Geometry of Data
This work offers a theoretical bridge between geometric and topological data analysis, potentially benefiting researchers in computational geometry and data science, though it appears incremental in linking existing concepts.
The paper connects geometric data analysis, which measures how much balls must be enlarged to intersect, to the traditional concept of curvature, enabling a reconceptualization of curvature and linking it to hyperconvexity while providing a geometric perspective on topological data analysis methods.
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.