Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling

Adamczak, Radosław; Litvak, Alexander E.; Pajor, Alain; Tomczak-Jaegermann, Nicole

doi:10.1007/s00365-010-9117-4

Cited by 87 publications

(134 citation statements)

References 28 publications

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“…The result is optimal, up to the factor log log 3m, as shown in [2]. As for Theorem 7, assuming unconditionality of the distributions of the rows, we can remove this factor.…”

Section: Resultsmentioning

confidence: 88%

“…The articles [1], [2] and [3] considered random matrices with independent columns, and investigated high dimensional geometric properties of the convex hull of the columns and the RIP for various models of matrices, including the log-concave Ensemble build with independent isotropic log-concave columns. It was shown that various properties of random vectors can be efficiently studied via operator norms and the parameter Γ n,m recalled below.…”

Section: Introductionmentioning

confidence: 99%

“…It was shown that various properties of random vectors can be efficiently studied via operator norms and the parameter Γ n,m recalled below. In order to control this parameter an efficient technique of chaining was developed in [1] and [2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Geometry of log-concave ensembles of random matrices and approximate reconstruction

Adamczak¹,

Latała²,

Litvak³

et al. 2011

Comptes Rendus. Mathématique

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We study the Restricted Isometry Property of a random matrix Γ with independent isotropic log-concave rows. To this end, we introduce a parameter Γ k,m that controls uniformly the operator norm of sub-matrices with k rows and m columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.

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“…The result is optimal, up to the factor log log 3m, as shown in [2]. As for Theorem 7, assuming unconditionality of the distributions of the rows, we can remove this factor.…”

Section: Resultsmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometry of log-concave ensembles of random matrices and approximate reconstruction

Adamczak¹,

Latała²,

Litvak³

et al. 2011

Comptes Rendus. Mathématique

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“…In particular, approximate message passing algorithms (AMP) for compressed sensing reconstruction [8], [4] can be recast in the form (1) for suitable choices of the matrix A and functions f . Generalizations of this algorithm were developed by several groups [17], [15], [14] Here, we will use the term AMP for the generalized recursion (1). We also notice that the universality of compressed sensing phase transitions can be conjectured from the results of the non-rigorous replica method [13], [16].…”

Section: Resultsmentioning

confidence: 97%

“…We say that the sequence is (C, d)-regular (or, for short, regular) polynomial sequence if (1) For each N , the entries (A ij (N )) 1≤i<j≤N are independent centered random variables. Further they are subgaussian with common scale factor C/N .…”

Section: A Universalitymentioning

confidence: 99%

Universality in polytope phase transitions and iterative algorithms

Bayati¹,

Lelarge²,

Montanari³

2012

2012 IEEE International Symposium on Information Theory Proceedings

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We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N ×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are known as 'approximate message passing' algorithms. We study the high-dimensional (large N ) behavior of the iterates of F for polynomial functions F, and prove that it is universal, i.e. it depends only on the first two moments of the entries of A.As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves a conjecture by David Donoho and Jared Tanner. I. MAIN RESULTSLet A ∈ R N ×N be a random Wigner matrix, i.e. a symmetric random matrix with i.i.d. entries (A ij ) i≤j satisfying E{A ij } = 0 and E{A 2 ij } = 1/N . A considerable effort has been devoted to studying the distribution of the eigenvalues of such a matrix [3]. The universality phenomenon is a striking recurring theme in these studies. Roughly speaking, many asymptotic properties of the joint eigenvalues distribution are independent of the entries distribution as long as it matches the first two moments, and satisfies certain tail conditions. We refer to [3] and references therein for a selection of such results. Universality is extremely useful because it allows to compute asymptotics for one entries distribution (e.g. Gaussian) and then export the results to a broad class of distributions.In this paper we are concerned with random matrix universality, albeit we do not focus onto eigenvalues properties. Given A ∈ R N ×N , and an initial condition x 0 ∈ R N independent of A, we consider the sequence (x t ) t∈N defined by letting, for t ≥ 0,Here, div denotes the divergence operator and, for each. The present paper is concerned with the asymptotic distribution of x t as N → ∞ with t fixed, and establishes the following results: Universality. As N → ∞, the finite-dimensional marginals of the distribution of x t are asymptotically insensitive to the distribution of the entries of A ij . State evolution. The entries of x t are asymptotically Gaussian with zero mean, and variance that can be explicitly computed through a recursion, known as state evolution Phase transitions in polytope geometry. As an application, we use state evolution to prove universality of a phase transition on polytope geometry, with connections to compressed sensing. This solves a conjecture put forward by David Donoho and Jared Tanner [6], [12].We think that the first two results provide a useful tool to establish a variety of universality in contexts in which the iteration (1) would not a priori seem relevant. The third of the above contributions demonstrates this. In fact, we shall consider a more general setting than the one just described, see Sections I-A and I-B.Iterations of the form (1) emerge in a number of contexts. In particular, approximate message passing algorithms (AMP) for compressed sensing ...

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Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions

Vladimirova

Girard

Nguyen

et al. 2020

Stat

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We propose the notion of sub‐Weibull distributions, which are characterized by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalizes the sub‐Gaussian and sub‐Exponential families to potentially heavier tailed distributions. Sub‐Weibull distributions are parameterized by a positive tail index θ and reduce to sub‐Gaussian distributions for and to sub‐Exponential distributions for . A characterization of the sub‐Weibull property based on moments and on the moment generating function is provided and properties of the class are studied. An estimation procedure for the tail parameter is proposed and is applied to an example stemming from Bayesian deep learning.

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Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling

Cited by 87 publications

References 28 publications

Geometry of log-concave ensembles of random matrices and approximate reconstruction

Geometry of log-concave ensembles of random matrices and approximate reconstruction

Universality in polytope phase transitions and iterative algorithms

Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions

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