# Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data

arXiv: https://arxiv.org/abs/2105.07536 · HTML: https://mlsanity.com/paper/2105.07536/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 16 May 2021.

## Builds on (most cited first, 13 of 13)

- [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)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [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)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [t-viSNE: Interactive Assessment and Interpretation of t-SNE Projections](https://mlsanity.com/paper/2002.06910/) (2020)
- [Approximated and User Steerable tSNE for Progressive Visual Analytics](https://mlsanity.com/paper/1512.01655/) (2015)
- [t-SNE, Forceful Colorings and Mean Field Limits](https://mlsanity.com/paper/2102.13009/) (2021)
- [Stochastic Neighbor Embedding under f-divergences](https://mlsanity.com/paper/1811.01247/) (2018)
- [Stochastic Neighbor Embedding separates well-separated clusters](https://mlsanity.com/paper/1702.02670/) (2017)
- [Tree-SNE: Hierarchical Clustering and Visualization Using t-SNE](https://mlsanity.com/paper/2002.05687/) (2020)
- [Spectral Clustering Revisited: Information Hidden in the Fiedler Vector](https://mlsanity.com/paper/2003.09969/) (2020)
- [Dimensionality Reduction has Quantifiable Imperfections: Two Geometric Bounds](https://mlsanity.com/paper/1811.00115/) (2018)

## Built on by (most cited first, 30 of 55)

- [Your Contrastive Learning Is Secretly Doing Stochastic Neighbor Embedding](https://mlsanity.com/paper/2205.14814/) (2022)
- [Think-Then-Generate: Reasoning-Aware Text-to-Image Diffusion with LLM Encoders](https://mlsanity.com/paper/2601.10332/) (2026)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Symmetric Graph Contrastive Learning against Noisy Views for Recommendation](https://mlsanity.com/paper/2408.02691/) (2024)
- [A Spectral Method for Assessing and Combining Multiple Data Visualizations](https://mlsanity.com/paper/2210.13711/) (2022)
- [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)
- [Not Just What's There: Enabling CLIP to Comprehend Negated Visual Descriptions Without Fine-tuning](https://mlsanity.com/paper/2602.21035/) (2026)
- [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)
- [Laplacian-based Cluster-Contractive t-SNE for High Dimensional Data Visualization](https://mlsanity.com/paper/2207.12214/) (2022)
- [GaitSADA: Self-Aligned Domain Adaptation for mmWave Gait Recognition](https://mlsanity.com/paper/2301.13384/) (2023)
- [Strada-LLM: Graph LLM for traffic prediction](https://mlsanity.com/paper/2410.20856/) (2024)
- [t-SNE Exaggerates Clusters, Provably](https://mlsanity.com/paper/2510.07746/) (2025)
- [What Makes Position Zero Special? A Mechanistic Study of Position Zero Attention Sinks in LLMs](https://mlsanity.com/paper/2603.06591/) (2026)
- [CWD30: A Comprehensive and Holistic Dataset for Crop Weed Recognition in Precision Agriculture](https://mlsanity.com/paper/2305.10084/) (2023)
- [Sailing in high-dimensional spaces: Low-dimensional embeddings through angle preservation](https://mlsanity.com/paper/2406.09876/) (2024)
- [Learning Joint ID-Textual Representation for ID-Preserving Image Synthesis](https://mlsanity.com/paper/2504.14202/) (2025)
- [Efficient and Robust Bayesian Selection of Hyperparameters in Dimension Reduction for Visualization](https://mlsanity.com/paper/2306.00357/) (2023)
- [Subject clustering by IF-PCA and several recent methods](https://mlsanity.com/paper/2306.05363/) (2023)
- [Towards Transparency: Exploring LLM Trainings Datasets through Visual Topic Modeling and Semantic Frame](https://mlsanity.com/paper/2406.06574/) (2024)
- [Areas of Improvement for Autonomous Vehicles: A Machine Learning Analysis of Disengagement Reports](https://mlsanity.com/paper/2408.00051/) (2024)
- [Enhanced document retrieval with topic embeddings](https://mlsanity.com/paper/2408.10435/) (2024)
- [Stochastic neighborhood embedding and the gradient flow of relative entropy](https://mlsanity.com/paper/2409.16963/) (2024)
- [ScholarBench: A Bilingual Benchmark for Abstraction, Comprehension, and Reasoning Evaluation in Academic Contexts](https://mlsanity.com/paper/2505.16566/) (2025)
- [Unsupervised Learning for Industrial Defect Detection: A Case Study on Shearographic Data](https://mlsanity.com/paper/2511.02541/) (2025)
- [BERT-JEPA: Reorganizing CLS Embeddings for Language-Invariant Semantics](https://mlsanity.com/paper/2601.00366/) (2026)
- [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 investigates the theoretical foundations of the t-distributed stochastic neighbor embedding (t-SNE) algorithm, a popular nonlinear dimension reduction and data visualization method. A novel theoretical framework for the analysis of t-SNE based on the gradient descent approach is presented. For the early exaggeration stage of t-SNE, we show its asymptotic equivalence to power iterations based on the underlying graph Laplacian, characterize its limiting behavior, and uncover its deep connection to Laplacian spectral clustering, and fundamental principles including early stopping as implicit regularization. The results explain the intrinsic mechanism and the empirical benefits of such a computational strategy. For the embedding stage of t-SNE, we characterize the kinematics of the low-dimensional map throughout the iterations, and identify an amplification phase, featuring the intercluster repulsion and the expansive behavior of the low-dimensional map, and a stabilization phase. The general theory explains the fast convergence rate and the exceptional empirical performance of t-SNE for visualizing clustered data, brings forth interpretations of the t-SNE visualizations, and provides theoretical guidance for applying t-SNE and selecting its tuning parameters in various applications.