# On the continuum limit of t-SNE for data visualization

arXiv: https://arxiv.org/abs/2604.12041 · HTML: https://mlsanity.com/paper/2604.12041/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 47 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 12 of 12)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [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)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [t-SNE, Forceful Colorings and Mean Field Limits](https://mlsanity.com/paper/2102.13009/) (2021)
- [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)
- [Convergence rates for Poisson learning to a Poisson equation with measure data](https://mlsanity.com/paper/2407.06783/) (2024)
- [t-SNE Exaggerates Clusters, Provably](https://mlsanity.com/paper/2510.07746/) (2025)
- [Attraction-Repulsion Swarming: A Generalized Framework of t-SNE via Force Normalization and Tunable Interactions](https://mlsanity.com/paper/2411.10617/) (2024)

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

- [On the Abundance of Critical Points of the t-SNE Energy](https://mlsanity.com/paper/2609.04379/) (2026)

## Abstract (arXiv)

This work is concerned with the continuum limit of a graph-based data visualization technique called the t-Distributed Stochastic Neighbor Embedding (t-SNE), which is widely used for visualizing data in a variety of applications, but is still poorly understood from a theoretical standpoint. The t-SNE algorithm produces visualizations by minimizing the Kullback-Leibler divergence between similarity matrices representing the high dimensional data and its low dimensional representation. We prove that as the number of data points n → ∞, after a natural rescaling and in applicable parameter regimes, the Kullback-Leibler divergence is consistent as the number of data points n → ∞ and the similarity graph remains sparse with a continuum variational problem that involves a non-convex gradient regularization term and a penalty on the magnitude of the probability density function in the visualization space. These two terms represent the continuum limits of the attraction and repulsion forces in the t-SNE algorithm. Due to the lack of convexity in the continuum variational problem, the question of well-posedeness is only partially resolved. We show that when both dimensions are 1, the problem admits a unique smooth minimizer, along with an infinite number of discontinuous minimizers (interpreted in a relaxed sense). This aligns well with the empirically observed ability of t-SNE to separate data in seemingly arbitrary ways in the visualization. The energy is also very closely related to the famously ill-posed Perona-Malik equation, which is used for denoising and simplifying images. We present numerical results validating the continuum limit, provide some preliminary results about the delicate nature of the limiting energetic problem in higher dimensions, and highlight several problems for future work.