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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 recentlyproposed 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 nonadaptive point sampling. In this paper the theory of dis tilled sensing is extended to highlyundersampled regimes, as in compressive sensing. A simple adaptive samplingandrefinement 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 realvalued 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 highlyunderdetermined 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 zeromean 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 noisefree. In these settings, rather than attempt This work was partially supported by the ARO, grant no. W911NF091 0383, the NSF, grant no. CCF0353079, and the AFOSR, grant no. FA9550 0910140....
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 Spring '10
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