Stability for UMAP

TL;DR

This paper establishes a stability theorem for the UMAP construction by analyzing its representation as an iterated pushout of Vietoris-Rips objects and the induced maps between associated ep-metric spaces.

Contribution

It introduces a novel stability result for UMAP systems by relating their construction to ep-metric spaces and pushout diagrams, providing theoretical guarantees.

Findings

Proves a stability theorem for UMAP's global components.

Shows the induced maps between ep-metric spaces are stable under certain conditions.

Connects UMAP stability to excision principles in topology.

Abstract

This paper displays the Healy-McInnes UMAP construction $V(X,N)$ as an iterated pushout of Vietoris-Rips objects associated to extended pseudo metric spaces (ep-metric spaces) defined by choices of neighbourhoods of the elements of a finite set $X$. An inclusion $X \subset Y$ in another finite set defines a map of UMAP systems $V(X,N) \to V(Y,N')$ in the presence of a compatible system of neighbourhoods $N'$ for $Y$. There is also an induced map of ep-metric spaces $(X,D) \to (Y,D')$, where $D$ and $D'$ are colimits (global averages) of the metrics defined by the neighbourhood systems for $X$ and $Y$. We prove a stablity result for the restriction of this ep-metric space map to global components. This stability result translates, via excision for path components, to a stability result for global components of the UMAP systems.