Abstract

Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in R n. 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 ∼ δn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R n. 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 ⌊ρd⌋-neighborly, and its skeleton Skel ⌊ρd⌋(P) is combinatorially equivalent to Skel ⌊ρd⌋(C). We display graphs of ρN. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: facial neighborliness and sectional neighborliness [9]; we study both. The weakest, (k, ɛ)-facial neighborliness, asks if the k-faces are all simplicial and if

Citations

917	Compressed sensing - Donoho - 2006
488	Near optimal signal recovery from random projections: Universal encoding strategies - Candès, Tao - 2006
60	Neighborly polytopes and sparse solution of underdetermined linear equations. Stanford dept. of statistics working paper - Donoho - 2004
25	Convex polytopes, volume 221 of Graduate Texts in Mathematics - Grünbaum - 2003
23	Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability - Dembo, Zeitouni - 2010
23	Non-linear angle-sum relations for polyhedral cones and polytopes - McMullen - 1975
20	Random projections of regular simplices - Affentranger, Schneider - 1992
14	Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem - Vershik, Sporyshev - 1992
13	Neighboring vertices on a convex polyhedron - Gale - 1956
9	Random projections of regular polytopes - Henk - 1999
7	Neighbourliness of centrally symmetric polytopes in high dimensions - Schneider - 1975
6	The non-neighbourliness of centrally symmetric convex polytopes having many vertices - Burton - 1991
6	Diagrams for centrally symmetric polytopes - McMullen, Shephard - 1968
5	Grassmann angles of convex polytopes - Grünbaum - 1968
5	On the geometrical moments of skew-regular simplices in hyperspherical space, with some applications in geometry and mathematical statistics - Ruben - 1960
3	For most large systems of underdetermined equations, the minimum ℓ 1 - norm solution is the sparsest solution - Donoho - 2004
2	An asymptotic estimate of the average number of steps of the parametric simplex method - Vershik, Sporyshev - 1986
1	A uniform approximation to the right normal tail integral - Bryc - 2002
1	A convergent asymptotic expansion for Mill’s ratio and the normal probability integral in terms of rational functions - Ruben - 1963