1 research outputs found
Computing Hulls And Centerpoints In Positive Definite Space
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.Comment: 16 pages, 2 figure