Probabilistic analysis of the Grassmann condition number

Abstract

We analyze the probability that a random m-dimensional linear subspace of R^n both intersects a regular closed convex cone C\subseteq R^n and lies within distance \alpha of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number \C(A) for the homogeneous convex feasibility problem \exists x\in C\setminus 0 : Ax=0. The Grassmann condition number is a geometric version of Renegar's condition number, that we have introduced recently in [SIOPT 22(3):1029-1041, 2012]. We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of A\in R^{m\times n} are chosen i.i.d. standard normal, then for any regular cone C, we have E[ln\C(A)]<1.5 ln(n)+1.5. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.