Klee sets and Chebyshev centers for the right Bregman distance

TL;DR

This paper explores Klee sets and Chebyshev centers within the context of Bregman distances, establishing new theoretical results on their properties and uniqueness, relevant to Information Geometry and Machine Learning.

Contribution

It proves that every Klee set with respect to the right Bregman distance is a singleton and establishes the uniqueness of Chebyshev centers, extending classical convex analysis results.

Findings

Every Klee set with respect to the right Bregman distance is a singleton.

Chebyshev centers are unique under the studied conditions.

Characterizations relate to classical results by Garkavi, Klee, Nielsen, and Nock.

Abstract

We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is specialized to the energy, one obtains classical notions from Approximation Theory and Convex Analysis. The contribution of this paper is twofold. First, we provide an affirmative answer to a recently-posed question on whether or not every Klee set with respect to the right Bregman distance is a singleton. Second, we prove uniqueness of the Chebyshev center and we present a characterization that relates to previous works by Garkavi, by Klee, and by Nielsen and Nock.