  t-SNE is one of the most commonly used force-based nonlinear dimensionality
reduction methods. 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 $\sim k^{-1/4} n^{-1/4}$. This is supported by numerical
results. The mean field ansatz extends to other force-based dimensionality
reduction methods.
