# Stochastic neighborhood embedding and the gradient flow of relative entropy

arXiv: https://arxiv.org/abs/2409.16963 · HTML: https://mlsanity.com/paper/2409.16963/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 23 references that are in the map.
- First posted on arXiv 25 Sep 2024.

## Builds on (most cited first, 8 of 8)

- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data](https://mlsanity.com/paper/2105.07536/) (2021)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [An Analysis of the t-SNE Algorithm for Data Visualization](https://mlsanity.com/paper/1803.01768/) (2018)
- [t-SNE, Forceful Colorings and Mean Field Limits](https://mlsanity.com/paper/2102.13009/) (2021)
- [Stochastic Neighbor Embedding separates well-separated clusters](https://mlsanity.com/paper/1702.02670/) (2017)
- [An Empirical Evaluation of the t-SNE Algorithm for Data Visualization in Structural Engineering](https://mlsanity.com/paper/2109.08795/) (2021)
- [Convergence analysis of t-SNE as a gradient flow for point cloud on a manifold](https://mlsanity.com/paper/2401.17675/) (2024)

## Built on by (most cited first, 1 of 1)

- [t-SNE Exaggerates Clusters, Provably](https://mlsanity.com/paper/2510.07746/) (2025)

## Abstract (arXiv)

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.