Probabilistic Analysis of the Grassmann Condition Number

We analyze the probability that a random m-dimensional linear subspace of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document} both intersects a regular closed convex cone \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C\subseteq\mathbb{R}^{n}$\end{document} and lies within distance α 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 [inline-graphic not available: see fulltext] for the homogeneous convex feasibility problem ∃x∈C∖0:Ax=0. The Grassmann condition number is a geometric version of Renegar’s condition number, which we have introduced recently in Amelunxen and Bürgisser (SIAM J. Optim. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A\in\mathbb {R}^{m\times n}$\end{document} are chosen i.i.d. standard normal, then for any regular cone C, we have [inline-graphic not available: see fulltext]. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.