Dimension reduction, widely used in science, maps high-dimensional data into low-dimensional space. We investigate a basic mathematical model underlying the techniques of stochastic neighborhood embedding (SNE) and its popular variant t-SNE. Distances between points in high dimensions are used to define a probability distribution on pairs of points, measuring how similar the points are. The aim is to map these points to low dimensions in an optimal way so that similar points are closer together. This is carried out by minimizing the relative entropy between two probability distributions. We consider the gradient flow of the relative entropy and analyze its long-time behavior. This is a self-contained mathematical problem about the behavior of a system of nonlinear ordinary differential equations. We find optimal bounds for the diameter of the evolving sets as time tends to infinity. In particular, the diameter may blow up for the t-SNE version, but remains bounded for SNE.

翻译：降维作为科学领域广泛使用的技术，将高维数据映射至低维空间。本研究探讨了随机邻域嵌入（SNE）及其常用变体t-SNE技术背后的基础数学模型。通过高维空间中点对之间的距离定义概率分布，以度量点之间的相似性。其目标是以最优方式将这些点映射至低维空间，使得相似点更紧密聚集。该过程通过最小化两个概率分布之间的相对熵实现。我们考察相对熵的梯度流并分析其长期行为。这是一个关于非线性常微分方程组行为的自洽数学问题。我们获得了演化集合直径随时间趋于无穷时的最优界。特别地，t-SNE版本的直径可能发散，而SNE版本的直径始终保持有界。