REVIEW 3 major objections 6 minor 4 references
Pearson Distance is not a Distance
T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper proves that Pearson distance, defined as one minus the Pearson correlation coefficient, is not a metric because it can violate the triangle inequality, while its square root and the square root of one minus the squared…
desk verdict Pearson distance is indeed not a metric, and the paper's explicit counterexamples are useful, but the mathematical message is mostly repackaged known results and one proof contains a repairable gap. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine of the positive results is the identity $\mathbb{E}(X-Y)^2=2(1-\rho_{xy})$ for zero-mean unit-variance variables, which identifies $\sqrt{1-\rho_{xy}}$ with the $L^2$ distance up to the constant $1/\sqrt2$, so the $L^2$ triangle inequality transfers directly. For the sign-invariant metric, the paper uses a product construction: if $X$ and $Y$ are independent vectors with the same correlation matrix, then the products $Z_i=X_iY_i$ have correlations $\rho_{ij}^2$, so $\sqrt{1-\rho_{ij}^2}$ is the $L^2$ distance of those products. The non-metric results rest on an explicit three-variable construction that reduces the triangle inequality to a trigonometric inequality, and the generalization section applies two metric-preserving-function rules: strictly convex functions through the origin do not preserve metrics, while strictly increasing concave functions through the origin do.
What would settle it
Compute the paper's construction at $\theta=\pi/6$: let $U,V,W$ be zero-mean with $\mathrm{var}(V)=\mathrm{var}(W)=1/4$, $\mathrm{var}(U)=3/4$, $\mathrm{corr}(V,W)=-3/4$, and $U$ independent of $V,W$; set $X=U+V$, $Y=U+W$, $Z=U$. Then $\rho_{xy}=\cos^4(\pi/6)=9/16$, $\rho_{xz}=\rho_{yz}=\cos(\pi/6)=\sqrt3/2$, so $1-\rho_{xy}=7/16\approx0.4375$ while $2(1-\rho_{xz})=2-\sqrt3\approx0.2679$, violating the triangle inequality. Directly computing these correlations settles whether the paper's central claim is right.
Extended reading notes
Core claim
The central claim is that the dissimilarity $1-\rho_{xy}$ fails the metric axiom. The counterexample uses zero-mean variables $X=U+V$, $Y=U+W$, $Z=U$, where $V$ and $W$ have variance $\sin^2\theta$ and correlation $-\cos^2\theta$, $U$ has variance $\cos^2\theta$, and $X,Y$ end up with unit variance. This yields $\rho_{xy}=\cos^4\theta$ and $\rho_{xz}=\rho_{yz}=\cos\theta$, so the triangle inequality for Pearson distance becomes $1-\cos^4\theta > 2(1-\cos\theta)$, which holds for a wide range of $\theta$, including all $\theta\le\pi/4$. The same construction shows $1-|\rho_{xy}|$ is also not a metric. In the positive direction, $\sqrt{1-\rho_{xy}}$ is a metric because $\mathbb{E}(X-Y)^2 = 2(1-\rho_{xy})$ and the $L^2$ norm satisfies the triangle inequality, and $\sqrt{1-\rho_{xy}^2}$ is a metric by a product-copy argument that reduces it to the first case.
Load-bearing premise
The load-bearing premise is that every random variable involved has a well-defined Pearson correlation, which requires finite second moments and nonzero variance; the paper normalizes to zero mean and unit variance without stating this regularity condition explicitly.
Editorial extensions
If this is right
- Clustering pipelines that use Pearson distance as a dissimilarity measure can produce incoherent clusters, because the triangle inequality is not guaranteed: two variables can both be near a third while being far from each other.
- Replacing Pearson distance with $\sqrt{1-\rho_{xy}}$ restores the metric property while preserving the ordering of pairwise dissimilarities, so users only need to apply a square root to their existing matrix.
- For analyses that treat anti-correlation as similarity, $1-|\rho_{xy}|$ has the same defect, and $\sqrt{1-\rho_{xy}^2}$ is the metric replacement.
- The metric-preservation results imply that further transformations, such as $(1-\rho_{xy})^{1/4}$, are also metrics, giving a family of correlation-based distances with different geometrical behaviour.
- Because the proofs require only finite second moments, the metric and non-metric conclusions apply to any collection of real-valued random variables with well-defined correlations, not just to specific distributions.
Reading between the lines
- The paper's counterexample is not a razor-thin edge case: the violation holds for all $\theta \le \pi/4$ and beyond, so for smoothly varying data one should expect triangle-inequality failures of Pearson distance to be common rather than rare in practice.
- A testable extension is to audit real datasets, such as gene-expression or multi-voxel brain-imaging inputs, for triangle-inequality violations under Pearson distance; if violations are frequent, the square-root replacement would change cluster assignments in a way current practice misses.
- The product-copy argument indicates that the metric property of $\sqrt{1-\rho^2}$ should hold empirically for any correlation matrix estimated from data with finite variances, independent of the underlying distribution.
- The family of concave transformations opens a modelling choice: different concave powers compress large dissimilarities differently, which changes cluster shape, so selecting among them is a substantive decision rather than a technicality.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies four correlation-based dissimilarity measures between random variables: Pearson dissimilarity 1-ρ_xy, |P|earson dissimilarity 1-|ρ_xy|, sqrt-Pearson dissimilarity √(1-ρ_xy), and P2earson dissimilarity √(1-ρ_xy^2). The main claims are that 1-ρ_xy and 1-|ρ_xy| are not metrics, while √(1-ρ_xy) and √(1-ρ_xy^2) are metrics. The negative results are supported by explicit counterexamples built from a latent variable construction (Results II and IV). The positive result for √(1-ρ_xy) is proved by an elementary L2 triangle-inequality argument (Result I). The positive result for √(1-ρ_xy^2) is proved by constructing Z_k = X_kY_k from two independent random vectors with identical correlation matrices and then applying Result I (Result III). The appendix sketches general results from Gower and Legendre (1986) on similarity matrices and from the metric-preserving function literature.
Significance. If the proof issue in Result III is corrected, the paper's main conclusions are correct and practically useful: it gives explicit, easily reproducible counterexamples showing that Pearson dissimilarity can violate the triangle inequality by large margins, and it identifies simple alternatives that are genuine metrics. The elementary proofs are accessible to a broad statistical audience, and the paper correctly connects the √(1-ρ) result to the older work of Gower and Legendre. The explicit counterexamples are falsifiable and parameterized over large ranges, which strengthens the message. The main weakness is the incorrect factorization in the proof of Result III, which is load-bearing for the claim that √(1-ρ^2) is a metric; this is repairable but must be fixed.
major comments (3)
- [Section 3, Result III proof] The displayed computation of the correlation of Z_i and Z_j contains an invalid factorization. The paper writes E(Z_iZ_j) = E(X_iY_iX_jY_j) = E(X_iY_i)E(X_jY_j) = ρ_ij^2. This is wrong under the stated assumptions: since the X-vector and Y-vector are independent, E(X_iY_i) = E(X_i)E(Y_i) = 0 and likewise E(X_jY_j) = 0, so the product on the right is 0, not ρ_ij^2. The correct derivation is E(X_iY_iX_jY_j) = E(X_iX_j)E(Y_iY_j) = ρ_ij ρ_ij = ρ_ij^2, using independence of the two vectors. Because this step is the only justification that corr(Z_i,Z_j)=ρ_ij^2, the proof of Result III as written is incomplete. The underlying claim is true and the gap is easily repaired, but the proof must be corrected.
- [Section 1, definition of dissimilarity] The paper defines a dissimilarity as satisfying strict positivity with 'equality iff i = j', but then lists 1-|ρ_ij| as a dissimilarity. For X and Y with Y = -X (after centering and scaling), one has |ρ_xy| = 1, so 1-|ρ_xy| = 0 even though the variables are not equal. This contradicts the stated axiom (ii). The authors should either relax the axiom (for example, to equality iff |correlation| = 1, treating variables as equivalent up to sign and affine transformation) or explicitly acknowledge that 1-|ρ| is not a dissimilarity in the paper's strict sense. This issue affects the framing of Result IV.
- [Appendix, Result II*] The appendix states that no strictly convex function of √(1-ρ_ij) or of √(1-ρ_ij^2) that passes through the origin is a metric, and claims this follows from Corazza's theorem M1. That theorem says a strictly convex function with g(0)=0 is not metric preserving, meaning there exists some metric for which g∘d violates the triangle inequality. It does not, by itself, imply that a counterexample exists within the restricted class of metrics of the form √(1-ρ_ij) or √(1-ρ_ij^2). Supporting Result II* requires an argument that for every such strictly convex g the construction of Results II/IV (or a similar one) yields a violation within the correlation class. As written, the appendix overstates what the cited theorem establishes.
minor comments (6)
- [Section 1, Result I proof] The proof assumes zero mean and unit variance without stating that this is without loss of generality. Since correlations and the dissimilarities 1-ρ and √(1-ρ) are invariant to standardization, the proof should explicitly say that the variables are standardized first; otherwise the identity E(X-Y)^2 = 2(1-ρ_xy) does not hold for variables with arbitrary means and variances.
- [Throughout] Section numbering is inconsistent: the introduction refers to 'section II', 'section III', and 'section IV', while the actual section headers are numbered 1, 2, 3, and 4. Use one numbering convention consistently.
- [Section 2, Result II proof] The text says 'This certainly holds for |θ| ≤ π/4', but θ is defined on (0, π/2); the intended statement is '0 < θ ≤ π/4'.
- [Section 3, Result IV proof] The first sentence says 'We use the construction from Theorem 2.2'; this should refer to 'Result II', not 'Theorem 2.2'.
- [Throughout] The manuscript contains numerous typographical errors and OCR artifacts, including 'New So uth Wales', 'clu ster', 'fuctions' in the Corazza reference, missing spaces after commas in the reference list, 'Not that' for 'Note that', and '1N' for '1, ..., N'. A thorough proofreading is needed.
- [Notation] The notation for correlations is inconsistent: both ρ_xy and ρx,y appear. Please choose one form and use it consistently.
Circularity Check
No significant circularity; the derivations are self-contained proofs or restatements of independent external theorems.
full rationale
The paper makes mathematical claims about dissimilarity measures and proves them directly. Result I is an elementary proof of the triangle inequality for sqrt(1 - rho_xy); Result II and Result IV are explicit counterexamples showing violations for 1 - rho_xy and 1 - |rho_xy|; Result III relies on a construction from Innocenti and Materassi (2008), a reference by other authors, and the appendix invokes independent results of Gower and Legendre (1986) and Corazza (1999). There is no fitting of parameters, no prediction made from data, and no load-bearing self-citation: Victor Solo's own prior work is not the source of any central premise. The invalid displayed factorization in the proof of Result III is a mathematical typographical or derivation error, not a circular step: the claimed conclusion does not depend on assuming the conclusion, and the intended identity can be repaired by using independence of the X-vector and Y-vector. Thus there is no circular reasoning warranting a nonzero score, and concerns about the typo belong to correctness, not circularity.
Assumptions & free parameters
assumptions (6)
- standard math Random variables can be standardized to zero mean and unit variance without changing Pearson correlation.
- standard math For any positive semidefinite correlation matrix there exist random variables realizing it (e.g., Gaussian construction).
- standard math For independent random vectors X and Y, E(X_i Y_i X_j Y_j) = E(X_i X_j) E(Y_i Y_j).
- standard math Gower-Legendre: if the similarity matrix [s_ij] is positive semidefinite, then sqrt(1 minus s_ij) is a metric.
- standard math Corazza: a strictly convex function with f(0)=0 is not metric preserving.
- domain assumption All random variables have finite, nonzero variance so Pearson correlation is defined.
Cite this review
Pith. "Pith review of Pearson Distance is not a Distance." pith.science (2026). https://pith.science/paper/O5V757YN
@misc{pith2026190806029,
author = {Pith},
title = {Pith review of: Pearson Distance is not a Distance},
year = {2026},
howpublished = {\url{https://pith.science/paper/O5V757YN}},
note = {Machine review of arXiv:1908.06029}
}
abstract
The Pearson distance between a pair of random variables $X,Y$ with correlation $\rho_{xy}$, namely, 1-$\rho_{xy}$, has gained widespread use, particularly for clustering, in areas such as gene expression analysis, brain imaging and cyber security. In all these applications it is implicitly assumed/required that the distance measures be metrics, thus satisfying the triangle inequality. We show however, that Pearson distance is not a metric. We go on to show that this can be repaired by recalling the result, (well known in other literature) that $\sqrt{1-\rho_{xy}}$ is a metric. We similarly show that a related measure of interest, $1-|\rho_{xy}|$, which is invariant to the sign of $\rho_{xy}$, is not a metric but that $\sqrt{1-\rho_{xy}^2}$ is. We also give generalizations of these results.
Reference graph
Works this paper leans on
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[1]
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Gibbons, F. and Roth, F. (2002), ‘Judging the quality of gene expre ssion-based clustering methods using gene annotation’, Genome Research 12, 15741581. Gower, J. (1966), ‘Some distance properties of latent root and ve ctor methods used in multivariate analysis’, Biometrika 53, 325–338. Gower, J. (1967), ‘A comparison of some methods of cluster analys is’...
work page 2002
Reviewed August 14, 2026 · model on record in the stance chip above.
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