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：降维技术广泛应用于科学领域，它将高维数据映射到低维空间。我们研究了随机邻域嵌入（SNE）及其流行变体t-SNE技术背后的基本数学模型。高维空间中点之间的距离被用于定义点对上的概率分布，以度量点之间的相似程度。其目标是以最优方式将这些点映射到低维空间，使得相似点更靠近。这是通过最小化两个概率分布之间的相对熵来实现的。我们考虑相对熵的梯度流，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。我们得到了演化集合直径随时间趋于无穷时的最优界。特别地，对于t-SNE版本，直径可能趋于无穷大，而对于SNE则保持有界。