Computing Hulls And Centerpoints In Positive Definite Space

TL;DR

This paper introduces algorithms for computing approximate convex hulls and centerpoints in positive definite matrix space, enabling analysis of complex data types like diffusion tensors and kernel maps.

Contribution

It develops the concept of horoball hulls and demonstrates their use in approximating convex hulls and centerpoints in positive definite space, a novel approach in this domain.

Findings

Approximate horoball hulls contain the true convex hull.

The method preserves geodesic extents, enabling diameter and width approximations.

Algorithms for robust centerpoints in positive definite space are proposed.

Abstract

In this paper, we present algorithms for computing approximate hulls and centerpoints for collections of matrices in positive definite space. There are many applications where the data under consideration, rather than being points in a Euclidean space, are positive definite (p.d.) matrices. These applications include diffusion tensor imaging in the brain, elasticity analysis in mechanical engineering, and the theory of kernel maps in machine learning. Our work centers around the notion of a horoball: the limit of a ball fixed at one point whose radius goes to infinity. Horoballs possess many (though not all) of the properties of halfspaces; in particular, they lack a strong separation theorem where two horoballs can completely partition the space. In spite of this, we show that we can compute an approximate "horoball hull" that strictly contains the actual convex hull. This approximate hull also preserves geodesic extents, which is a result of independent value: an immediate corollary is that we can approximately solve problems like the diameter and width in positive definite space. We also use horoballs to show existence of and compute approximate robust centerpoints in positive definite space, via the horoball-equivalent of the notion of depth.