Average-case Hardness of RIP Certification
This addresses a fundamental computational bottleneck for constructing and verifying RIP matrices, which is important for researchers in compressed sensing and machine learning, but the results are incremental as they build on existing hardness frameworks.
The paper tackles the problem of efficiently certifying the restricted isometry property (RIP) for matrices, which is crucial for sparse recovery in compressed sensing and statistical learning, and shows that average-case certification is computationally hard in optimal parameter regimes, based on a new assumption about detecting dense subgraphs.
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.