Measuring dissimilarity between convex cones by means of max-min angles

TL;DR

This paper introduces a new dissimilarity measure between convex cones based on max-min angles, relating it to the Pompeiu-Hausdorff distance, and applies it to few-shot image-set classification.

Contribution

It proposes a novel angle-based dissimilarity measure for convex cones, with algorithms for polyhedral cones, and demonstrates its application in few-shot learning.

Findings

The measure relates closely to Pompeiu-Hausdorff distance.

A cutting-plane algorithm approximates the measure for polyhedral cones.

The measure aids in classifying image sets in few-shot learning scenarios.

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.