Probabilistic frames: An overview

TL;DR

This paper introduces probabilistic frames as a generalization of finite frames, explores their properties, and surveys their applications across various fields like directional statistics and convex geometry.

Contribution

It provides foundational properties of probabilistic frames and characterizes a subclass via potential minimization, connecting them to diverse mathematical areas.

Findings

Probabilistic frames generalize finite frames as probability measures.

A subclass characterized by potential function minimization.

Survey of applications in statistics, geometry, and design theory.

Abstract

Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of probabilistic frames, and we characterize one of their subclasses in terms of minimizers of some appropriate potential function. In addition, we survey a range of areas where probabilistic frames, albeit, under different names, appear. These areas include directional statistics, the geometry of convex bodies, and the theory of t-designs.