Geometry of Data

TL;DR

This paper explores the geometric principles underlying data analysis by linking topological intersection concepts with curvature and hyperconvexity, offering a new perspective on topological data analysis.

Contribution

It introduces a novel connection between curvature, hyperconvexity, and topological data analysis, providing a geometric framework for understanding data intersections.

Findings

Reconceptualizes curvature in the context of data analysis

Links topological intersection principles to geometric curvature

Provides a geometric perspective on topological data analysis

Abstract

Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric concept of curvature. This enables us, on one hand, to reconceptualize curvature and link it to the geometric notion of hyperconvexity. On the other hand, we can then also understand methods of topological data analysis from a geometric perspective.