Critical angles between two convex cones I. General theory

Abstract

The concept of critical (or principal) angle between two linear subspaces has applications in statistics, numerical linear algebra, and other areas. Such concept has been abundantly studied in the literature, both from a theoretical and computational point of view. Part I of this work is an attempt to build a general theory of critical angles for a pair of closed convex cones. The need of such theory is motivated, among other reasons, by some specific problems arising in regression analysis of cone-constrained data, see Tenenhaus (Psychometrika 53:503–524, 1988). Angle maximization and/or angle minimization problems involving a pair of convex cones are at the core of our discussion. Such optimization problems are nonconvex in general and their numerical resolution offer a number of challenges. Part II of this work focusses on the practical computation of the maximal and/or minimal angle between specially structured cones.