Equilibrium Distribution for t-Distributed Stochastic Neighbor Embedding with Generalized Kernels

Yi Gu; Antonio Auffinger

stat.MLMay 30, 2025

Equilibrium Distribution for t-Distributed Stochastic Neighbor Embedding with Generalized Kernels

Authors: Yi Gu, Antonio Auffinger

Organizations: Department of Mathematics, Northwestern University

Abstract

We extend the results by Auffinger and Fletcher in 2023 on the large-sample limit of t-SNE to a broader class of input and output kernels. We prove that under natural assumptions on the input distribution and the decay of the output kernel, if perplexity scales linearly with data size, the empirical measures and values of minimizers converge to an equilibrium distribution. We conclude with numerical examples comparing different kernels.

Figures & tables

Output kernel	Type of decay	Restriction on $s$
$(1+r^{2}/\alpha)^{-\alpha}$	Regular, $\beta=2\alpha$	$s\leq 2\alpha$
$(1+r^{4})^{-1}$	Regular, $\beta=4$	$s\leq 4$
$e^{-r^{2}}$ and $e^{-r^{4}}$	Rapid	All $s$
$e^{2-2\sqrt{1+r^{2}}}$	Rapid	All $s$
$e^{-r^{2}-0.8\sin(r^{2})}$	Rapid	All $s$

Table 1: Examples satisfying Conditions 2 and 4 .

Name	$k(r)$
Student-type	$(1+r^{2}/\alpha)^{-\alpha}$ , $\alpha\in\{1/2,1,2\}$
Gaussian	$e^{-r^{2}}$
Quartic exponential	$e^{-r^{4}}$
Quartic rational	$(1+r^{4})^{-1}$
Smooth exponential	$e^{2-2\sqrt{1+r^{2}}}$
Rippled Gaussian	$e^{-r^{2}-0.8\sin(r^{2})}$

Table 2: The eight output kernels used in the simulations.

Figure 1: Two-dimensional outputs for $3000$ observations from $N(0,I_{50})$ , with $\textbf{Perp}=150$ . Rows vary the input kernel; columns vary the output kernel. The same observations and initial coordinates are used in every panel. Panel (a) is t-SNE and panel (b) is symmetric SNE. Axes show output coordinates after centering and orthogonal alignment, without a change of scale.

Figure 2: The same kernel combinations as in Figure 1 , for the balanced three-component Gaussian mixture in $\mathbb{R}^{50}$ . There are $3000$ observations and $\textbf{Perp}=150$ . Colors identify the generating component. All panels reuse the same sample and initialization.

Figure 3: MNIST: the same $3000$ images, represented by their $784$ pixel values, embedded in two dimensions under twelve kernel combinations. There are $300$ images per digit and $\textbf{Perp}=150$ . Colors denote digit labels and are fixed across panels. The layout follows Figure 1 .

Figure 4: Fashion-MNIST: $3000$ images in $\mathbb{R}^{784}$ , with $300$ images per clothing class and $\textbf{Perp}=150$ . Rows, columns, initialization, and display conventions match Figure 3 . Class colors are shown in the legend.

Figure 5: Original SNE, Student-type $\alpha=1/2$ output, and quartic rational output on the four main datasets. All panels use Gaussian input, $n=3000$ , and $\textbf{Perp}=150$ . The first column uses ( 6.2 ); the others use the joint objective. Colors agree with the preceding figures. Full-view insets, where present, include remote points outside the central view.

Figure 6: Changing perplexity with the t-SNE kernels fixed. Each row uses the same $3000$ image vectors and initial coordinates as the corresponding main grid. The central views and any full-view insets retain the original output scales. Class colors follow Figures 3 and 4 .

Dimensionality Reduction

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

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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

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