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• W2010669260 abstract "Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in Rn. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly projected cross polytopes in the proportional-dimensional case where d ∼ δn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rn. We derive ρN(δ) > 0 with the property that, for any ρ < ρN(δ), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 ≤ k ≤ ρd. This implies that P is centrally ⌊ ρ d ⌋-neighborly, and its skeleton Skel⌊ ρ d ⌋(P) is combinatorially equivalent to Skel⌊ ρ d⌋(C). We display graphs of ρN. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: weak neighborliness and sectional neighborliness [9]; we study both. Weak (k,ε)-neighborliness asks if the k-faces are all simplicial and if the number of k-dimensional faces fk(P) ≥ fk(C)(1 – ε). We characterize and compute the critical proportion ρW(δ) > 0 such that weak (k,ε) neighborliness holds at k significantly smaller than ρW · d and fails for k significantly larger than ρW · d. Sectional (k,ε)-neighborliness asks whether all, except for a small fraction ε, of the k-dimensional intrinsic sections of P are k-dimensional cross polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ρS(δ) > 0 guaranteeing this property for k/d ∼ ρ < ρS(δ). We display graphs of ρS and ρW." @default.
• W2010669260 created "2016-06-24" @default.
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• W2010669260 date "2005-12-22" @default.
• W2010669260 modified "2023-10-09" @default.
• W2010669260 title "High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension" @default.
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• W2010669260 doi "https://doi.org/10.1007/s00454-005-1220-0" @default.
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