Branch points and stability

TL;DR

This paper explores the mathematical structure of branch points in data sets, demonstrating that a specific map related to data inclusion is a controlled homotopy equivalence, with control defined via Hausdorff distance.

Contribution

It introduces a method using upper bounds to analyze the stability of branch points and proves a controlled homotopy equivalence for the map of branch points under data inclusion.

Findings

The hierarchy poset and branch point poset admit a calculus of least upper bounds.

The map of branch points is a controlled homotopy equivalence.

Control is expressed via Hausdorff distance.

Abstract

The hierarchy poset and branch point poset for a data set both admit a calculus of least upper bounds. A method involving upper bounds is used to show that the map of branch points associated to the inclusion of data sets is a controlled homotopy equivalence, where the control is expressed by an upper bound relation that is constrained by Hausdorff distance.