Stability Properties of Neighbourly Random Polytopes

Mankiewicz, Piotr; Tomczak-Jaegermann, Nicole

doi:10.1007/s00454-008-9092-8

Cited by 1 publication

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“…We refer the reader to the books [16] and [31] for classical details on neighborly polytopes. (Some new quantitative invariants related to neighborliness were recently developed in [22]. )…”

Section: Neighborly Polytopesmentioning

confidence: 99%

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

et al. 2010

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This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X 1 , . . . , ±X N ∈ R n , (N ≥ n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property (RIP) with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a sub-exponential tail inequality possess this property RIP with overwhelming probability. We show that such "sensing" matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ 1 minimization with m ≤ Cn/ log 2 (cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K ⊂ R n is a convex body and X 1 , . . . , X N ∈ K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m ∼ n/ log 2 (cN/n).AMS Classification: primary 52A20, 94A12, 52B12, 46B09 secondary 15A52, 41A45, 94B75

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Section: Neighborly Polytopesmentioning

confidence: 99%