Minimization of the Probabilistic p-frame Potential

TL;DR

This paper explores optimal point arrangements on the sphere for potential functions, linking frame theory and probabilistic distributions, and revealing connections to statistical shape analysis.

Contribution

It characterizes optimal configurations in both deterministic and probabilistic settings using frame theory, introducing new insights into their approximation properties.

Findings

Optimal configurations are characterized by their approximation properties.

Probabilistic optimal distributions are described via special classes of probabilistic frames.

Connections between statistical shape analysis and frame theory are established.

Abstract

We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory. Furthermore, we consider similar optimal configurations in terms of random distributions of points on the sphere. In this probabilistic setting, we characterize these optimal distributions by means of special classes of probabilistic frames. Our work also indicates some connections between statistical shape analysis and frame theory.