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∈Rn, with at most k non-zero co-ordinates, (1−δ)∥x∥2≤∥Ax∥2≤(1+δ)∥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 δ. 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 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 SSE-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). 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 indeed very
small, i.e. the matrix is in fact \emph{strongly RIP}, certifying that the
matrix exhibits \emph{weak RIP} itself is SSE-Hard.
In order to prove the hardness result, we prove a variant of the Cheeger's
Inequality for sparse vectors