t-SNE, forceful colorings, and mean field limits

t-SNE is a commonly used force-based nonlinear dimensionality reduction method. This paper has two contributions: the first is forceful colorings, an idea that is also applicable to other force-based methods (UMAP, ForceAtlas2,...). In every equilibrium, the attractive and repulsive forces acting on a particle cancel out: however, both the size and the direction of the attractive (or repulsive) forces acting on a particle are related to its properties: the force vector can serve as an additional feature. Secondly, we analyze the case of t-SNE acting on a single homogeneous cluster (modeled by affinities coming from the adjacency matrix of a random k-regular graph); we derive a mean-field model that leads to interesting questions in classical calculus of variations. The model predicts that, in the limit, the t-SNE embedding of a single perfectly homogeneous cluster is not a point but a thin annulus of diameter ∼k-1/4n-1/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim k^{-1/4} n^{-1/4}$$\end{document}. This is supported by numerical results. The mean field ansatz extends to other force-based dimensionality reduction methods. query Please check the edit made in the article title.