The restricted isometry property (RIP) for design matrices gives guarantees for optimal recovery in sparse linear models. It is of high interest in compressed sensing and statistical learning. This property is particularly important for computationally efficient recovery methods. As a consequence, even though it is in general NP-hard to check that RIP holds, there have been substantial efforts to find tractable proxies for it. These would allow the construction of RIP matrices and the polynomial-time verification of RIP given an arbitrary matrix. We consider the framework of average-case certifiers, that never wrongly declare that a matrix is RIP, while being often correct for random instances. While there are such functions which are tractable in a suboptimal parameter regime, we show that this is a computationally hard task in any better regime. Our results are based on a new, weaker assumption on the problem of detecting dense subgraphs.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Estimating information from structured data is acentral theme in statistics that by now has found applications in a wide array of disciplines.

This increase in data sources has led to applications that are increasingly high-dimensional. To be both statistically and computationally efficient in this setting, it is then important to develop approaches that can exploit the structure within the data.