Improved bounds for sparse recovery from adaptive measurements

It is shown here that adaptivity in sampling results in dramatic improvements in the recovery of sparse signals in white Gaussian noise. An adaptive sampling-and-refinement procedure called distilled sensing is discussed and analyzed, resulting in fundamental new asymptotic scaling relationships in terms of the minimum feature strength required for reliable signal detection or localization (support recovery). In particular, reliable detection and localization using non-adaptive samples is possible only if the feature strength grows logarithmically in the problem dimension. Here it is shown that using adaptive sampling, reliable detection is possible provided the feature strength exceeds a constant, and localization is possible when the feature strength exceeds any (arbitrarily slowly) growing function of the problem dimension

Improved bounds for sparse recovery from adaptive measurements

It is shown here that adaptivity in sampling results in dramatic improvements in the recovery of sparse signals in white Gaussian noise. An adaptive sampling-and-refinement procedure called distilled sensing is discussed and analyzed, resulting in fundamental new asymptotic scaling relationships in terms of the minimum feature strength required for reliable signal detection or localization (support recovery). In particular, reliable detection and localization using non-adaptive samples is possible only if the feature strength grows logarithmically in the problem dimension. Here it is shown that using adaptive sampling, reliable detection is possible provided the feature strength exceeds a constant, and localization is possible when the feature strength exceeds any (arbitrarily slowly) growing function of the problem dimension

Improved Bounds for Universal One-Bit Compressive Sensing

Unlike compressive sensing where the measurement outputs are assumed to be real-valued and have infinite precision, in "one-bit compressive sensing", measurements are quantized to one bit, their signs. In this work, we show how to recover the support of sparse high-dimensional vectors in the one-bit compressive sensing framework with an asymptotically near-optimal number of measurements. We also improve the bounds on the number of measurements for approximately recovering vectors from one-bit compressive sensing measurements. Our results are universal, namely the same measurement scheme works simultaneously for all sparse vectors. Our proof of optimality for support recovery is obtained by showing an equivalence between the task of support recovery using 1-bit compressive sensing and a well-studied combinatorial object known as Union Free Families.Comment: 14 page

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 $p=q=1$ or $p=q=2$, this task can be accomplished using $m=O(k \log (n/k))$ non-adaptive measurements [CRT06] and that this bound is tight [DIPW10,FPRU10,PW11]. 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 $C=1+eps$ and $p=q=2$ we show - A scheme with $m=O((1/eps)k log log (n eps/k))$ measurements that uses $O(log* k \log \log (n eps/k))$ rounds. This is a significant improvement over the best possible non-adaptive bound. - A scheme with $m=O((1/eps) k log (k/eps) + k \log (n/k))$ 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. As an independent application, we show how to solve the problem of finding a duplicate in a data stream of $n$ items drawn from ${1, 2, ..., n-1}$ using $O(log n)$ bits of space and $O(log log n)$ passes, improving over the best possible space complexity achievable using a single pass.Comment: 18 pages; appearing at FOCS 201

Adaptive Compressed Sensing for Support Recovery of Structured Sparse Sets

This paper investigates the problem of recovering the support of structured signals via adaptive compressive sensing. We examine several classes of structured support sets, and characterize the fundamental limits of accurately recovering such sets through compressive measurements, while simultaneously providing adaptive support recovery protocols that perform near optimally for these classes. We show that by adaptively designing the sensing matrix we can attain significant performance gains over non-adaptive protocols. These gains arise from the fact that adaptive sensing can: (i) better mitigate the effects of noise, and (ii) better capitalize on the structure of the support sets.Comment: to appear in IEEE Transactions on Information Theor