Metrics for Inter-Dataset Similarity with Example Applications in Synthetic Data and Feature Selection Evaluation -- Extended Version

Anton D. Lautrup; Arthur Zimek; Muhammad Rajabinasab

REVIEW 4 major objections 6 minor 45 references

Metrics for Inter-Dataset Similarity with Example Applications in Synthetic Data and Feature Selection Evaluation -- Extended Version

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Two PCA-based metrics, the difference in explained variance and the angle between first principal components, are claimed to give deterministic, model-independent measurements of inter-dataset similarity.

desk verdict Useful PCA-based heuristic for dataset similarity, but Eq. (4.7) needs an explicit normalization precondition and the noise-invariance proof is only expectation-level. read the letter →

arxiv 2501.09591 v1 pith:K6GIFYU4 submitted 2025-01-16 cs.LG

Muhammad Rajabinasab , Anton D. Lautrup , Arthur Zimek This is my paper · ORCID

classification cs.LG MSC 62H25

keywords inter-datasetsimilarityprincipalcomponentanalysisexplainedvariancedifferenceanglesyntheticdataevaluationfeatureselectionmodel-agnosticmetrictabular

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

The paper sets out to measure how similar two numerical tabular datasets are by comparing their principal component structure instead of training any model. It proposes two metrics: the difference in explained variance ($\Delta\lambda$), a sum of absolute differences between eigenvalue spectra with a data-dimension-dependent normalization, and the angle difference ($\Delta\theta$) between the first principal components of the two datasets. Both metrics are deterministic for a given sample and need no labels, kernels, pretrained embeddings, or downstream tasks, which the paper argues distinguishes them from common similarity measures. The paper then argues that these metrics work in two practical roles: tracking and scoring synthetic-data quality during training, and, through an averaged angle difference called AAD, selecting the feature-selection result that best preserves the dataset's structure without running a classifier. A sympathetic reader would care because cheap, model-free dataset similarity would make synthetic-data auditing and feature-selection comparison more reproducible and less dependent on arbitrary modeling choices.

What carries the argument

The carrying object is the eigenvalue decomposition of the sample covariance matrix, specifically the ordered eigenvalue spectrum $\lambda_1 \ge \cdots \ge \lambda_p$ and the first eigenvector $a_1$. The eigenvalues encode how much variance each principal direction explains, so $\Delta\lambda$ measures differences in how variance is distributed, while $a_1$ encodes the dominant direction of variation, so $\Delta\theta$ measures whether the two populations vary along different axes. The normalization in $\Delta\lambda$ is derived from a bound comparing a fully concentrated spectrum with a noise-like uniform spectrum, while the factor $2/\pi$ in $\Delta\theta$ maps the minimum angle between the two first principal components onto the unit interval. AAD assembles these angle differences into a feature-selection criterion by setting each unselected feature to zero and averaging the resulting angle cost over all unselected features.

What would settle it

Run the paper's synthetic-data benchmark twice on the same generated datasets, once with mean-subtracted data and once with data standardized by standard deviation; if the rankings or values of $\Delta\lambda$ and $\Delta\theta$ change, the metric's output depends on preprocessing choice, and the claimed normalization to $[0,1]$ is not intrinsic to the data.

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Extended reading notes

Core claim

The central discovery is that inter-dataset similarity for numerical tabular data can be read off the PCA decomposition of the two datasets: if two datasets share the same underlying statistics, their empirical covariance matrices and therefore their eigenvalue sequences and leading eigenvectors should agree, and any divergence is quantified by the two proposed metrics. The eigenvalue-based metric is $\Delta\lambda = \frac{d}{d+p-2}\sum_i |\lambda_i - \lambda_i'|$, where $d$ is the number of variables and $p$ the number of components considered, and it captures how the total variance is distributed across principal directions. The angle metric is $\Delta\theta = \frac{2}{\pi}\min[\arccos(a_1 \cdot a_1'), \arccos(a_1 \cdot (-a_1'))]$, with the minimum taken because principal components are only defined up to sign, and it captures whether the dominant axes of variation point in the same directions. The paper further claims that averaging the angle differences contributed by zeroing out each unselected feature gives AAD, a model-agnostic score whose minimum across feature-selection methods agrees with the best classifier-based or clustering-based result in most of the benchmark datasets examined.

Load-bearing premise

The unit-interval guarantee for $\Delta\lambda$ holds only when the eigenvalue spectra of both datasets are normalized to sum to one, and the paper never states this precondition in the metric definition; without a fixed standardization convention, the same pair of datasets yields different similarity values.

Editorial extensions

If this is right

Synthetic-data quality can be monitored during training: the proposed metrics fall as a generative model improves and plateau when quality converges, offering a possible early-stopping signal.
Because the metrics are deterministic and independent of downstream tasks, evaluations using them avoid the variability introduced by classifier training, kernel choice, or embedding models.
Under standardization, the metrics are invariant to translation and scaling, and orthogonal rotations change only $\Delta\theta$ by the rotation angle, so they can sanity-check whether a supposedly synthetic dataset is merely an altered copy of real data.
AAD provides a model-agnostic way to rank feature-selection results: picking the result with the minimum AAD agrees with the best classifier-based or clustering-based result on most of the datasets tested.
AAD decreases as more features are selected, reflecting the marginal contribution of each feature to the dataset's principal structure.

Reading between the lines

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

A natural extension the authors gesture at but do not develop is to use AAD as a stopping rule in forward feature selection, since the reported downward trend suggests a plateau once redundant features are added.
Because $\Delta\theta$ only reads the first principal component, two datasets with identical first eigenvectors but different later components could be rated very similar; a version averaging angles over several components would be a direct stress test of the method.
The preprocessing sensitivity noted in Section 7 implies that reproducible use of these metrics requires publishing a fixed standardization convention, since switching between mean-subtracted and standardized data changes the measured values.
If a consistent preprocessing convention is fixed, the same machinery applies beyond tabular data to any numerical representation, including images or text embeddings.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a useful heuristic paper with a real gap between its theoretical claims and its definitions. The two metrics are simple and cheap, and the two applications show they track something meaningful. But Eq. (4.7) as written is not a well-defined function on datasets: the unit-interval bound in Eq. (4.8) only works if the eigenvalue spectra are normalized to sum to 1, and neither Section 3 nor the metric statement says to do that. Under standardization, eigenvalues sum to d, so Δλ can be d times larger than intended. The appendix's noise-invariance proof is expectation-only; finite-sample noise changes the covariance matrix, so the statement "random noise does not change the elements" is wrong as written. These are fixable, but they are the load-bearing parts of the "fully deterministic" claim.

What is actually new: paired metrics Δλ and Δθ, a wrapper AAD for feature selection evaluation, and empirical evidence that they behave sensibly on real datasets. The correlation analysis with SynthEval metrics is a good sanity check; Fig. 3's tracking of TVAE training is a nice demonstration. The paper is honest in Section 7 about preprocessing dependence, which mitigates the normalization issue somewhat, but the definition should state the convention up front.

Soft spots beyond the normalization: Δθ uses only the first PC, so "holistic" is oversold; the AAD decreases automatically as more features are selected because fewer features are zeroed, so comparing feature selection methods with different subset sizes needs care; the paper doesn't position against the principal-angle/subspace-similarity literature; and the GitHub link is not in the manuscript.

Who it's for: people building or benchmarking tabular synthetic data generators, and people comparing feature selection methods without wanting to train classifiers. It's a heuristic toolkit, not a deep theoretical result. A serious referee should see it, but the normalization and the proof need fixing first.

Referee Report

4 major / 6 minor

Summary. The paper proposes two PCA-based metrics for inter-dataset similarity: Δλ, the normalized difference between eigenvalue spectra, and Δθ, the angle between first principal components. After reviewing PCA background, the authors analyze the metrics' normalization and invariance, then apply them to two tasks: evaluating synthetic data utility and evaluating feature selection results via the derived Average Angle Difference (AAD). The empirical sections report experiments on synthetic data from several generators (using SynthEval) and on 20 UCI datasets with four feature selection methods. The central claims are that the metrics are deterministic, model-agnostic, and capture holistic dataset structure, and that AAD can select the best feature selection result without a downstream model.

Significance. If the technical issues are resolved, the paper offers a simple, deterministic, and computationally light complement to model-dependent inter-dataset similarity measures. The authors provide an implementation on GitHub and evaluate on a substantial number of benchmark datasets, and the correlation analysis in Section 5.1 is a useful way to position the new metrics against existing utilities and privacy metrics. The proposed metrics have no fitted parameters (beyond the choice of p for Δλ), which is a genuine practical advantage. However, the paper's primary quantitative claims currently depend on unstated preprocessing and normalization choices, and the feature-selection application needs a sharper statement of what AAD actually measures.

major comments (4)

[Section 4.1, Eqs. (4.7) and (4.8)] The normalization of Δλ is not well-defined as stated. The derivation of the bound in Eq. (4.8) assumes that each eigenvalue spectrum sums to 1, with the 'pure noise' spectrum given by λ'_i = 1/d. But Section 3 defines PCA on the sample covariance matrix S = (1/(n−1)) X̃ᵀX̃, whose eigenvalues sum to the total variance; for mean-centered data this sum is an arbitrary scale-dependent quantity, and for z-score standardized data the correlation-matrix eigenvalues sum to d. Without an explicit convention (e.g., standardizing and dividing all eigenvalues by d), the claimed unit-interval range is not guaranteed: for standardized data with p = d, the worst-case spectra λ = (d,0,…,0) and λ' = (1,…,1) give Δλ = d, not a value in [0,1]. Section 7 concedes that switching between mean-subtracted and standard-deviation-normalized data changes the results. The definition in Eq. (4.7) must specify the preprocessing and the normalization of the λ_i, and the numerical values in Table 2 and the figures must be recomputed under that convention.
[Appendix, Eqs. (.2) and (.3)] The proof that additive noise does not change the covariance matrix is only valid in expectation, not for finite samples. In Eq. (.2), the sample mean of the noise terms is claimed to be zero, but for a finite sample Σ_i δx_i is not exactly zero; similarly, the cross terms Σ_i x_i δy_i, Σ_i y_i δx_i, and the term η Σ_i δx_i δy_i in Eq. (.3) do not vanish exactly. The statement 'it has been shown that random noise does not change the elements of the covariance matrix' is therefore too strong. The privacy sanity-check claims in Section 5.3 rely on this result, so the appendix should either state the result as an expectation/probabilistic statement or provide a finite-sample bound.
[Section 6.2 and 6.3, Eq. (6.10)] The AAD criterion in Eq. (6.10) is partly mechanical: as more features are selected, the complement set F^C shrinks and fewer features are set to zero, so the average angle difference tends to decrease regardless of feature quality. Figure 6 confirms a clear downward trend as the number of selected features increases. Consequently, the claim that 'the minimum AAD value can successfully capture the best feature selection result' is not supported unless all compared feature selection outputs have the same cardinality, or unless the metric is adjusted for cardinality. The experimental section should state the selected-feature counts for each method, compare methods at equal cardinality, or otherwise demonstrate that AAD is not simply favoring larger feature subsets.
[Abstract and Section 4.2] The paper positions the metrics as providing a 'holistic perspective on the entire dataset', but Δθ in Eq. (4.9) uses only the first principal component of each dataset. While Δλ summarizes the full spectrum when p is chosen large enough, Δθ discards all principal components beyond the first, and in near-degenerate spectra the first eigenvector is not even uniquely defined. If the holistic claim is to be retained, the paper should either generalize Δθ to multiple principal components or explicitly qualify that Δθ measures only the dominant direction of variation.

minor comments (6)

[Section 5.3] The text refers to 'Proofs and further details are supplied in Appendix 7', but the appendix has no numbered sections; the reference should simply say 'Appendix'.
[Section 4.1] The definition of Δλ never states how the number of components p is chosen, even though both the sum and the normalization factor in Eq. (4.7) depend on p. The paper should discuss the choice of p and the sensitivity of results to it.
[Appendix, Eq. (.9)] The notation in the rotation invariance proof is inconsistent with Section 3: the projection is written as Z = XA = YQQᵀB = YB, but earlier Eq. (3.6) defines Z = X̃A. Clarify whether the data are centered and whether X in the appendix is already centered.
[Table 2] The 'noisy (10%)' row reports Δλ = 0.00 and Δθ = 0.00 exactly, which conflicts with the finite-sample noise invariance caveat raised in the appendix unless the values are rounded to two significant figures; this should be clarified.
[Figure 4] The labels 'pca_eigval' and 'pca_eigvec' are used without defining which proposed metric each corresponds to; the caption should state that these are Δλ and Δθ.
[References] Reference [40] is a non-archival blog post; for a peer-reviewed venue it should be replaced by a citable source on PCA applications.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PCA-based metrics are defined directly from the inputs; the validation experiments use external benchmarks, and the flagged preprocessing ambiguity is a correctness issue rather than a circular reduction.

full rationale

The paper's metric definitions are self-contained functions of PCA outputs: Δλ (Eq. 4.7) is a normalized L1 distance between eigenvalue spectra, Δθ (Eq. 4.9) is the angle between first principal components, and AAD (Eq. 6.10) is the average of Δθ over the unselected features. No parameter is fitted to a target, no downstream model is used in the definitions, and no conclusion is obtained by renaming a fitted input. The self-citations in the paper ([19] SynthEval, [30] FSDEM, and the PCA-related background [10], [17], [31]) are used as benchmarking infrastructure or related work, not as load-bearing derivational premises; in particular, no uniqueness theorem or prior construction is imported to force the metric choices. The feature-selection evaluation is genuinely empirical: the minimum-AAD selection is compared against F1-score and clustering accuracy on 20 datasets, rather than being declared correct by definition. The downward trend in Fig. 6 is partly a consequence of averaging over a shrinking set of unselected features, but it is not a formal identity (the average can increase if the newly included feature has a below-average zeroing angle), so it does not constitute a circular prediction. The paper's own Section 7 limitation statement—that switching between mean-subtracted and standardized data changes results—is a serious correctness gap: Eq. (4.8) derives the normalization factor under spectra that sum to 1, whereas Section 3's covariance eigenvalues sum to total variance (or d after standardization), so the claimed unit-interval range of Δλ is not guaranteed as stated. This undermines the quantitative claims, but it is a flaw in the definition's normalization, not a circular derivation, and therefore does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The metrics rest on PCA, so the main axioms are domain assumptions about PCA statistics sufficiency and standardization of data. The appendix adds an assumption about noise terms vanishing per sample, which is false for finite samples.

free parameters (1)

p (number of principal components used in Δλ) = not specified in the paper
The metric Δλ sums over p components; the choice of p changes the metric's range and meaning, and the paper does not provide a selection rule.

assumptions (4)

domain assumption Two datasets that share the same fundamental statistics will have similar PCA representations (Section 4, opening paragraph).
This is the core premise that makes Δλ and Δθ meaningful; it is plausible but not proven, and the converse (similarity in these statistics implies similarity of datasets) is used in the applications.
domain assumption The first principal component is sufficient for angle comparison, as 'the remaining are orthogonal' (Section 4.2).
Orthogonality of PCs within each dataset does not imply alignment of the remaining PCs across datasets, so discarding them is a strong assumption.
ad hoc to paper Eigenvalue spectra are normalized to sum to 1 in the derivation of the normalization bound (Section 4.1, Eq. 4.8).
The bound assumes λ1≈1, λ2≈0 etc. and λ'i≈1/d, which presumes spectra sum to unity; this requires standardized data and eigenvalue normalization, which the metric definition omits.
ad hoc to paper Noise cross-terms such as Σ_i x_i δy_i vanish for finite samples (Appendix, Eq. .3).
The proof asserts these terms 'disappear' because the noise is centered at zero; this holds only in expectation over the noise distribution, not for a particular finite sample.

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Cite this review

Pith. "Pith review of Metrics for Inter-Dataset Similarity with Example Applications in Synthetic Data and Feature Selection Evaluation -- Extended Version." pith.science (2026). https://pith.science/paper/K6GIFYU4

@misc{pith2026250109591,
  author       = {Pith},
  title        = {Pith review of: Metrics for Inter-Dataset Similarity with Example Applications in Synthetic Data and Feature Selection Evaluation -- Extended Version},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6GIFYU4}},
  note         = {Machine review of arXiv:2501.09591}
}

read the original abstract

Measuring inter-dataset similarity is an important task in machine learning and data mining with various use cases and applications. Existing methods for measuring inter-dataset similarity are computationally expensive, limited, or sensitive to different entities and non-trivial choices for parameters. They also lack a holistic perspective on the entire dataset. In this paper, we propose two novel metrics for measuring inter-dataset similarity. We discuss the mathematical foundation and the theoretical basis of our proposed metrics. We demonstrate the effectiveness of the proposed metrics by investigating two applications in the evaluation of synthetic data and in the evaluation of feature selection methods. The theoretical and empirical studies conducted in this paper illustrate the effectiveness of the proposed metrics.

Figures

Figures reproduced from arXiv: 2501.09591 by the authors.

**Figure 2.** Metric Variation with Varying Sample Size and Number of Attributes. The proposed metrics and other state-of-the-art metrics (correlation matrix difference, propensity mean squared error, and the Kolmogorov-Smirnov test) do not conform to the optimum value, due to noise for fewer samples and sparsity from additional dimensions even when the compared samples stem from the same multivariate distribution. the empirical … view at source ↗

**Figure 3.** TVAE Training Loss Alongside the Proposed Metrics. After an initial period (the model learns characteristics of the data), the metrics drop, indicating a new alignment of the real and synthetic data. Afterwards the metrics steadily improve, aligned with the decreased loss. This demonstrates that the metrics are meaningful for measuring synthetic data quality. Confidence interval is 95%, based on 10 independent runs.… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Correlation Hierarchy of Metrics. The heat-map is produced by taking the correlations of results of the metrics on 64 synthetic datasets made using different generative models. Closely associated metrics have little value in the same benchmarks since they describe the …

**Figure 5.** Figure 5: Experimental Results for Feature selection. The experimental results on 20 different datasets, 4 different feature selection methods and compared to two different model-dependent metric. effectiveness of AAD as a model-agnostic metric for evaluation of feature selectio…

**Figure 6.** Figure 6: In this figure, we show the change in AAD in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 6.** Figure 6: Downwards Trend of Angle Difference Results on 20 different datasets show that AAD decreases as more features are selected. Scale of AAD is adjusted to each dataset. selected for the experiments (note that the scale is adjusted for each dataset). We start with selectin…

**Figure 7.** Figure 7: Extended Experimental Results For Feature Selection Results of experiments on 20 different datasets and for the proposed metrics, F1-score, and clustering accuracy. Color scale is reversed for the first two heat-maps as lower values are desired for those and higher val…

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