46 research outputs found
Hidden cliques and the certification of the restricted isometry property
International audienceCompressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this property with optimal parameters are mainly obtained via probabilistic arguments. Deciding whether a given matrix satisfies the restricted isometry property is a non-trivial computational problem. Indeed, we show in this paper that restricted isometry parameters cannot be approximated in polynomial time within any constant factor under the assumption that the hidden clique problem is hard. Moreover, on the positive side we propose an improvement on the brute-force enumeration algorithm for checking the restricted isometry property
Average-case Hardness of RIP Certification
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
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
The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
This paper deals with the computational complexity of conditions which
guarantee that the NP-hard problem of finding the sparsest solution to an
underdetermined linear system can be solved by efficient algorithms. In the
literature, several such conditions have been introduced. The most well-known
ones are the mutual coherence, the restricted isometry property (RIP), and the
nullspace property (NSP). While evaluating the mutual coherence of a given
matrix is easy, it has been suspected for some time that evaluating RIP and NSP
is computationally intractable in general. We confirm these conjectures by
showing that for a given matrix A and positive integer k, computing the best
constants for which the RIP or NSP hold is, in general, NP-hard. These results
are based on the fact that determining the spark of a matrix is NP-hard, which
is also established in this paper. Furthermore, we also give several complexity
statements about problems related to the above concepts.Comment: 13 pages; accepted for publication in IEEE Trans. Inf. Theor
Optimal detection of sparse principal components in high dimension
We perform a finite sample analysis of the detection levels for sparse
principal components of a high-dimensional covariance matrix. Our minimax
optimal test is based on a sparse eigenvalue statistic. Alas, computing this
test is known to be NP-complete in general, and we describe a computationally
efficient alternative test using convex relaxations. Our relaxation is also
proved to detect sparse principal components at near optimal detection levels,
and it performs well on simulated datasets. Moreover, using polynomial time
reductions from theoretical computer science, we bring significant evidence
that our results cannot be improved, thus revealing an inherent trade off
between statistical and computational performance.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1127 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org