Averagecase Hardness of RIP Certification
Abstract
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 NPhard 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 polynomialtime verification of RIP given an arbitrary matrix. We consider the framework of averagecase 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.09646
 Bibcode:
 2016arXiv160509646W
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Computational Complexity;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning