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.
翻译:t-SNE是最常用的基于武力的非线性维度减少方法之一。 本文有两种贡献: 第一是强烈的颜色, 这个想法也适用于其他基于武力的方法( UMAP, ForceAtlas2,...)。 在每一个平衡中, 微粒上作用的有吸引力和令人厌恶的力量: 然而, 微粒上作用的吸引( 或令人厌恶的) 力量的大小和方向都与其特性有关: 力矢量可以作为附加特性。 第二, 我们分析t- SNE在单一的组合( 由随机K- 经常图的相邻矩阵所建的亲近关系所建成) 中发挥作用的案例; 我们产生一个平均的场模型, 导致在典型的变异性微积中产生有趣的问题。 该模型预测, 在极限中, 单个完全同质的聚集体的 t- SNE嵌入不是点, 而是直径$sim k ⁇ -1/4} n_ 1/4} 。 这得到数字结果的支持。 平均的字段 atz 延伸至其他以力基的裁减方法。