Some Theoretical Limitations of t-SNE

Rupert Li; Elchanan Mossel

cs.LGApr 14, 2026

Some Theoretical Limitations of t-SNE

Authors: Rupert Li, Elchanan Mossel

Abstract

t-SNE has gained popularity as a dimension reduction technique, especially for visualizing data. It is well-known that all dimension reduction techniques may lose important features of the data. We provide a mathematical framework for understanding this loss for t-SNE by establishing a number of results in different scenarios showing how important features of data are lost by using t-SNE.

Figures & tables

Panel of Figure 1 — Figure 1. t-SNE and PCA on points clustered around the 10 vertices of a regular 9-simplex. Here, the vertices are the ten elementary basis vectors in $\mathbb{R}^{10}$ , and each cluster has 100 points sampled i.i.d. from a spherical Gaussian with standard deviation $0.2$ in each direction.

Panel of Figure 2 — Figure 2. t-SNE on points drawn from the unit sphere $\mathbb{S}^{1}\subset\mathbb{R}^{2}$ .

Panel of Figure 3 — Figure 3. t-SNE on points drawn from the unit sphere $\mathbb{S}^{2}\subset\mathbb{R}^{3}$ .

Panel of Figure 4 — Figure 4. t-SNE on points drawn from the unit sphere $\mathbb{S}^{4}\subset\mathbb{R}^{5}$ .

Panel of Figure 5 — Figure 5. t-SNE on points drawn from the unit sphere $\mathbb{S}^{19}\subset\mathbb{R}^{20}$ .

Panel of Figure 6 — Figure 6. t-SNE on points drawn from the unit sphere $\mathbb{S}^{99999}\subset\mathbb{R}^{100000}$ .

Panel of Figure 7 — Figure 7. t-SNE on 1000 points drawn from the unit sphere $\mathbb{S}^{19}\subset\mathbb{R}^{20}$ conditioned on the first coordinate having magnitude at least $20^{-0.1}\approx 0.74$ .

Dimensionality Reduction

Dimensionality reduction maps high-dimensional data to compact representations for visualization, prediction, compression, and scientific modeling.

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On the continuum limit of t-SNE for data visualization

Jeff Calder, Zhonggan Huang, Ryan Murray, Adam Pickarski

School of Mathematics, University of Minnesota · Department of Mathematics, University of Utah · Department of Mathematics, North Carolina State University

Data Visualization Dimensionality Reduction

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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 \to \infty

, after a natural rescaling and in applicable parameter regimes, the Kullback-Leibler divergence is consistent as the number of data points

n \to \infty

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.

Abstract

Dimensionality Reduction

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