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A Euclidean approximate sparse recovery system consists of parameters k,N, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, the system approximates x by ^x=D(Φ x), which must satisfy ||^x - x||₂≤ C ||x - x_k||₂, where x_k denotes the optimal k-term approximation to x. (The output ^x may have more than k terms). For each vector x, the system must succeed with probability at least 3/4. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, D.

In this paper, we give a system with m=O(k log(N/k)) measurements--matching a lower bound, up to a constant factor--and decoding time k log^{O(1) N, matching a lower bound up to log(N) factors. We also consider the encode time (i.e., the time to multiply Φ by x), the time to update measurements (i.e., the time to multiply Φ by a 1-sparse x), and the robustness and …