In sparse approximation theory, the fundamental problem is to reconstruct a signal A inR^n from linear measurements <A,psi_i> with respect to a dictionary of psi_i's. Recently, there is focus on the novel direction of Compressed Sensing where the reconstruction can be done with very few-O(k log n)-linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients. In particular, these results prove that there exists a single O(k log n) x n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications.In 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 number of measurements polynomial in k, log n, 1/ε, we can reconstruct any compressible signal upto 1+epsilon error. This is the first known polynomial time explicit construction of any such measurement matrix. Our result not only improves the error guarantee from O(1) to 1 + ε but also improve the reconstruction time from poly(n) to poly(k log n).

Our second result is a randomized construction of a measurement matrix of size O(k polylog n) that works for each signal with high probability and gives per-instance approximation guarantee rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantee; 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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