arxiv: 2604.12041 · v1 · submitted 2026-04-13 · 📊 stat.ML · cs.LG· math.AP· math.ST· stat.TH

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On the continuum limit of t-SNE for data visualization

Jeff Calder , Zhonggan Huang , Ryan Murray , Adam Pickarski

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Pith reviewed 2026-05-10 15:05 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.APmath.STstat.TH

keywords t-SNEcontinuum limitKullback-Leibler divergencevariational problemdata visualizationnon-convex optimizationPerona-Malik equationgradient regularization

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The pith

As the dataset size tends to infinity, t-SNE's Kullback-Leibler objective converges after rescaling to a continuum variational problem with non-convex gradient regularization and a density penalty.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that t-SNE, which minimizes a discrete Kullback-Leibler divergence between high- and low-dimensional similarity matrices, admits a continuum limit as the number of points grows large. After a natural rescaling that keeps the similarity graph sparse, this limit becomes a variational energy whose two principal terms are the limits of the algorithm's attraction and repulsion forces. A reader would care because the limiting energy is non-convex and only partially well-posed, which accounts for t-SNE's observed tendency to produce separated visualizations in seemingly arbitrary configurations. The analysis further reveals that in one dimension the problem possesses a unique smooth minimizer together with infinitely many discontinuous minimizers, and that the energy is closely related to the Perona-Malik equation. These facts supply a theoretical explanation for the method's empirical behavior and identify concrete directions for future analysis.

Core claim

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 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, well-posedness is only partially resolved: when both dimensions are one the problem admits a unique smooth minimizer along with an infinite number of discontinuous minimizers interpreted in a relaxed sense.

What carries the argument

The continuum variational problem whose integrand combines a non-convex gradient regularization term with a penalty on the magnitude of the probability density function in the visualization space.

If this is right

The discrete attraction and repulsion forces of t-SNE correspond exactly to the gradient-regularization and density-penalty terms in the continuum limit.
In one dimension there exists a unique smooth minimizer and infinitely many discontinuous minimizers, matching the empirical capacity of t-SNE to separate data in arbitrary ways.
The limiting energy is closely related to the Perona-Malik equation, suggesting shared analytical and numerical features with image-denoising problems.
Numerical experiments with increasing data size confirm that the discrete t-SNE objective approaches the predicted continuum energy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The non-convexity of the limit may explain why t-SNE embeddings can depend strongly on initialization and hyper-parameter choices in practice.
In dimensions greater than one, studying relaxed or measure-valued minimizers could clarify how the method produces distinct cluster separations.
Numerical schemes developed for the Perona-Malik equation might be adapted to compute or regularize t-SNE embeddings at large scale.

Load-bearing premise

After natural rescaling the similarity graph remains sparse and the continuum limit of the discrete objective exists in the applicable parameter regimes.

What would settle it

Numerical simulations in which the t-SNE embeddings for successively larger point sets fail to approach the minimizers of the stated continuum energy would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 2604.12041 by Adam Pickarski, Jeff Calder, Ryan Murray, Zhonggan Huang.

**Figure 2.1.** Figure 2.1: Evolution of the t-SNE embedding over the iterations of gradient descent, taken from [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗

**Figure 2.2.** Figure 2.2: Embeddings obtained by using tSNE on points distributed on a sphere, taken from [ [PITH_FULL_IMAGE:figures/full_fig_p008_2_2.png] view at source ↗

**Figure 3.1.** Figure 3.1: The function Θ(v) for η(z) = 3 4 (1 − z 2 )+. v → ∞ (since ∫︁ η(z)dz = 1). The other equalities in the fourth assertion in the statement also follow the same line of reasoning, and the fourth statement is immediate. Next, using that the integrand in (3.8) is an even function of z, a direct computation yields Θ ′′(v) = 2 ∫︂ ∞ 0 η(z) 6vz2 − 2v 3 z 4 (1 + v 2z 2) 3 dz = 4v ∫︂ ∞ 0 η(z) d dz z 3 (1 + v 2z 2) … view at source ↗

**Figure 3.2.** Figure 3.2: Random Initialization. Top row compares Tn and T while the bottom row compares T ′ n and T ′ . We increase the separation between the Gaussian clusters from left to right. We consider a family of densities of the form ρX(x) = p(Gσ(x − c) + Gσ(x + c)) + 1 2 (1 − 2p), on the interval [−1, 1], where Gσ is the standard normal probability density with variance σ 2 . This is a mixture of two Gaussians with mea… view at source ↗

**Figure 3.3.** Figure 3.3: Identity Initialization. Top row compares Tn and T while the bottom row compares T ′ n and T ′ . We increase the separation between the Gaussian clusters from left to right. −1.0 −0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t-SNE Continuum Limit −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t-SNE Continuum Limit −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t-SNE Continuum Limit −1.0 … view at source ↗

**Figure 3.4.** Figure 3.4: Continuum Limit Initialization. Top row compares Tn and T while the bottom row compares T ′ n and T ′ . We increase the separation between the Gaussian clusters from left to right. away from the discontinuities, the absolute value of the slope |T ′ n (x)| matches moderately well with the continuum limit |T ′ (x)|, though we see wild fluctuations. Figures 3.3 and 3.4 show the same results except with the … view at source ↗

**Figure 3.5.** Figure 3.5: Comparison of t-SNE loss during optimization for random initialization, identity ini [PITH_FULL_IMAGE:figures/full_fig_p035_3_5.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a continuum limit for t-SNE's KL objective as a non-convex variational energy with a gradient term and density penalty, plus a 1D result on multiple minimizers.

read the letter

The main point is that t-SNE's discrete objective converges, under a natural rescaling and when the similarity graph stays sparse, to a continuum variational problem whose two main terms come from the attraction and repulsion forces. One is a non-convex gradient regularizer and the other penalizes large density values in the low-dimensional space. They also show that in one dimension the problem has a unique smooth minimizer but infinitely many discontinuous ones in a relaxed sense, and they note the link to the Perona-Malik equation. That last observation is useful because it ties the embedding behavior to a known ill-posed model from image processing. The 1D multiplicity result is new and lines up with how t-SNE can separate points in flexible ways. The asymptotic analysis itself looks direct and avoids circular arguments by working from the graph KL divergence outward. Numerical checks are mentioned and seem to support the limit in simple settings. The main limitation is that everything is conditioned on parameter regimes where the rescaled graph remains sparse. Standard t-SNE uses fixed perplexity, so the bandwidth and embedding scaling have to be tuned just right; if they are off, the limit functional changes or the graph becomes nonlocal. The paper does not give a full map of those regimes, especially when the original data dimension is greater than one. Well-posedness is only partial beyond 1D, and the higher-dimensional numerics are described as preliminary. This is the sort of work that would interest people studying theoretical foundations of embedding algorithms or continuum limits of graph-based methods. A reader who already cares about variational problems in machine learning or the Perona-Malik equation would get concrete value from the explicit energy and the 1D analysis. I would send it to peer review. The core derivation is new enough and the approach careful enough that referees should see it, even if the higher-dimensional cases need more work to be fully convincing.

Referee Report

3 major / 2 minor

Summary. The paper analyzes the continuum limit of t-SNE by proving that, as n→∞ after natural rescaling and in regimes where the similarity graph remains sparse, the discrete KL divergence between high- and low-dimensional similarities converges to a continuum variational energy consisting of a non-convex gradient regularization term (from attraction) and a density penalty (from repulsion). It establishes that the 1D limit problem admits a unique smooth minimizer together with infinitely many discontinuous minimizers (in a relaxed sense), links the energy to the Perona-Malik equation, supplies numerical validation of the limit, and offers preliminary observations on the higher-dimensional case.

Significance. If the consistency result is fully established, the work supplies a rigorous asymptotic foundation for t-SNE that explains its empirical clustering behavior and connects a widely used visualization method to a known ill-posed variational problem. The explicit 1D uniqueness statement and the numerical checks are concrete strengths; the partial higher-D analysis correctly identifies open questions rather than overclaiming.

major comments (3)

[Abstract, §1] Abstract and §1: the consistency statement is conditioned on 'applicable parameter regimes' in which the rescaled similarity graph remains sparse, yet no explicit bounds on perplexity, bandwidth scaling, or embedding dimension are supplied to delineate these regimes. This assumption is load-bearing for the limit derivation and for applicability to standard fixed-perplexity t-SNE.
[§4] §4 (1D well-posedness): while uniqueness of the smooth minimizer is shown, the relaxed formulation for discontinuous minimizers is only sketched; a precise statement of the relaxation (e.g., which space and which notion of convergence) is needed to confirm that the infinite family of minimizers is indeed attained by the discrete t-SNE dynamics.
[§5] §5 (higher dimensions): the preliminary numerical observations on non-uniqueness and ill-posedness are useful, but the absence of even a partial well-posedness result or a clear statement of the technical obstacles (beyond non-convexity) leaves the central claim about the continuum limit incomplete for d>1.

minor comments (2)

Notation for the rescaled bandwidth and embedding coordinates should be introduced once and used consistently; several passages reuse symbols without redefinition.
The numerical section would benefit from a table or plot showing the rate at which the discrete KL approaches the continuum energy as n increases, rather than qualitative agreement only.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and indicate the revisions we plan to incorporate.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1: the consistency statement is conditioned on 'applicable parameter regimes' in which the rescaled similarity graph remains sparse, yet no explicit bounds on perplexity, bandwidth scaling, or embedding dimension are supplied to delineate these regimes. This assumption is load-bearing for the limit derivation and for applicability to standard fixed-perplexity t-SNE.

Authors: We agree that the manuscript would benefit from clearer delineation of the regimes. In the revision we will add explicit asymptotic conditions in the abstract and §1 (e.g., perplexity scaling slower than n and bandwidth chosen so that the rescaled graph remains locally finite) under which the sparsity assumption holds for typical data distributions. Deriving fully sharp, distribution-independent bounds is technically demanding and lies beyond the present scope; we will flag this as future work. revision: partial
Referee: [§4] §4 (1D well-posedness): while uniqueness of the smooth minimizer is shown, the relaxed formulation for discontinuous minimizers is only sketched; a precise statement of the relaxation (e.g., which space and which notion of convergence) is needed to confirm that the infinite family of minimizers is indeed attained by the discrete t-SNE dynamics.

Authors: The referee correctly identifies that the relaxed formulation is only sketched. We will revise §4 to give a precise statement: the energy is extended to the space of probability measures equipped with weak-* convergence, and we will prove that the relaxed functional is the Gamma-limit of the discrete energies. This will make the connection to the discrete t-SNE dynamics rigorous. revision: yes
Referee: [§5] §5 (higher dimensions): the preliminary numerical observations on non-uniqueness and ill-posedness are useful, but the absence of even a partial well-posedness result or a clear statement of the technical obstacles (beyond non-convexity) leaves the central claim about the continuum limit incomplete for d>1.

Authors: We acknowledge that the higher-dimensional analysis remains preliminary. In the revision we will expand §5 with an explicit list of obstacles (lack of a maximum principle, possible mass concentration, and failure of standard compactness arguments) and will state that no partial well-posedness result is currently available for d>1. The continuum-limit derivation itself is independent of well-posedness and holds under the same sparsity assumption. revision: partial

Circularity Check

0 steps flagged

No circularity: direct asymptotic analysis of discrete KL to continuum energy

full rationale

The paper derives the continuum limit of the t-SNE objective via standard asymptotic analysis of the discrete Kullback-Leibler divergence as n→∞ under rescaling, showing convergence to a variational problem with gradient regularization and density penalty terms. This proceeds from the finite-n graph-based objective to integral functionals without any parameter fitting, self-referential definitions, or load-bearing self-citations; the 1D well-posedness is established directly from the limiting energy, and higher-dimensional issues are left open. The sparsity assumption defines the regime of applicability but does not reduce the claimed consistency result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The limit proof relies on standard assumptions for graph Laplacians and kernel densities in the large-n regime; no new free parameters or invented entities are introduced beyond the rescaling already stated in the abstract.

axioms (2)

domain assumption Data points are sampled from a continuous probability density allowing a well-defined continuum limit of the similarity graph
Invoked to pass from discrete KL divergence to the continuum variational problem as n→∞.
domain assumption The similarity kernel bandwidth and perplexity parameters are chosen so that the graph remains sparse in the limit
Required for the rescaling to produce a non-trivial continuum energy rather than a trivial or dense limit.

pith-pipeline@v0.9.0 · 5615 in / 1397 out tokens · 34331 ms · 2026-05-10T15:05:04.182120+00:00 · methodology

discussion (0)

Reference graph

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