Operatopes, Operanoids, and Noncommutative Zonoids

TL;DR

This paper introduces operatopes, operanoids, and noncommutative zonoids as new classes of convex bodies derived from operator norm balls, expanding the geometric framework with free probability and applications in statistics.

Contribution

It generalizes zonotopes to noncommutative zonoids using operator norms, introduces operanoids as their probabilistic counterparts, and explores asymptotic properties and applications.

Findings

Operanoids are the expectation of random affine images of operator norm balls.

Noncommutative zonoids form a new class of convex bodies in high dimensions.

Framework of free probability helps analyze properties of these bodies.

Abstract

We study a class of convex bodies called operatopes that are obtained by taking Minkowski sums of affine images of an operator norm ball. This notion generalizes that of zonotopes which are Minkowksi sums of line segments. Taking the limit of the number of line segments to infinity yields the class of convex bodies called zonoids, which can also be viewed as the expectation of a random line segment. Expanding on this interpretation, we analogously define operanoids as the expectation of a random affine image of an operator norm ball. In studying the properties of operanoids when the dimension of the operator norm ball grows, we arrive at a new asymptotic regime for limits of convex bodies. This leads to the more general class of convex bodies called noncommutative zonoids, and we use the framework of free probability theory to illustrate basic properties and examples. Finally, we discuss applications of operanoids and noncommutative zonoids in statistics and stochastic processes.