Geometry of Geometric Data Set I

TL;DR

This paper explores the geometric structure of data sets, extending and analyzing the observable and box distances, proving key properties like non-separability and completeness within this framework.

Contribution

It introduces the extension of observable and box distances to geometric data sets and proves their fundamental properties, including non-separability and completeness.

Findings

Observable distance between geometric data sets is non-separable.

Box distance between geometric data sets is complete.

Extension of metric concepts to geometric data sets is rigorous.

Abstract

Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between geometric data sets. In this paper, we begin by proving the non-separability of the observable distance between geometric data sets. We then extend the box distance between mm-spaces to that between geometric data sets and prove its completeness and non-separability.