On the Power of Adaptivity in Sparse Recovery
Author(s)Indyk, Piotr; Price, Eric C.; Woodruff, David P.
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The goal of (stable) sparse recovery is to recover a k-sparse approximation x* of a vector x from linear measurements of x. Specifically, the goal is to recover x* such that ∥x-x*∥[subscript p] ≤ C min, k-sparse x, ∥x-x'∥[subscript q] for some constant C and norm parameters p and q. It is known that, for p = q=l or p = q = 2, this task can be accomplished using m = O(k log(n/k)) non-adaptive measurements  and that this bound is tight , , . In this paper we show that if one is allowed to perform measurements that are adaptive, then the number of measurements can be considerably reduced. Specifically, for C = 1+∈ and p = q = 2 we show · A scheme with m= O(1/∈ log log (n∈/k)) measurements that uses O(log* k · log log(n∈/k)) rounds. This is a significant improvement over the best possible non-adaptive bound. · A scheme with m = O(1/∈k log(k/∈) + k log(n/k)) measurements that uses two rounds. This improves over the best possible non-adaptive bound. To the best of our knowledge, these are the first results of this type.
Departmentmove to dc.description.sponsorship; Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011
Institute of Electrical and Electronics Engineers (IEEE)
Indyk, Piotr, Eric Price, and David P. Woodruff. “On the Power of Adaptivity in Sparse Recovery.” IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011. 285–294.
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