Equilibrium Distributions for t-distributed Stochastic Neighbour Embedding

TL;DR

This paper analyzes the convergence of the empirical distribution of t-SNE outputs to an equilibrium distribution, characterized as a solution to a variational problem, under certain input data assumptions.

Contribution

It provides a theoretical framework for understanding the limiting behavior of t-SNE embeddings as the number of data points grows.

Findings

Empirical measures converge to an equilibrium distribution.

Equilibrium characterized by a variational problem.

Conditions on input data ensure convergence.

Abstract

We study the empirical measure of the output of the t-distributed stochastic neighbour embedding algorithm when the initial data is given by n independent, identically distributed inputs. We prove that under certain assumptions on the distribution of the inputs, this sequence of measures converges to an equilibrium distribution, which is described as a solution of a variational problem.