On the continuum limit of t-SNE for data visualization

• URL: http://arxiv.org/abs/2604.12041v1
• Date: Mon, 13 Apr 2026 20:32:28 GMT
• Title: On the continuum limit of t-SNE for data visualization
• Abstract summary: This work is concerned with the limit of a graph-based data visualization technique called the t-Distributed Neighbor Embedding (t-SNE)<n>We prove that as the number of data points $nto infty$, the Kullback-Leibler divergence is consistent as the number data points $nto infty$.
• Score: 1.8549313085249324
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Manifold limit for the training of shallow graph convolutional neural networks [1.2744523252873352]
  We study the consistency of the training of shallow graph convolutional neural networks (GCNNs) on proximity graphs of sampled point clouds.<n>We prove $$-convergence of regularized empirical risk minimization functionals and corresponding convergence of their global minimizers.
  arXiv  Detail & Related papers  (2026-01-09T18:59:20Z)
• Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization [49.85875869048434]
  We propose a CP-based low-rank tensor function parameterized by neural networks for implicit neural representation.<n>To achieve sparser CP decomposition, we introduce a variational Schatten-p quasi-norm to prune redundant rank-1 components.<n>For smoothness, we propose a regularization term based on the spectral norm of the Jacobian and Hutchinson's trace estimator.
  arXiv  Detail & Related papers  (2025-06-27T11:23:10Z)
• Large data limits and scaling laws for tSNE [1.2085509610251701]
  We show that embeddings of the original tSNE algorithm cannot have any consistent limit as $n to in$. We propose a rescaled model which mitigates the decay of the attractive energy, and which does have a consistent limit.
  arXiv  Detail & Related papers  (2024-10-16T21:43:02Z)
• Learning Discretized Neural Networks under Ricci Flow [48.47315844022283]
  We study Discretized Neural Networks (DNNs) composed of low-precision weights and activations.<n>DNNs suffer from either infinite or zero gradients due to the non-differentiable discrete function during training.
  arXiv  Detail & Related papers  (2023-02-07T10:51:53Z)
• Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery [52.21846313876592]
  Low-rank tensor function representation (LRTFR) can continuously represent data beyond meshgrid with infinite resolution. We develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. Our method substantiates the superiority and versatility of our method as compared with state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-12-01T04:00:38Z)
• High-dimensional limit theorems for SGD: Effective dynamics and critical scaling [6.950316788263433]
  We prove limit theorems for the trajectories of summary statistics of gradient descent (SGD) We show a critical scaling regime for the step-size, below which the effective ballistic dynamics matches gradient flow for the population loss. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate.
  arXiv  Detail & Related papers  (2022-06-08T17:42:18Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)
• Quantitative Propagation of Chaos for SGD in Wide Neural Networks [39.35545193410871]
  In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Gradient Descent (SGD) We show 'propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios. We identify two under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand.
  arXiv  Detail & Related papers  (2020-07-13T12:55:21Z)
• Conditional gradient methods for stochastically constrained convex minimization [54.53786593679331]
  We propose two novel conditional gradient-based methods for solving structured convex optimization problems. The most important feature of our framework is that only a subset of the constraints is processed at each iteration. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees.
  arXiv  Detail & Related papers  (2020-07-07T21:26:35Z)
• Dynamic Regret of Convex and Smooth Functions [93.71361250701075]
  We investigate online convex optimization in non-stationary environments. We choose the dynamic regret as the performance measure. We show that it is possible to further enhance the dynamic regret by exploiting the smoothness condition.
  arXiv  Detail & Related papers  (2020-07-07T14:10:57Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• The Heavy-Tail Phenomenon in SGD [7.366405857677226]
  We show that depending on the structure of the Hessian of the loss at the minimum, the SGD iterates will converge to a emphheavy-tailed stationary distribution. We translate our results into insights about the behavior of SGD in deep learning.
  arXiv  Detail & Related papers  (2020-06-08T16:43:56Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.