Computational Complexity of Certifying Restricted Isometry Property

Given a matrix A with n rows, a number k < n, and 0 < δ < 1, A is (k, δ)-RIP (Restricted Isometry Property) if, for any vector x ∈ R, with at most k non-zero co-ordinates, (1− δ)‖x‖2 ≤ ‖Ax‖2 ≤ (1 + δ)‖x‖2 In other words, a matrix A is (k, δ)-RIP if Ax preserves the length of x when x is a k-sparse vector. In many applications, such as compressed sensing and sparse recovery, it is desirable to construct RIP matrices with a large k and a small δ. It is known that, with high probability, random constructions produce matrices that exhibit RIP. This motivates the problem of certifying whether a randomly generated matrix exhibits RIP with suitable parameters. In this paper, we prove that it is hard to approximate the RIP parameters of a matrix assuming the Small-Set-Expansion Hypothesis. Specifically, we prove that for any arbitrarily large constant C > 0 and any arbitrarily small constant 0 < δ < 1, there exists some k such that given a matrix M , it is Small-Set-Expansion-hard to distinguish the following two cases: • (Highly RIP) M is (k, δ)-RIP. • (Far away from RIP) M is not (k/C, 1− δ)-RIP. Most of the previous results on the topic of hardness of RIP certification only hold for certification when δ = o(1); i.e, when the matrix exhibits strong RIP. In practice, it is of interest to understand the complexity of certifying a matrix with δ being close to √ 2 − 1, as it suffices for many real applications to have matrices with δ = √ 2− 1. Our hardness result holds for any constant δ. Specifically, our result proves that even if δ is ∗[email protected] †[email protected]