Neighborliness of randomly projected simplices in high dimensions

Let A be a d x n matrix and T = T(n-1) be the standard simplex in Rn. Suppose that d and n are both large and comparable: d approximately deltan, delta in (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 Rn. We derive rhoN(delta) > 0 with the property that, for any rho < rhoN(delta), with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0 < or = k < or = rhod. This implies that P is left floor rhod right floor-neighborly, and its skeleton Skel(left floor rhod right floor)(P) is combinatorially equivalent to Skel( left floor rhod right floor)(T). We also study a weaker notion of neighborliness where the numbers of k-dimensional faces f(k)(P) > or = f(k)(T)(1-epsilon). Vershik and Sporyshev previously showed existence of a threshold rhoVS(delta) > 0 at which phase transition occurs in k/d. We compute and display rhoVS and compare with rhoN. Corollaries are as follows. (1) The convex hull of n Gaussian samples in Rd, with n large and proportional to d, has the same k-skeleton as the (n-1) simplex, for k < rhoN (d/n)d(1 + oP(1)). (2) There is a "phase transition" in the ability of linear programming to find the sparsest nonnegative solution to systems of underdetermined linear equations. For most systems having a solution with fewer than rhoVS(d/n)d(1 + o(1)) nonzeros, linear programming will find that solution.