Measuring dissimilarity between convex cones by means of max-min angles
Welington de Oliveira, Valentina Sessa, David Sossa
TL;DR
This work defines a max-min angle $Θ(P,Q)$ between convex cones and a Pompeiu-Hausdorff–inspired dissimilarity $Dis_r(P,Q)$, unifying geometric intuition with a rigorous distance on the cone space. It derives an equivalent min-max formulation using the support function $F_Q(u)$ and develops both global and local cutting-plane strategies to compute $Θ(P,Q)$ for polyhedral cones, leveraging a master problem that is either globally solved or addressed via local NLP methods. The authors present closed-form results for special cone classes (linear subspaces and revolution cones) and propose a polyhedral framework with a Kelley-style cutting-plane algorithm, including warm-start and convergence insights. An application to Few-Shot image-set classification demonstrates that representing image sets as polyhedral cones can yield accurate discrimination with limited data, validating the practical relevance of the approach for geometry-driven machine learning tasks.
Abstract
This work introduces a novel dissimilarity measure between two convex cones, based on the max-min angle between them. We demonstrate that this measure is closely related to the Pompeiu-Hausdorff distance, a well-established metric for comparing compact sets. Furthermore, we examine cone configurations where the measure admits simplified or analytic forms. For the specific case of polyhedral cones, a nonconvex cutting-plane method is deployed to compute, at least approximately, the measure between them. Our approach builds on a tailored version of Kelley's cutting-plane algorithm, which involves solving a challenging master program per iteration. When this master program is solved locally, our method yields an angle that satisfies certain necessary optimality conditions of the underlying nonconvex optimization problem yielding the dissimilarity measure between the cones. As an application of the proposed mathematical and algorithmic framework, we address the image-set classification task under limited data conditions, a task that falls within the scope of the \emph{Few-Shot Learning} paradigm. In this context, image sets belonging to the same class are modeled as polyhedral cones, and our dissimilarity measure proves useful for understanding whether two image sets belong to the same class.