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/epsilon, 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 + \epsilon 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.
Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2005/2005-40.pdf