131 research outputs found
Computational Complexity of Certifying Restricted Isometry Property
Given a matrix with rows, a number , and , is
-RIP (Restricted Isometry Property) if, for any vector , with at most non-zero co-ordinates, In many applications, such as
compressed sensing and sparse recovery, it is desirable to construct RIP
matrices with a large and a small . 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 and any arbitrarily small
constant , there exists some such that given a matrix , it
is SSE-Hard to distinguish the following two cases:
- (Highly RIP) is -RIP.
- (Far away from RIP) is not -RIP.
Most of the previous results on the topic of hardness of RIP certification
only hold for certification when . In practice, it is of interest
to understand the complexity of certifying a matrix with being close
to , as it suffices for many real applications to have matrices
with . Our hardness result holds for any constant
. Specifically, our result proves that even if 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
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