# Clustering with t-SNE, provably

arXiv: https://arxiv.org/abs/1706.02582 · HTML: https://mlsanity.com/paper/1706.02582/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2017.

## Built on by (most cited first, 30 of 51)

- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data](https://mlsanity.com/paper/2105.07536/) (2021)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [Your Contrastive Learning Is Secretly Doing Stochastic Neighbor Embedding](https://mlsanity.com/paper/2205.14814/) (2022)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [A Probabilistic Graph Coupling View of Dimension Reduction](https://mlsanity.com/paper/2201.13053/) (2022)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [t-SNE, Forceful Colorings and Mean Field Limits](https://mlsanity.com/paper/2102.13009/) (2021)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Multidimensional Scaling: Approximation and Complexity](https://mlsanity.com/paper/2109.11505/) (2021)
- [Enhancing cluster analysis via topological manifold learning](https://mlsanity.com/paper/2207.00510/) (2022)
- [A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction](https://mlsanity.com/paper/2305.19043/) (2023)
- [DeepGB-TB: A Risk-Balanced Cross-Attention Gradient-Boosted Convolutional Network for Rapid, Interpretable Tuberculosis Screening](https://mlsanity.com/paper/2508.02741/) (2025)
- [Assessing and improving reliability of neighbor embedding methods: a map-continuity perspective](https://mlsanity.com/paper/2410.16608/) (2024)
- [The Shape of Attraction in UMAP: Exploring the Embedding Forces in Dimensionality Reduction](https://mlsanity.com/paper/2503.09101/) (2025)
- [Quantum Architecture Search with Unsupervised Representation Learning](https://mlsanity.com/paper/2401.11576/) (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 Is Not Optimized to Reveal Clusters in Data](https://mlsanity.com/paper/2110.02573/) (2021)
- [Outlier Detection in Large Radiological Datasets using UMAP](https://mlsanity.com/paper/2407.21263/) (2024)
- [t-SNE Exaggerates Clusters, Provably](https://mlsanity.com/paper/2510.07746/) (2025)
- [Semantic properties of English nominal pluralization: Insights from word embeddings](https://mlsanity.com/paper/2203.15424/) (2022)
- [Sailing in high-dimensional spaces: Low-dimensional embeddings through angle preservation](https://mlsanity.com/paper/2406.09876/) (2024)
- [AVIDA: Alternating method for Visualizing and Integrating Data](https://mlsanity.com/paper/2206.00135/) (2022)
- [MN-Pair Contrastive Damage Representation and Clustering for Prognostic Explanation](https://mlsanity.com/paper/2301.06077/) (2023)
- [Preserving local densities in low-dimensional embeddings](https://mlsanity.com/paper/2301.13732/) (2023)
- [From structure mining to unsupervised exploration of atomic octahedral networks](https://mlsanity.com/paper/2306.12272/) (2023)
- [Few-shot 1/a Anomalies Feedback : Damage Vision Mining Opportunity and Embedding Feature Imbalance](https://mlsanity.com/paper/2307.12676/) (2023)
- [DatasetEquity: Are All Samples Created Equal? In The Quest For Equity Within Datasets](https://mlsanity.com/paper/2308.09878/) (2023)
- [Universal Deoxidation of Semiconductor Substrates Assisted by Machine-Learning and Real-Time-Feedback-Control](https://mlsanity.com/paper/2312.01662/) (2023)

## Abstract (arXiv)

t-distributed Stochastic Neighborhood Embedding (t-SNE), a clustering and visualization method proposed by van der Maaten & Hinton in 2008, has rapidly become a standard tool in a number of natural sciences. Despite its overwhelming success, there is a distinct lack of mathematical foundations and the inner workings of the algorithm are not well understood. The purpose of this paper is to prove that t-SNE is able to recover well-separated clusters; more precisely, we prove that t-SNE in the `early exaggeration' phase, an optimization technique proposed by van der Maaten & Hinton (2008) and van der Maaten (2014), can be rigorously analyzed. As a byproduct, the proof suggests novel ways for setting the exaggeration parameter α and step size h. Numerical examples illustrate the effectiveness of these rules: in particular, the quality of embedding of topological structures (e.g. the swiss roll) improves. We also discuss a connection to spectral clustering methods.