On the continuum limit of t-SNE for data visualization

2026-04-13

AI Summary

This work addresses the unclear theoretical foundations of t-SNE in large-scale settings by establishing, for the first time, a continuous variational formulation of t-SNE in the limit as the number of data points tends to infinity. The authors derive the continuum limits of the attractive and repulsive forces inherent in t-SNE and uncover a profound connection between this limiting problem and the Perona–Malik equation. Through a combination of variational analysis, probabilistic density modeling, sparse graph limit theory, and non-convex optimization techniques, they rigorously prove that the Kullback–Leibler divergence, under appropriate scaling, converges to a continuous energy functional. In the one-dimensional case, the structure of the solution is fully characterized, and numerical experiments corroborate the predicted asymptotic behavior, while also highlighting significant theoretical challenges that remain in higher dimensions.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• t-SNE
• continuum limit
• Kullback-Leibler divergence
• variational problem
• well-posedness

Innovation

• continuum limit
• t-SNE
• non-convex variational problem
• Kullback-Leibler divergence
• Perona-Malik equation