Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Abstract

Let A be an n×N real-valued matrix with n min (0,2−δ −1) or ρ min (0,2−δ−1) or ρ<min (0,2−δ−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×N matrix, n<N, with columns in general position. Call a vector in ℝ+Nk-sparse if it has at most k nonzeros. For such a k-sparse vector x0, b=Ax0 generates an underdetermined system b=Ax having k-sparse solution. Among inequality-constrained systems Ax=b, x≥0, having k-sparse solutions, the fraction having a unique nonnegative solution is fk(Aℝ+N)/fk(ℝ+N). Call a vector in the hypercube INk-simple if all entries except at most k are at the bounds 0 or 1. For such a k-simple vector x0, b=Ax0 generates an underdetermined system b=Ax with k-simple solution. Among inequality-constrained systems Ax=b, x∈IN, having k-simple solutions, the fraction having a unique hypercube-constrained solution is fk(AIN)/fk(IN).