# On the Abundance of Critical Points of the t-SNE Energy

arXiv: https://arxiv.org/abs/2609.04379 · HTML: https://mlsanity.com/paper/2609.04379/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Visualizing and Understanding Neural Models in NLP](https://mlsanity.com/paper/1506.01066/) (2015)
- [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)
- [Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1412.4172/) (2014)
- [On Probabilistic Embeddings in Optimal Dimension Reduction](https://mlsanity.com/paper/2408.02433/) (2024)
- [Convergence analysis of t-SNE as a gradient flow for point cloud on a manifold](https://mlsanity.com/paper/2401.17675/) (2024)
- [Large data limits and scaling laws for tSNE](https://mlsanity.com/paper/2410.13063/) (2024)
- [t-SNE Exaggerates Clusters, Provably](https://mlsanity.com/paper/2510.07746/) (2025)
- [T-SNE Is Not Optimized to Reveal Clusters in Data](https://mlsanity.com/paper/2110.02573/) (2021)
- [Accelerating hyperbolic t-SNE](https://mlsanity.com/paper/2401.13708/) (2024)
- [On the continuum limit of t-SNE for data visualization](https://mlsanity.com/paper/2604.12041/) (2026)
- [Some Theoretical Limitations of t-SNE](https://mlsanity.com/paper/2604.13295/) (2026)

## Abstract (arXiv)

This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical examples, several of which are reproduced in this article, suggest a complicated energy landscape with many local minimizers that do not respect the topology or clustering structure of the underlying data. This work seeks to provide first steps towards a rigorous explanation of these phenomena. Specifically, for a general family of energies, which include both the original t-SNE algorithm and recently identified large data limits, and for densities in feature space which obey a continuous symmetry, we construct infinite families of distinct critical points. These critical points are based upon identifying pairs of discrete symmetries, one in the original feature space and the other in the target embedding space, which are preserved under gradient dynamics. These critical configurations exhibit many characteristics, such as topology breaking and spurious clustering, which are often observed empirically. Finally, numerical and analytical examples are given throughout as a means of illustrating the approach.