High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension

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 proportionaldimensional case where d n, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rn . We derive N () > 0 with the property that, for any < N (), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 k d. This implies that P is centrally d -neighborly, and its skeleton Skel d (P) is combinatorially equivalent to Skel d (C). We display graphs of N . Two weaker notions of neighborliness are also important for understanding sparse sol...

• Type: Journal
• Year: 2006
• Where: DCG
• Authors: David L. Donoho
• Keywords: Centrosymmetric Polytopes, DCG 2006, Polytopes, Sparse Solutions