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REVIEW 5 major objections 5 minor 10 references

Riemannian Principal Component Analysis

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Riemannian PCA built on UMAP local distances preserves cluster geometry better than standard PCA.

desk verdict A simple UMAP-weighted PCA recipe is presented as Riemannian PCA, but the Riemannian structure is never actually defined; the heuristic itself is reproducible and may be useful. read the letter →

arxiv 2506.00226 v1 pith:TA33BBC3 submitted 2025-05-30 stat.ML cs.LGmath.STstat.COstat.TH

Oldemar Rodríguez This is my paper
classification stat.MLcs.LGmath.STstat.COstat.TH MSC 62H2562R30
keywords RiemannianPCAUMAPlocaldistancemanifoldstatisticsdimensionreductioncorrelationcircleprincipalgeodesicanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to generalize principal component analysis so it can be applied to any data table, not just points in flat Euclidean space. The proposed Riemannian PCA (R-PCA) uses the local similarities that the UMAP algorithm builds between nearby points as a Riemannian metric, then redefines vector subtraction, the mean, covariance, and correlation in terms of that metric. With this machinery, the algorithm finds principal directions that respect local distances while still maximizing projected variance. On a simulated dataset with nested circular and parabolic clusters, R-PCA keeps the clusters visibly intact in the principal plane and explains 43.38% of the variance, against 25.58% for standard PCA; on the Olivetti Faces image dataset it separates the 40 individuals more cleanly and explains 39.59% of inertia versus 39.22%. If the construction is sound, it would extend manifold-based statistics to any kind of data.

What carries the argument

The load-bearing mechanism is the symmetric UMAP graph and its edge weights $B_{ij} = S_{\mathrm{UMAP}}(x_i, x_j)$, interpreted as local geometric affinities. From these weights the paper defines a subtraction operation $x_\alpha \ominus x_\beta = (1 - S_{\mathrm{UMAP}}(x_\alpha, x_\beta))(x_\alpha - x_\beta)$, a UMAP distance $d_{\mathrm{UMAP}}$, a Riemannian mean $g$ as the minimizer of the sum of squared UMAP distances, and a covariance matrix $S$ built from the rescaled residuals $x_i \ominus g$. Eigendecomposition of the corresponding correlation matrix yields the R-PCA principal directions and variances. The nerve theorem and Čech complex are invoked to justify that the UMAP graph reflects the data's topological structure, which is what licenses calling the local affinities a manifold structure.

What would settle it

Compute the Riemannian mean from Definition 6 on a small dataset by direct numerical minimization of the sum of squared UMAP distances and check whether it ever coincides with one of the data points, as Definition 7 asserts; then run R-PCA with the UMAP similarity matrix replaced by a random symmetric matrix with the same row sums and see whether the reported cluster-preservation advantage survives. If the advantage does survive, the improvement is not coming from the manifold structure.

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Extended reading notes

Core claim

The paper's central claim is that equipping any data table with UMAP-induced local distances converts it into a Riemannian manifold in a workable sense, and that rebuilding PCA on top of those distances yields a method—Riemannian PCA—that outperforms standard PCA in preserving the geometric structure of clusters and in explaining variance. The key step is Definition 5, where the difference between two rows is rescaled by $\rho_{\alpha\beta} = 1 - S_{\mathrm{UMAP}}(x_\alpha, x_\beta)$, so that close points (similarity near 1) are pulled together and distant points are pushed apart. Definitions 6 and 7 then define a Riemannian mean and a covariance matrix from these rescaled differences, and the principal directions are the eigenvectors of the resulting correlation matrix. The paper demonstrates the method on a 2900-by-10 simulated table and on the Olivetti Faces dataset, reporting better cluster separation and higher explained inertia for R-PCA in both settings.

Load-bearing premise

The method's load-bearing premise is that the UMAP similarity scores between points really do define a legitimate local Riemannian metric, so that rescaling each difference by $1 - S_{\mathrm{UMAP}}$ is a valid manifold subtraction and the minimizer of the resulting sum of squared distances is one of the data points; if either part fails, R-PCA is just ordinary PCA on a UMAP-shrunk data matrix.

Editorial extensions

If this is right

  • R-PCA can be applied to any data table, because UMAP supplies the local distance structure even when no manifold is known in advance.
  • In the simulated 10-dimensional data, the R-PCA principal plane keeps the nested circular clusters and the parabolic cluster visually distinct, whereas standard PCA produces more diffuse clusters.
  • The R-PCA correlation circle highlights the original variables x and y more sharply than the PCA correlation circle on the simulated data.
  • On Olivetti Faces, R-PCA separates the 40 individuals more clearly in the principal plane and explains 39.59% of inertia in the first plane versus 39.22% for PCA.
  • Because the same recipe could be reused with any affinity graph, the framework opens a route to Riemannian versions of other multivariate methods such as regression and k-means.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A comparison the paper does not report: replacing UMAP affinities with t-SNE affinities or a Gaussian kernel would show whether the cluster-preservation gain is specific to UMAP or generic to any smooth local similarity.
  • Because the covariance is built from rescaled Euclidean differences, R-PCA may be equivalent to a weighted PCA on the original data; spelling out that equivalence would clarify what the 'Riemannian' label adds.
  • The same 'local affinities then covariance eigendecomposition' recipe could be applied to supervised problems, with the metric informed by class labels, but the paper does not explore that direction.
  • Direct minimization of the Riemannian-mean objective on a small example would verify whether the mean always lands on a data point, as Definition 7 assumes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes 'Riemannian Principal Component Analysis' (R-PCA), an extension of PCA that, instead of assuming Euclidean structure, equips a data table with local distances derived from the UMAP nearest-neighbor graph. The method defines a UMAP-induced subtraction (Definition 5), a Riemannian mean (Definition 6), a covariance and correlation matrix (Definition 7), and then extracts eigenvectors from the correlation matrix (Algorithm 1). The authors report results on a simulated 10-dimensional dataset with five clusters and on the Olivetti Faces dataset, claiming that R-PCA preserves cluster geometry better than PCA and explains more variance in the first principal plane. Section 1 argues that this generalizes PGA to arbitrary data tables without an intrinsic manifold structure.

Significance. If the central claim were supported, R-PCA would be a notable contribution: it would provide a principled way to apply PCA to arbitrary data equipped with local metrics, extending PGA beyond datasets with known manifold structure. The manuscript also points to a Python package, which is a practical asset. However, the mathematical premise is not established: the quantity d_UMAP is not shown to be a Riemannian distance, the 'Riemannian mean' is defined only over data points without proof, and the reported empirical advantages are assessed visually and through a non-comparable explained-variance measure. The paper does not provide quantitative clustering metrics, sensitivity analyses, or external validation, so the claimed outperformance is not demonstrated. The contribution as presented is a heuristic reweighting of Euclidean PCA using UMAP affinities rather than a Riemannian generalization.

major comments (5)
  1. [Section 3, Definition 5 and Algorithm 1 (steps 3–7)] The quantity d_UMAP(x_alpha, x_beta) = ||(1 - S_UMAP(x_alpha, x_beta))(x_alpha - x_beta)|| is not a Riemannian distance. The manuscript does not define a smooth manifold with a smoothly varying inner product on tangent spaces, and S_UMAP can equal 1 for distinct points, so d_UMAP(x_alpha, x_beta) can be zero. The rescaling factor (1 - S_UMAP) depends on the pair (alpha,beta), so the triangle inequality can fail; for example, two points connected with high affinity can have small d_UMAP, while a third point is far from both. Therefore the covariance matrix in Definition 7 is a reweighted Euclidean covariance built from UMAP affinities, not a Riemannian covariance. This undermines the central claim in Sections 3 and 5 that the method generalizes PCA to Riemannian manifolds.
  2. [Section 3, Definition 7 (and Algorithm 1, step 7)] The statement 'note that g must be equal to x_lambda for some lambda' is a restriction, not a proved theorem. Definition 6 defines the Riemannian mean as a minimizer over the manifold M induced by the simplicial complex, but Algorithm 1, step 7 computes the mean over the finite data set. There is no proof that the minimizer of the sum of d_UMAP^2 over any larger set is attained at one of the observed data points, and the manuscript gives no argument that restricting to data points preserves the intended Fréchet mean. Since every subsequent quantity (covariance, correlation, principal directions) is centered at g, the output of the algorithm depends on this unproved restriction, which is load-bearing for the method's validity.
  3. [Section 3, paragraph after Algorithm 1; Sections 4.1 and 4.2] The recommended choice k = floor(n/c), where c is the number of clusters, introduces a circularity into the empirical evaluation. The same cluster count c that is later used to judge success (e.g., c = 5 for Data10D and c = 40 for Olivetti Faces) sets the UMAP neighborhood size and therefore controls the local affinities that define the R-PCA projection. A method whose free parameter is tuned using the cluster labels cannot be said to 'preserve' those clusters independently. The paper reports no sensitivity analysis over k or over the UMAP parameters rho_i and sigma_i, so the claimed cluster preservation may be a direct consequence of the chosen parameterization.
  4. [Sections 4.1, 4.2 and 5] The empirical comparison between PCA and R-PCA is not quantitative in a valid sense. The results are assessed visually (Figures 5–8 and 10–11), and the reported explained inertia (43.38% vs. 25.58%; 39.59% vs. 39.22%) is not a meaningful basis for comparison because the two methods compute variances in different spaces and use different normalization (Algorithm 1 extracts eigenvectors of the correlation matrix, while Definition 7 defines a covariance matrix). No clustering metric such as adjusted Rand index or normalized mutual information is reported, and no repeated runs or confidence intervals are given. Therefore the conclusion in Section 5 that 'R-PCA outperforms the standard PCA in preserving the geometric structure of the clusters' is not supported by the presented evidence.
  5. [Algorithm 1, steps 8–10; Definition 7] The algorithm mixes covariance and correlation without justification. Definition 7 defines the variance-covariance matrix S, but Algorithm 1, step 9 computes the correlation matrix R and step 10 extracts eigenvectors and eigenvalues of R. This choice changes the principal directions and variances, especially when variable scales differ, and the manuscript does not state whether the reported results use S or R. This ambiguity affects the reproducibility of every experiment and the interpretation of the explained variance values.
minor comments (5)
  1. [Section 2, Definition 4] The definition of a Riemannian metric is incomplete: it omits the smoothness condition on the family of inner products and uses 'Fixed x' without defining how the inner product varies from point to point.
  2. [Section 2, UMAP graph construction] The paragraph describing the Nerve Theorem and the Čech complex suggests a one-to-one relation between data and simplicial complexes, but no precise statement is given that connects the UMAP symmetrization in equation (1) to a Riemannian metric; the reader is left to infer the link.
  3. [Section 3, Definition 5] The phrase 'we define the the addition induced by the UMAP algorithm' contains a typo, and the definition of addition as x_alpha = x_beta ⊕ x_gamma if x_gamma = x_alpha ⊖ x_beta is not shown to be well-defined or consistent across different pairs.
  4. [Section 4.2] The figures for Olivetti Faces (Figures 9–11) are discussed qualitatively; adding quantitative measures of cluster separation would make the comparison more informative.
  5. [References] Reference [5] contains garbled text ('6, 352–357' after the arXiv identifier), and the GitHub link in reference [9] has a space in the repository name ('riemannian stats'), which likely breaks the URL.

Circularity Check

2 steps flagged · score 6.0 of 10

R-PCA is UMAP-reweighted PCA renamed 'Riemannian', and its reported cluster-preservation advantage is partly forced by choosing k from the same cluster count used for evaluation.

  1. fitted input called prediction [Section 3 (k recommendation), Section 4.1 (k=580), Section 5 (conclusion)]
    "In practice, in algorithm 1 it is recommended to use at least k=⌊n/c⌋ (the whole part) where c is the number of clusters that the data table is suspected to have, or the number of clusters that you want to study, that is, k is at least the average number of individuals that each cluster has. ... Here we are computing R-PCA with k=⌊2900/5⌋=580."

    The same cluster count c that is later used to judge success is fed into the algorithm as the hyperparameter k, which controls the UMAP neighborhood graph and therefore determines S_UMAP, ρ, dUMAP, the mean g, and the covariance. In Section 5 the paper then reports that 'R-PCA outperforms the standard PCA in preserving the geometric structure of the clusters.' The claimed advantage is thus partly a consequence of tuning the method with the target number of clusters, not an independent prediction from the data alone.

  2. renaming known result [Definition 5, Definitions 6-7, Algorithm 1 steps 3-10]
    "Definition 5. Let x_α and x_β rows of X, we define the subtraction induced by the UMAP algorithm as x_α ⊖ x_β = ρ_αβ (x_α − x_β), where ρ_αβ =1−S_UMAP(x_α,x_β). ... Definition 7 ... S = 1/n Σ_i (x_i ⊖ g)(x_i ⊖ g)^t ... Algorithm 1: Compute the Riemannian mean g using D ... Compute the Riemannian variance-covariance matrix S."

    No smooth manifold, tangent space, or Riemannian metric tensor is ever constructed; dUMAP is just a pointwise scalar rescaling of Euclidean distances by UMAP affinities. The covariance and eigenvectors are exactly those of PCA on the rescaled vectors (1−S_UMAP(x_i,g))(x_i−g). Calling this 'Riemannian Principal Component Analysis' renames UMAP-weighted PCA rather than deriving a generalization from a Riemannian structure. Any geometric property of the output is therefore tied by definition to the initial UMAP weights, not to an independently verified manifold structure.

full rationale

The paper does not rely on a load-bearing self-citation chain: the only self-citation is a software link in reference [9], and the methodological citations to UMAP, PGA, and t-SNE are external. The circularity is structural instead. Definition 5 reduces the claimed Riemannian distance to a UMAP-affinity rescaling of Euclidean distance, and Algorithm 1 diagonalizes the covariance built from exactly those rescaled differences; no manifold or metric tensor is constructed, so the word 'Riemannian' is a renaming of the reweighting. In the empirical section, k is set to floor(n/c) using the same cluster count c that is later used to declare superior cluster preservation, so the main empirical advantage is partly an artifact of the input choice. These issues make the central claims partially circular by construction, but the algorithm is fully specified and could in principle be tested with independent hyperparameter selection on external benchmarks, so the score is moderate rather than maximal.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The central method depends on UMAP's graph affinities as a stand-in for a Riemannian metric, on the unproved existence of a data-point mean, and on a hand-chosen neighbor count k. These are the quantities that, if wrong, would invalidate the claimed R-PCA result.

free parameters (2)
  • k (number of UMAP nearest neighbors) = 580 for Data10D; not explicitly stated for Olivetti, recommended as floor(n/c)
    Controls the locality of UMAP's metric and therefore the weights in the R-PCA covariance. It is set from the suspected or known number of clusters, so it is chosen by hand for each dataset.
  • rho_i and sigma_i (UMAP local connectivity and scale) = computed per point from k-NN distances
    These define the edge weights S_UMAP and hence the R-PCA scaling; they are fitted to the data by UMAP's optimization, not derived from first principles.
assumptions (4)
  • domain assumption The data table is assumed to lie on a Riemannian manifold, and UMAP's k-NN graph with local metrics gives a valid approximation of that manifold.
    Assumed in Section 2 when UMAP is used to endow the data with local distance notions; no proof that the fuzzy simplicial set weights are a Riemannian metric.
  • standard math The Nerve Theorem and Cech complex construction justify a one-to-one correspondence between data and the simplicial complex, enabling the definition of vector subtraction on the manifold.
    Invoked in Section 2 after Theorem 1 and Corollary 1; the theorem itself is standard, but the conclusion drawn for data operations is an extrapolation.
  • domain assumption The Riemannian mean g in Definition 6 exists, is unique, and is one of the data points x_lambda.
    Stated in Definition 7 without proof; needed to compute the covariance with (x_i ⊖ g) = rho_{i lambda}(x_i - g).
  • domain assumption The symmetrized UMAP adjacency matrix B = A + A^T - A∘A^T defines a valid global similarity that can serve as (1 - rho) in a metric.
    Used in Definition 5; the fuzzy union operation is from the UMAP paper but is not shown to satisfy triangle inequality or any metric axiom.
invented entities (2)
  • UMAP-induced subtraction ⊖
    purpose: Defines x_alpha ⊖ x_beta = rho_alpha_beta (x_alpha - x_beta) to build a covariance matrix on the manifold.
    Introduced ad hoc in Definition 5; no external falsifiable prediction or independent validation.
  • Riemannian mean constrained to data points
    purpose: Serves as the center g for the covariance computation.
    Definition 6 and 7 introduce this without proof of existence or uniqueness; it is not validated against any known geometric quantity.

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Cite this review

Pith. "Pith review of Riemannian Principal Component Analysis." pith.science (2026). https://pith.science/paper/TA33BBC3

@misc{pith2026250600226,
  author       = {Pith},
  title        = {Pith review of: Riemannian Principal Component Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TA33BBC3}},
  note         = {Machine review of arXiv:2506.00226}
}
read the original abstract

This paper proposes an innovative extension of Principal Component Analysis (PCA) that transcends the traditional assumption of data lying in Euclidean space, enabling its application to data on Riemannian manifolds. The primary challenge addressed is the lack of vector space operations on such manifolds. Fletcher et al., in their work {\em Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape}, proposed Principal Geodesic Analysis (PGA) as a geometric approach to analyze data on Riemannian manifolds, particularly effective for structured datasets like medical images, where the manifold's intrinsic structure is apparent. However, PGA's applicability is limited when dealing with general datasets that lack an implicit local distance notion. In this work, we introduce a generalized framework, termed {\em Riemannian Principal Component Analysis (R-PCA)}, to extend PGA for any data endowed with a local distance structure. Specifically, we adapt the PCA methodology to Riemannian manifolds by equipping data tables with local metrics, enabling the incorporation of manifold geometry. This framework provides a unified approach for dimensionality reduction and statistical analysis directly on manifolds, opening new possibilities for datasets with region-specific or part-specific distance notions, ensuring respect for their intrinsic geometric properties.

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Reference graph

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