On the Power of Adaptivity in Sparse Recovery

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*∥_p ≤ C min, k-sparse x, ∥x-x´∥_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 [3] and that this bound is tight [9], [12], [28]. 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.