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arxiv: 2605.13520 · v2 · pith:7IQ5ROQXnew · submitted 2026-05-13 · ❄️ cond-mat.stat-mech · cs.LG

Beyond Explained Variance: A Cautionary Tale of PCA

Gionni Marchetti This is my paper

Pith reviewed 2026-05-20 21:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.LG
keywords PCAt-SNEpersistent homologyfossil teethmanifold learningdimensionality reductionKuehneotheriumcosine distances
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The pith

PCA scatterplots can falsely indicate clusters in data that actually forms a ring-like manifold.

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

The paper examines how principal component analysis can produce misleading two-dimensional visualizations for data that lies on a nonlinear low-dimensional manifold. Using a dataset of fossil teeth from the early mammal Kuehneotherium, the authors show that a published PCA plot appears to show clusters, yet t-SNE and persistent homology analysis instead recover a ring structure with intrinsic dimension one and no evident groups. They introduce a simple generative model in which points are drawn uniformly from a unit circle; under this model the pairwise cosine distances obey an arcsine distribution that matches the U-shaped pattern seen in the data. This geometric account provides independent support for the manifold interpretation and highlights a general limitation of relying on explained variance alone when choosing visualization methods.

Core claim

For the Kuehneotherium fossil teeth data, PCA produces an apparent clustering in the region of negative PC2, whereas t-SNE and persistent homology recover a single ring of intrinsic dimension one; a generative model that samples points uniformly from the unit circle reproduces the observed U-shaped distribution of pairwise cosine distances.

What carries the argument

The generative probabilistic-geometric model of uniform sampling from a unit circle, which directly yields the arcsine distribution for pairwise cosine distances.

If this is right

  • PCA scatterplots alone are insufficient for revealing the geometry of data on nonlinear manifolds such as circles.
  • Manifold-aware methods like t-SNE combined with topological tools can expose ring structures and low intrinsic dimension where PCA suggests clusters.
  • The arcsine law for cosine distances offers a simple, model-based check on whether data lie on a circular manifold.

Where Pith is reading between the lines

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

  • The same caution about PCA may apply to other biological or morphological datasets that are suspected to vary continuously around a closed shape.
  • Distance-distribution diagnostics could be added to standard pipelines to flag when a circular or periodic embedding is plausible before visualization.
  • Persistent homology loops detected in the data could be tested against the specific persistence diagram expected from uniform sampling on a circle.

Load-bearing premise

The fossil teeth data can be treated as points sampled uniformly from a circle in an appropriate embedding space.

What would settle it

A statistical test showing that the empirical distribution of pairwise cosine distances deviates significantly from the arcsine law predicted by uniform circle sampling.

Figures

Figures reproduced from arXiv: 2605.13520 by Gionni Marchetti.

Figure 1
Figure 1. Figure 1: FIG. 1: All computations are performed on the standardized fossil teeth data. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: , while the corresponding estimated parameters are reported in Table I. For completeness, we also re￾port the Akaike Information Criterion (AIC) [10, 47] and the Bayesian Information Criterion (BIC) [10, 48] for the two models. We obtain AIC ≈ 6011 and BIC ≈ 6042 for the GMM, whereas AIC ≈ 5127 and BIC ≈ 5159 for the BMM. Accordingly, the Beta mixture model provides the preferred fit to the empirical data,… view at source ↗
read the original abstract

We address shortcomings of principal component analysis (PCA) for visualizing high-dimensional data lying on a nonlinear low-dimensional manifold via two-dimensional scatterplots, focusing on a fossil teeth dataset from the early mammalian insectivore Kuehneotherium. While the PCA scatterplot reported by Jolliffe and Cadima (Philosophical Transactions of the Royal Society A, 2016) shows clustering in the region where PC2 < 0, our analysis based on t-SNE and persistent homology (PH) reveals a ring-like structure with no evident clustering and intrinsic dimensionality equal to one. We further propose a generative probabilistic-geometric model in which the data are sampled uniformly from a unit circle. Under this model, pairwise cosine distances follow an arcsine distribution, in qualitative agreement with the observed U-shaped distribution, thereby independently supporting the analysis based on t-SNE and persistent homology.

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.

Referee Report

3 major / 2 minor

Summary. The paper claims that PCA can mislead when visualizing high-dimensional data on nonlinear manifolds, using a fossil teeth dataset from Kuehneotherium. While PCA scatterplots from Jolliffe and Cadima (2016) suggest clustering where PC2 < 0, t-SNE and persistent homology reveal a ring-like structure with no clustering and intrinsic dimension 1. The authors introduce a generative model of uniform sampling from a unit circle, under which pairwise cosine distances follow an arcsine distribution that qualitatively matches the observed U-shaped distribution and independently supports the topological findings.

Significance. If the ring structure and geometric model are confirmed, this provides a concrete cautionary example of PCA limitations for manifold data in statistical mechanics and paleobiology. The integration of t-SNE, persistent homology, and a probabilistic-geometric model is a strength, offering a template for interpreting distance distributions beyond linear variance explained.

major comments (3)
  1. Abstract: the statement that the arcsine distribution 'independently supports' the t-SNE/PH analysis is undercut by the fact that the generative model is constructed from the observed ring structure; without separate verification of uniform angular sampling in the embedding space, the support is not independent.
  2. Generative probabilistic-geometric model: the derivation of the arcsine distribution for cosine distances assumes uniform sampling from a unit circle, but no quantitative test (e.g., Rayleigh test for uniformity or KS statistic against empirical distances) is provided to check conformity of the fossil teeth data to this geometry.
  3. t-SNE and persistent homology sections: the claims of ring-like structure and intrinsic dimensionality equal to one rest on visual inspection of outputs without reported quantitative metrics such as persistence diagram summaries, stability across hyperparameters, or error bars.
minor comments (2)
  1. Figure captions for t-SNE and PH plots should explicitly note the parameters (e.g., perplexity, homology dimension) used to identify the ring and dimension-1 features.
  2. The reference to the original PCA application could be expanded with the exact dataset size or preprocessing steps to allow direct reproduction of the clustering observation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. We have carefully reviewed each major comment and provide point-by-point responses below, indicating where revisions will be made to address the concerns raised.

read point-by-point responses

  1. Referee: Abstract: the statement that the arcsine distribution 'independently supports' the t-SNE/PH analysis is undercut by the fact that the generative model is constructed from the observed ring structure; without separate verification of uniform angular sampling in the embedding space, the support is not independent.

    Authors: We agree that the phrasing 'independently supports' may suggest a stronger separation than is warranted, given that the generative model is motivated by the ring structure observed in the t-SNE and persistent homology results. The arcsine distribution offers a geometric and probabilistic explanation consistent with the topological findings, but it does not provide fully independent verification without additional checks on angular uniformity. We will revise the abstract to remove 'independently' and instead describe the model as providing 'further geometric support consistent with' the t-SNE and PH analysis. revision: yes

  2. Referee: Generative probabilistic-geometric model: the derivation of the arcsine distribution for cosine distances assumes uniform sampling from a unit circle, but no quantitative test (e.g., Rayleigh test for uniformity or KS statistic against empirical distances) is provided to check conformity of the fossil teeth data to this geometry.

    Authors: The referee correctly notes the absence of quantitative goodness-of-fit or uniformity tests in the current manuscript. Although the qualitative match between the theoretical arcsine distribution and the empirical U-shaped histogram is evident, formal statistical validation would increase rigor. In the revision we will add a Kolmogorov-Smirnov test comparing the observed cosine distances to the arcsine distribution and report the test statistic and p-value. We will also discuss the feasibility of estimating angular coordinates from the embedding to perform a uniformity test such as the Rayleigh test. revision: yes

  3. Referee: t-SNE and persistent homology sections: the claims of ring-like structure and intrinsic dimensionality equal to one rest on visual inspection of outputs without reported quantitative metrics such as persistence diagram summaries, stability across hyperparameters, or error bars.

    Authors: We acknowledge that the current presentation relies primarily on visual assessment of the t-SNE plots and persistence diagrams. To address this, the revised manuscript will include quantitative summaries of the persistence diagrams (such as the lifespan of the dominant H1 feature), results demonstrating stability of the ring structure across a range of t-SNE hyperparameters (e.g., perplexity values), and any available measures of variability or bootstrap-based assessments supporting the intrinsic dimension of one. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation chain is self-contained

full rationale

The paper identifies a ring-like structure and intrinsic dimension one via t-SNE and persistent homology on the fossil teeth data. It then proposes a generative model of uniform sampling from a unit circle and derives the arcsine distribution for pairwise cosine distances as a mathematical consequence. This distribution is noted to qualitatively match the observed U-shaped pattern in the data. No step reduces by construction to its inputs: the geometric model is not defined using the distance distribution, the arcsine result follows from standard geometry rather than a fit, and the match serves as an external consistency check rather than a renamed input. No self-citations, uniqueness theorems, or smuggled ansatzes appear in the provided text. The central claims rest on independent visualization and topological methods plus a derived geometric implication, satisfying the criteria for a non-circular analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The analysis assumes the data manifold is one-dimensional and ring-shaped; the generative model is postulated without external validation beyond the qualitative distance match.

axioms (1)
  • domain assumption High-dimensional data lies on a nonlinear low-dimensional manifold
    Invoked to explain why PCA (linear) is insufficient and why topological methods are needed.
invented entities (1)
  • Generative model of uniform sampling from a unit circle no independent evidence
    purpose: To reproduce the observed U-shaped cosine distance distribution and support the ring structure
    New probabilistic-geometric construction introduced in the paper; no independent evidence such as a predicted observable outside the current dataset is provided.

pith-pipeline@v0.9.0 · 5671 in / 1314 out tokens · 33587 ms · 2026-05-20T21:14:21.503883+00:00 · methodology

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