Neighborliness of randomly projected simplices in high dimensions

Abstract

Let A be a d ؋ n matrix and T ‫؍‬ T n؊1 be the standard simplex in R n . Suppose that d and n are both large and comparable: d Ϸ ␦n, ␦ ʦ (0, 1). We count the faces of the projected simplex AT when the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R n . 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 ‫؍‬ AT is exactly the same as for T, for 0 < k < d. This implies that P is d-neighborly, and its skeleton Skeld(P) is combinatorially equivalent to Skel d(T). We also study a weaker notion of neighborliness where the numbers of k-dimensional faces f k(P) > f k(T)(1 ؊ ). Vershik and Sporyshev previously showed existence of a threshold VS(␦) > 0 at which phase transition occurs in k͞d. We compute and display VS and compare with N. Corollaries are as follows. (1) neighborly polytopes ͉ convex hull of Gaussian sample ͉ underdetermined systems of linear equations ͉ uniformly distributed random projections ͉ phase transitions

Sparse nonnegative solution of underdetermined linear equations by linear programming

“…The fact that it ''crosses the line'' ϭ 1͞2 for ␦ near 0.425 is noteworthy; this means that whereas a polytope can only be d͞2 neighborly, it can be Ͼd͞2 weakly neighborly! In fact, we know VS (␦) 3 1 as ␦ 3 1 (12,14). …”

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“…In ref. 12 we revisit the Vershik-Sporyshev model, asking about neighborliness of the resulting high-dimensional random polytopes. It proves the following.…”

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