May the force be with you

TL;DR

This paper reveals that force-based dimensionality reduction methods inherently compute a vector field that contains valuable information, and proposes a Morse theory-inspired refinement strategy to enhance their effectiveness, demonstrated with t-SNE.

Contribution

It introduces the concept of the vector field in force-based methods and proposes a novel refinement strategy leveraging Morse theory for improved results.

Findings

Vector fields provide high-quality additional information.

Morse theory-based refinement improves dimensionality reduction.

Demonstrated effectiveness on synthetic and real datasets.

Abstract

Modern methods in dimensionality reduction are dominated by nonlinear attraction-repulsion force-based methods (this includes t-SNE, UMAP, ForceAtlas2, LargeVis, and many more). The purpose of this paper is to demonstrate that all such methods, by design, come with an additional feature that is being automatically computed along the way, namely the vector field associated with these forces. We show how this vector field gives additional high-quality information and propose a general refinement strategy based on ideas from Morse theory. The efficiency of these ideas is illustrated specifically using t-SNE on synthetic and real-life data sets.