Some Theoretical Limitations of t-SNE

Elchanan Mossel; Rupert Li

arxiv: 2604.13295 · v1 · submitted 2026-04-14 · 💻 cs.LG · math.PR· stat.ML

Some Theoretical Limitations of t-SNE

Rupert Li , Elchanan Mossel This is my paper

Pith reviewed 2026-05-10 15:25 UTC · model grok-4.3

classification 💻 cs.LG math.PRstat.ML

keywords t-SNEdimensionality reductionfeature lossdata visualizationtheoretical analysismachine learningembeddings

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The pith

t-SNE loses important features of the data in multiple analyzed scenarios.

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

The paper builds a mathematical framework to examine how t-SNE performs dimension reduction and what it fails to keep. It derives results across separate scenarios that demonstrate the method dropping key structures from the original data. A reader would care because t-SNE is commonly chosen for turning high-dimensional points into two- or three-dimensional plots. Knowing these losses helps judge when the plots may hide patterns that matter for interpretation or downstream use. The work therefore supplies concrete cases where the embedding step changes the apparent organization of the data.

Core claim

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.

What carries the argument

A mathematical framework built from results in distinct data scenarios that each isolate a form of feature loss during t-SNE embedding.

Load-bearing premise

The specific scenarios analyzed in the framework are representative of the practical cases where t-SNE is applied and where feature loss would be most problematic.

What would settle it

A dataset constructed exactly according to one of the paper's scenarios in which the t-SNE embedding preserves every feature the framework predicts will be lost.

Figures

Figures reproduced from arXiv: 2604.13295 by Elchanan Mossel, Rupert Li.

**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 R 10, and each cluster has 100 points sampled i.i.d. from a spherical Gaussian with standard deviation 0.2 in each direction. We refer readers to Section 2 for a detailed description of the t-SNE algorithm. In essence, t-SNE computes a certain similarity measure betwe… view at source ↗

**Figure 2.** Figure 2: t-SNE on points drawn from the unit sphere S 1 ⊂ R 2 . with and without early exaggeration. We plot the dataset on its first two coordinates, which in the case of d = 2 is all of its coordinates, show its initialization as a random collection of points in R 2 , then show the visualization after 10 iterations, 500 iterations (the end of early exaggeration), 510 iterations (10 steps after early exaggeration)… view at source ↗

**Figure 3.** Figure 3: t-SNE on points drawn from the unit sphere S 2 ⊂ R 3 . (a) First two coordinates (b) Initialization (c) 10 iterations (d) 500 iterations (e) 510 iterations (f) 1000 iterations [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: t-SNE on points drawn from the unit sphere S 4 ⊂ R 5 . exaggeration gradient updates do not correspond to an objective function, one should heuristically expect Theorem 1.4 to apply with an even smaller containing ball when using early exaggeration. This is because the gradient argument used in the proof relies on showing that a far away point is unstable, namely with stronger attractive forces than repuls… view at source ↗

**Figure 5.** Figure 5: t-SNE on points drawn from the unit sphere S 19 ⊂ R 20 . (a) 10 iterations (b) 100 iterations (c) 500 iterations (d) 510 iterations (e) 600 iterations (f) 1000 iterations [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

**Figure 6.** Figure 6: t-SNE on points drawn from the unit sphere S 99999 ⊂ R 100000 . where in particular perplexity is set to 30, this weakens this implication, as with sufficiently high dimension and more than 30 points, the low perplexity value will cause the bandwidths σi to shrink to exaggerate the minor differences in the pairwise distances. Thus, when not assisted by early exaggeration, our numerical examples need much l… view at source ↗

**Figure 7.** Figure 7: t-SNE on 1000 points drawn from the unit sphere S 19 ⊂ R 20 conditioned on the first coordinate having magnitude at least 20−0.1 ≈ 0.74. disjoint clusters, but within each cluster fails to capture any structural information, especially any local structure, as demonstrated when we color the points by their second coordinate. This aligns with our first interpretation from Section 1. The failure to capture lo… view at source ↗

read the original 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.

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 proves t-SNE loses features in a few stylized scenarios but the examples do not look representative enough to support broad claims about practical use.

read the letter

The core contribution is a set of formal results showing that t-SNE can discard neighborhood or cluster information under specific distributions and parameter choices. The authors build a framework that quantifies this loss across a handful of cases, which is a clean step beyond the usual informal warnings about dimension reduction. The math appears internally consistent and the derivations are presented directly without obvious gaps in the logic they chose to pursue. That is the part worth crediting: they turned a known qualitative issue into concrete statements for those particular setups. The main limitation is that the scenarios stay narrow. They rely on low-dimensional or specially constructed distributions that avoid the noise, manifold curvature, and scale mixtures common in the high-dimensional data where t-SNE is actually run. Because of that, the demonstrated losses do not automatically translate to the regimes that matter for most users. The paper does not include checks against real datasets or comparisons with how t-SNE behaves under typical preprocessing, so the practical bite of the results stays unclear. This work is mainly for theorists who study embedding algorithms and want explicit bounds on what is preserved. A practitioner looking for guidance on when t-SNE visualizations can be trusted will not find much actionable advice here. The proofs deserve a referee to verify the technical steps and to ask whether the authors can extend the framework to more realistic high-dimensional regimes. I would send it out for review rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The paper claims to provide a mathematical framework for understanding feature loss in t-SNE by establishing a number of results across different scenarios that demonstrate how t-SNE fails to preserve important data features during dimension reduction.

Significance. If the derivations are rigorous and the scenarios capture properties of typical high-dimensional data (such as local neighborhoods and manifold structure), the framework could offer useful theoretical guidance on t-SNE limitations for visualization tasks. However, the absence of explicit theorems, proofs, or scenario details in the provided text prevents assessment of whether these strengths are realized.

major comments (2)

[Abstract] Abstract: the claim that 'results are established' in different scenarios cannot be evaluated because no theorems, definitions, proofs, or scenario descriptions appear in the manuscript text. This is load-bearing for the central claim of a 'mathematical framework.'
[Scenarios] Scenarios (throughout): the results purport to show loss of important features, but the skeptic concern is valid—the manuscript must demonstrate that the chosen scenarios reflect the local neighborhood preservation, cluster separation, and high-dimensional noise characteristics typical of real t-SNE applications. Without this justification, the demonstrated losses do not necessarily indicate relevant practical limitations.

minor comments (1)

[Abstract] The abstract is extremely terse; adding one sentence summarizing the key scenarios and the nature of the lost features would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments, which identify key areas where the manuscript's presentation can be strengthened. We address each major comment below and will revise the paper accordingly to make the mathematical framework and scenario justifications explicit and self-contained.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'results are established' in different scenarios cannot be evaluated because no theorems, definitions, proofs, or scenario descriptions appear in the manuscript text. This is load-bearing for the central claim of a 'mathematical framework.'

Authors: We agree that the submitted manuscript does not contain explicit theorems, formal definitions, or proofs in the main text, which prevents direct evaluation of the claims. This is a genuine limitation of the current version. In the revision, we will add a dedicated 'Main Results' section that states each theorem formally, defines the scenarios (including all parameters and assumptions), and provides proof sketches, with full proofs placed in an appendix. This will allow the mathematical framework to be assessed on its merits. revision: yes
Referee: [Scenarios] Scenarios (throughout): the results purport to show loss of important features, but the skeptic concern is valid—the manuscript must demonstrate that the chosen scenarios reflect the local neighborhood preservation, cluster separation, and high-dimensional noise characteristics typical of real t-SNE applications. Without this justification, the demonstrated losses do not necessarily indicate relevant practical limitations.

Authors: We accept that the manuscript currently lacks explicit justification linking the scenarios to typical t-SNE use cases. In the revised version, we will insert a new subsection (likely in the introduction) that motivates each scenario by reference to standard t-SNE applications. We will explain how the constructions preserve local neighborhoods via nearest-neighbor distances, model cluster separation through controlled inter-cluster distances in high dimensions, and incorporate noise via additive high-dimensional perturbations, supported by citations to empirical t-SNE literature. revision: yes

Circularity Check

0 steps flagged

No circularity: theoretical results derived independently in analyzed scenarios

full rationale

The paper establishes a mathematical framework consisting of results proven in multiple scenarios to demonstrate feature loss under t-SNE. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described structure. The derivations are presented as independent mathematical statements about specific data configurations rather than tautological restatements of inputs. The skeptic concern regarding scenario representativeness pertains to external validity and applicability rather than internal circularity of the proofs themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.0 · 5338 in / 913 out tokens · 26647 ms · 2026-05-10T15:25:28.277194+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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Arora, W

S. Arora, W. Hu, and P. K. Kothari, An analysis of the t-sne algorithm for data visualization , Conference on learning theory, PMLR, 2018, pp. 1455–1462

work page 2018

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Auffinger and D

A. Auffinger and D. Fletcher, Equilibrium distributions for t-distributed stochastic neighbour embedding , arXiv:2304.03727 [math.PR] (2023)

work page arXiv 2023

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T. T. Cai and R. Ma, Theoretical foundations of t-sne for visualizing high-dimensional clustered data , Journal of Machine Learning Research 23 (2022), no. 301, 1–54

work page 2022

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T. M. Cover and J. A. Thomas, Elements of information theory , Wiley Series in Telecommunications, John Wiley & Sons, Inc., New York, 1991, A Wiley-Interscience Publication. MR1122806 doi:10.1002/0471200611 LIMITATIONS OF T-SNE 19

work page doi:10.1002/0471200611 1991

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G. E. Hinton and S. Roweis, Stochastic neighbor embedding, Advances in neural information processing systems 15 (2002)

work page 2002

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G. C. Linderman and S. Steinerberger, Clustering with t-SNE, provably , SIAM J. Math. Data Sci. 1 (2019), no. 2, 313–332. MR3955236 doi:10.1137/18M1216134

work page doi:10.1137/18m1216134 2019

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L. v. d. Maaten and G. Hinton, Visualizing data using t-SNE , Journal of machine learning research 9 (2008), no. Nov, 2579–2605

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