In sparse approximation theory, the fundamental problem is to reconstruct a signalAinR^{n}from linear measurements <A,ψ_{i}> with respect to a dictionary of ψ_{i}'s. Recently, there is focus on the novel direction ofCompressed Sensingwhere the reconstruction can be done with very few-O(klogn)-linear measurements over a modified dictionary if the signal iscompressible, that is, its information is concentrated inkcoefficients with the original dictionary. In particular, these results prove that there exists a singleO(klogn) ×nmeasurement matrix such that any such signal can be reconstructed from these measurements, with error at mostO(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying MathematicsIn this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in

k, logn, 1/ε, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee fromO(1) to 1 + ε and improves the reconstruction time frompoly(n) topoly(klogn).Our second result is a randomized construction of

O(kpolylog(n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory, Streaming algorithms and Complexity Theory for this case.Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement.

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