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.

Guys, to tell you the truth when I heard the name tSNE and the full form being t-distributed stochastic neighborhood embedding, I was scared. Gradually, I could find my way through the papers, various blogs, and realized it is not so ominous as the name suggests. I just thought I will try to share the intuitions that I gathered. Let me put it down in a question, answer format like it came to my mind. Less number of attributes are simpler to manage.