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$$\norm{p}{x-x^*} \le C \min_{k\text{-sparse } x'} \norm{q}{x-x'}$$for some constant $C$ and norm parameters $p$ and $q$. It is known that, for $p=q=1$ or $p=q=2$, this task can be accomplished using $m=O(k \log (n/k))$ {\em non-adaptive}measurements~\cite{CRT06:Stable-Signal} and that this bound is tight~\cite{DIPW, FPRU, PW11}. In this paper we show that if one is allowed to perform measurements that are {\em adaptive}, then the number of measurements can be considerably reduced. Specifically, for $C=1+\epsilon$ and $p=q=2$ we show\begin{itemize}\item A scheme with $m=O(\frac{1}{\eps}k \log \log (n\eps/k))$ measurements that uses $O(\log^* k \cdot \log \log (n\eps/k))$ rounds. This is a significant improvement over the best possible non-adaptive bound. \item A scheme with $m=O(\frac{1}{\eps}k \log (k/\eps) + k \log (n/k))$ measurements that uses {\em two} rounds. This improves over the best possible non-adaptive bound. \end{itemize} To the best of our knowledge, these are the first results of this type.