the set S = { i : x i 6 = 0 } , where |S| = s ¿ n , and consider observing x according to the linear observation model y = Ax + w, (1) where A is an m × n real-valued matrix (possibly random) that satisfies E £ k A k 2 F / ≤ n , and where w i iid ∼ N (0 ,σ 2 ) for some σ ≥ . This model is central to the emerging field of compressive sensing (CS), which deals primarily with recovery of x in highly-underdetermined settings (that is, where the number of measurements m ¿ n ). Initial results in CS establish a rather surprising result— using certain observation matrices A for which the number of rows is a constant multiple of s log n , it is possible to recover x exactly from { y,A } , and in addition, the recovery can be accomplished by solving a tractable convex optimization [1]– [3]. Matrices A for which this exact recovery is possible are easy to construct in practice. For example, matrices whose entries are i.i.d. realizations of certain zero-mean distributions (Gaussian, symmetric Bernoulli, etc.) are sufficient to allow this recovery with high probability [2]–[4]. In practice, however, it is rarely the case that observations are perfectly noise-free. In these settings, rather than attempt This work was partially supported by the ARO, grant no. W911NF-09-1- 0383, the NSF, grant no. CCF-0353079, and the AFOSR, grant no. FA9550- 09-1-0140....
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compress-acssc-2009 - 1 Compressive Distilled Sensing:...

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