Probabilistic Analysis of the Grassmann Condition Number

• Published in: Foundations of computational mathematics Vol. 15; no. 1; pp. 3 - 51
• Main Authors: Amelunxen, Dennis, Bürgisser, Peter
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.02.2015; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Computational mathematics; Computer Science; Convex programming; Distribution (Probability theory); Economics; Feasibility; Foundations; Linear and Multilinear Algebras; Linear programming; Manifolds; Manifolds (Mathematics); Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Number theory; Numerical Analysis; Origins; Perturbation (Mathematics); Probabilistic analysis; Probability; Subspaces; Texts; Topological manifolds; United States; 60D05; Tube formulas; Spherically convex sets; 52A22; 52A55; Perturbation; Convex programming; Average analysis; 90C31; Grassmann manifold; 90C25; 90C22; Condition number
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9178-4

We analyze the probability that a random m -dimensional linear subspace of both intersects a regular closed convex cone  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 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 are chosen i.i.d. standard normal, then for any regular cone  C , we have . The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.