The dimension of almost spherical sections of convex bodies

Figiel, T.; Lindenstrauss, Joram; Milman, Vitali

doi:10.1007/bf02392234

Cited by 340 publications

(175 citation statements)

References 23 publications

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“…Existing results for recovering sparse signals 1 At first sight, this seems somewhat surprising as faces of different people look so different to human eyes. That is probably because human brain has adapted to distinguish highly correlated visual signals such as faces or voices.…”

Section: A a Motivating Examplementioning

confidence: 99%

“…On the theoretical side, the progress has been propelled by powerful tools and results from multiple mathematical areas such as measure concentration [1]- [3], statistics [4]- [6], combinatorics [7], and coding theory [8]. On the practical side, a lot of excitement has been generated by remarkable successes in real-world applications in areas such as signal (image or speech) processing [9], communications [10], computer vision and pattern recognition [11]- [13] etc.…”

Section: Introductionmentioning

confidence: 99%

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Dense error correction via l1-minimization

Wright

2009

2009 IEEE International Conference on Acoustics, Speech and Signal Processing

159

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This paper studies the problem of recovering a non-negative sparse signal x ∈ R n from highly corrupted linear measurements y = Ax + e ∈ R m , where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any non-negative, sufficiently sparse signal x can be recovered by solving an 1 -minimization problem:More precisely, if the fraction ρ of errors is bounded away from one and the support of x grows sublinearly in the dimension m of the observation, then as m goes to infinity, the above 1 -minimization succeeds for all signals x and almost all sign-and-support patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100% of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of iid Gaussian vectors with nonzero mean and small variance, which we call the "cross-and-bouquet" model. Simulations and experimental results corroborate the findings, and suggest extensions to the result.

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Section: A a Motivating Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dense error correction via l1-minimization

Wright

2009

2009 IEEE International Conference on Acoustics, Speech and Signal Processing

159

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“…We believe that additional applications of this notion and its variants will be discovered in the future. In particular, it seems worthwhile to study the interaction between metric type and metric cotype (such as in Kwapien's theorem [35]), the possible "Markov" variants of metric cotype (à la Ball [4]) and their relation to the Lipschitz extension problem, and the relation between metric cotype and the nonlinear Dvoretzky theorem (see [10], [5] for information about the nonlinear Dvoretzky theorem, and [22] for the connection between cotype and Dvoretzky's theorem).…”

Section: Introductionmentioning

confidence: 99%

Metric cotype

2008

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We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L p coarsely or uniformly embeds into L q . We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matoušek.

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“…One can use the map Ω to define a (2n)-dimensional Banach space X with a Euclidean subspace E with dim E = n so that X/E is also Euclidean. It then follows from results of Figiel, Lindenstrauss and Milman [5] (see also [4]) that X has type 2 constant T 2 (X) ≤ C(1 + log n). But then Maurey's extension theorem [9] produces a projection P : X → E with P ≤ T 2 (X).…”

Section: Remarksmentioning

confidence: 81%

“…We would like to take this opportunity to resolve a question raised in [5] concerning such twisted sums. In [5], Figiel, Lindenstrauss and Milman ask for an estimate on the Banach-Mazur distance…”

Section: Remarksmentioning

confidence: 99%