Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications
Let A be an n x N real-valued matrix with n   delta and k(n)/(n) -> rho as n -> infinity. If each matrix A(n,Nn) has its columns in general position, then fk(AI(N))/f(k)(I(N)) tends to zero or one depending on whether rho > min(0, 2 - delta(-1)) or rho   min(0, 2 - delta(-1)) or rho < min(0, 2 - delta(-1)). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermined systems of linear equations. Below, let A be a fixed n x N matrix, n < N, with columns in general position.
zonotope random polytopes random cones random matrices wendel's theorem winder-cover theorem arrangements of hyperplanes compressed sensing threshold phenomena unique solution of underdetermined systems of linear equations