78 research outputs found

    The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing

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    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

    Certifying the restricted isometry property is hard

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    This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P \neq NP

    Sparse bayesian polynomial chaos approximations of elasto-plastic material models

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    In this paper we studied the uncertainty quantification in a functional approximation form of elastoplastic models parameterised by material uncertainties. The problem of estimating the polynomial chaos coefficients is recast in a linear regression form by taking into consideration the possible sparsity of the solution. Departing from the classical optimisation point of view, we take a slightly different path by solving the problem in a Bayesian manner with the help of new spectral based sparse Kalman filter algorithms

    Computational Complexity of Certifying Restricted Isometry Property

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    Given a matrix AA with nn rows, a number k<nk<n, and 0<δ<10<\delta < 1, AA is (k,δ)(k,\delta)-RIP (Restricted Isometry Property) if, for any vector xRnx \in \mathbb{R}^n, with at most kk non-zero co-ordinates, (1δ)x2Ax2(1+δ)x2(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 kk and a small δ\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 Small-Set-Expansion-Hypothesis. Specifically, we prove that for any arbitrarily large constant C>0C>0 and any arbitrarily small constant 0<δ<10<\delta<1, there exists some kk such that given a matrix MM, it is SSE-Hard to distinguish the following two cases: - (Highly RIP) MM is (k,δ)(k,\delta)-RIP. - (Far away from RIP) MM is not (k/C,1δ)(k/C, 1-\delta)-RIP. Most of the previous results on the topic of hardness of RIP certification only hold for certification when δ=o(1)\delta=o(1). In practice, it is of interest to understand the complexity of certifying a matrix with δ\delta being close to 21\sqrt{2}-1, as it suffices for many real applications to have matrices with δ=21\delta = \sqrt{2}-1. Our hardness result holds for any constant δ\delta. Specifically, our result proves that even if δ\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 SSE-Hard. In order to prove the hardness result, we prove a variant of the Cheeger's Inequality for sparse vectors

    High-Girth Matrices and Polarization

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    The girth of a matrix is the least number of linearly dependent columns, in contrast to the rank which is the largest number of linearly independent columns. This paper considers the construction of {\it high-girth} matrices, whose probabilistic girth is close to its rank. Random matrices can be used to show the existence of high-girth matrices with constant relative rank, but the construction is non-explicit. This paper uses a polar-like construction to obtain a deterministic and efficient construction of high-girth matrices for arbitrary fields and relative ranks. Applications to coding and sparse recovery are discussed
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