Computational Complexity of Certifying Restricted Isometry Property
Abstract
Given a matrix $A$ with $n$ rows, a number $k<n$, and $0<\delta < 1$, $A$ is $(k,\delta)$RIP (Restricted Isometry Property) if, for any vector $x \in \mathbb{R}^n$, with at most $k$ nonzero coordinates, $$(1\delta) \x\_2 \leq \A x\_2 \leq (1+\delta)\x\_2$$ In many applications, such as compressed sensing and sparse recovery, it is desirable to construct RIP matrices with a large $k$ and a small $\delta$. Given the efficacy of random constructions in generating useful RIP matrices, the problem of certifying the RIP parameters of a matrix has become important. In this paper, we prove that it is hard to approximate the RIP parameters of a matrix assuming the SmallSetExpansionHypothesis. Specifically, we prove that for any arbitrarily large constant $C>0$ and any arbitrarily small constant $0<\delta<1$, there exists some $k$ such that given a matrix $M$, it is SSEHard to distinguish the following two cases:  (Highly RIP) $M$ is $(k,\delta)$RIP.  (Far away from RIP) $M$ is not $(k/C, 1\delta)$RIP. Most of the previous results on the topic of hardness of RIP certification only hold for certification when $\delta=o(1)$. In practice, it is of interest to understand the complexity of certifying a matrix with $\delta$ being close to $\sqrt{2}1$, as it suffices for many real applications to have matrices with $\delta = \sqrt{2}1$. Our hardness result holds for any constant $\delta$. Specifically, our result proves that even if $\delta$ is indeed very small, i.e. the matrix is in fact \emph{strongly RIP}, certifying that the matrix exhibits \emph{weak RIP} itself is SSEHard. In order to prove the hardness result, we prove a variant of the Cheeger's Inequality for sparse vectors.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.5791
 Bibcode:
 2014arXiv1406.5791N
 Keywords:

 Computer Science  Computational Complexity