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Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For large bandwidths, the Gaussian RBF kernel's feature space becomes a scaled copy of the original Euclidean data geometry, with a residual that shrinks like (ρ/σ)².

desk verdict Real extension of the flat-limit story, but the abstract oversells PCA convergence by dropping the distinct-eigenvalue assumption. read the letter →

arxiv 2607.21823 v1 pith:74UGUQJM submitted 2026-07-23 cs.LG

Sergio A. Alvarez This is my paper
classification cs.LG MSC 46E2247A5562H2568T05
keywords GaussianRBFkernelreproducingHilbertspacePCAbandwidthasymptoticsdataeccentricitynear-isometrycenteredalignmentspectralperturbationtheory
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 proves that the infinite-dimensional feature space of a Gaussian RBF kernel is, up to one overall scale factor, asymptotically identical to the original Euclidean data space in the large-bandwidth limit. For any finite dataset, the Gaussian and linear kernel feature embeddings of the same points have pairwise distances that agree to within a residual of order (ρ/σ)², where ρ is the dataset's eccentricity — the ratio of its diameter to its median pairwise distance. Because kernel PCA is a purely metric construction, the paper further shows that Gaussian kernel PCA eigenvalues, eigenprojections, and projected scores converge to those of ordinary linear PCA as bandwidth grows. A reader should care because ρ gives a simple, data-dependent rule of thumb for when a Gaussian kernel will behave like a linear one, which matters for choosing bandwidths in practice.

What carries the argument

The central mechanism is the Taylor expansion of the Gaussian kernel entries e^{-|x_i-x_j|²/(2 d_X² σ²)} around the flat limit, which yields the double-centered Gaussian Gram matrix as 1/σ_X² times the double-centered Euclidean Gram matrix plus O(σ^{-4}) terms. A key lemma identifies the double-centered Euclidean Gram matrix with the double-centered linear Gram matrix, so the RKHS distances of the Gaussian kernel approach those of the linear kernel. The data eccentricity ρ = diam(X)/d_X controls the uniform size of the exponential arguments, and analytic perturbation theory for symmetric matrices converts the Gram-matrix expansion into convergence statements for kernel PCA.

What would settle it

Measure the relative metric dissonance of Theorem 2 at σ = 2^7 on a dataset with near-duplicate points (large η) using extended-precision arithmetic; the theorem predicts the residual is bounded by (ρ/σ)²η times the diameter. Finding any pair of points whose scaled Gaussian distance deviates by more than that bound would refute the quantitative claim. For Theorem 3, construct a dataset whose centered linear Gram matrix has a repeated eigenvalue (for example, a centrally symmetric configuration) and check whether the scaled Gaussian top eigenvectors converge to a fixed linear eigenvector — the

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Extended reading notes

Core claim

Theorem 2 states that for any finite dataset X, the Gaussian RBF RKHS feature representation scaled by σ_X = d_X σ is a relative (ρ/σ)²η-near-isometry to the linear kernel feature representation as σ→∞, where ρ is the diameter-to-median-distance eccentricity and η is the diameter-to-minimum-nonzero-distance ratio. Theorem 3 then states that, assuming the centered linear Gram matrix has no repeated eigenvalues, the scaled Gaussian eigenvalues σ_X² λ_i, the eigenprojections, and the scaled Gaussian principal components σ_X Φ(x_i)·V_k all converge to their linear PCA counterparts. The proof achieves this by expanding the Gaussian Gram matrix in powers of 1/σ_X and showing that its double-center

Load-bearing premise

The load-bearing premise is that the data set is finite and non-degenerate — no exact duplicate points and a positive median pairwise distance — and, for the PCA result, that the centered linear Gram matrix has no repeated eigenvalues; if either fails, the stated convergence of individual eigenvectors does not follow.

Editorial extensions

If this is right

  • At bandwidths σ ≫ ρ, any kernel method that depends only on RKHS pairwise distances — such as centered kernel alignment — will behave like its linear-kernel counterpart, with error shrinking as 1/σ².
  • Gaussian kernel PCA at large bandwidth is equivalent to classical PCA on the original data, so a Gaussian kernel cannot extract nonlinear structure in this regime; nonlinearity only appears at smaller bandwidths.
  • The eccentricity ρ provides a practical, data-specific scale: across the 30 real datasets tested, the onset of 1/σ² convergence occurs near σ ≈ ρ for metric dissonance, eigenvalue differences, and top-eigenprojection dissonance.
  • Eigenprojection convergence is fastest for the top principal directions; convergence for lower eigenvalues can be delayed substantially, especially when the linear eigenvalues are repeated.
  • The O((ρ/σ)²) rate is robust enough that the residual and onset bandwidth both grow approximately quadratically with ρ, as measured by slopes near 2 in log-log regressions.

Reading between the lines

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

  • If the finite-sample Gram-matrix isometry holds pointwise, a natural extension would be to transfer the statement to out-of-sample points or to the infinite-dimensional RKHS; the paper does not prove that transfer, leaving it as a plausible but unverified step.
  • The eccentricity ρ could be repurposed as a heuristic for bandwidth selection: choose σ just above ρ to stay in the nonlinear regime, or far above ρ to approximate linear behavior — a consequence the paper hints at but does not develop into a full selection rule.
  • The bound's multiplicative constant η appears conservative: the paper's own experiments show clean 1/σ² slopes even on low-eccentricity datasets, suggesting that the true constant may be much smaller than (ρ/σ)²η in typical cases.
  • The repeated-eigenvalue case is a natural next target: the paper shows eigenprojections still converge, but a full characterization of individual eigenvector behavior when the linear spectrum is degenerate is left open.
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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

4 major / 5 minor

Summary. The paper studies the large-bandwidth limit of Gaussian RBF kernels on a fixed finite data set. It claims that, after scaling by σ_X = d_X σ, the Gaussian RBF RKHS feature representation becomes a relative O((ρ/σ)^2 η)-near-isometry to the linear (Euclidean) feature representation, where ρ is the ratio of diameter to median pairwise distance and η is a data-degeneracy ratio. It then uses analytic perturbation theory to argue that, when the centered linear Gram matrix has simple eigenvalues, Gaussian kernel PCA eigenvalues, eigenprojections, and scaled principal components converge to their linear PCA counterparts. The paper includes experiments on 30 OpenML data sets that show 1/σ² convergence and a positive correlation between convergence onset and ρ.

Significance. If the proof is made rigorous, the result would be a clean and useful statement about Gaussian RBF kernels: for σ large relative to the data diameter, metric RKHS structure and kernel PCA approach linear behavior at a quantified rate. The eccentricity measure ρ is a simple, data-dependent scale that could guide bandwidth selection. The paper has notable strengths: the first-order expansion K_G = ε²(K_lin + O(ε²)) is correct, no fitted constants appear in the theoretical bound, and the experiments cover a broad range of data sets with explicit asymptotic-slope checks and reproducible experimental details. The main weakness is that the proof of the central geometric theorem is sketched heuristically in places, and the abstract states the kernel PCA result more broadly than the theorem's assumptions warrant.

major comments (4)
  1. [§3.3, Eqs. (12)–(14)] The proof of Theorem 2 is not yet rigorous as written. Eq. (13) contains |(σ_X d_G)^2 − d_E^2|, but Eq. (14) is claimed to follow by dividing by a distance sum and then is displayed with σ_X² multiplying the Gaussian distance rather than σ_X; the divisor in the preceding sentence also uses σ_X² d_G. Thus the displayed inequality does not follow from Eq. (13). The intended correction is likely to replace σ_X² by σ_X, but this is load-bearing because Theorem 2 is about the σ_X-scaled embedding. In addition, the higher-order terms are handled by 'doubling the multiplicative factor in Eq. 13', which is a heuristic, not a bound. A Taylor theorem with remainder, using finiteness of the data set, should be used to supply a genuine O((ρ/σ)^4) remainder and a rigorous distance-difference bound.
  2. [Abstract and Theorem 3, §3.4] The abstract's unconditional statement that Gaussian RBF 'eigenvalues, eigenprojections, and principal components all converge' to linear PCA is stronger than the theorem, which explicitly assumes that K_lin has no repeated eigenvalues. The proof itself notes that in the repeated-eigenvalue case only summed eigenprojections converge and individual eigenvectors may not. The experiments corroborate this caveat: eigenprojection convergence for groups of top 2 and top 5 eigenvalues is slower and less strongly correlated with ρ (Tables 3–4, Fig. 8). The abstract, Section 1 contribution statement, and Section 5 conclusions should all be qualified by the simple-spectrum assumption, or restated in terms of eigenspace projections.
  3. [§3.4, final paragraph of proof of Theorem 3] The convergence of the scaled principal components σ_X Φ(x_i)·V_k is asserted rather than proved. Convergence of eigenvalues and eigenvectors of the rescaled Gram matrix does not by itself imply convergence of the PCA projections, because the feature map Φ(·) depends on σ and the displayed formula involves the uncentered Gaussian kernel K(x_j,x_i), which tends to 1 rather than to the linear kernel. The proof needs an explicit argument — for example, using the zero-sum property of centered eigenvectors, or working with the centered kernel — to show that σ_X Σ_j α_j K(x_j,x_i) converges to the corresponding linear PCA score. Without this, the principal-component part of Theorem 3 is unsupported.
  4. [§3.3, Eq. (12) and Definition 1 / Theorem 2] The bound in Theorem 2 is stated for 'any finite data set', but ρ and η are undefined or infinite for degenerate data: if median pairwise distance d_X = 0, or if there are not at least two distinct points, σ_X and η are not well-defined. The theorem should explicitly assume a nondegenerate data set with d_X > 0 and at least one nonzero pairwise distance. This is a boundary-condition issue rather than a flaw in the main asymptotics, but it affects the universality of the statement.
minor comments (5)
  1. [§3.3, Eq. (12)] The text says 'four parenthesized terms' in Eq. (12), but the right-hand side has three summands; moreover α_ii = α_jj = 0, so only the −2α_ij^4 term is nonzero. The subsequent bound is still valid with a smaller constant, but the counting and simplification should be corrected.
  2. [§4.1.3, Eq. (16)] The phrase 'shifting the result to 0 at σ=0' is unclear: the experimental grid is σ=2^p with p = −4,...,8 and does not include σ=0. The normalization denominators in Eqs. (17)–(19) are not f(0) as defined in Eq. (16). Please clarify what f(0) denotes and how the shift is computed in practice.
  3. [Table 2 and §4.2] Several convergence-onset values are reported as '256+', indicating right-censoring at the largest reliably tested bandwidth. The least-squares regressions that relate log2 ρ to log2 σ*_0 appear to treat these censored values as 256. This can bias the reported slopes and r² values; a censored regression or an explicit note on how '256+' was handled would be appropriate.
  4. [Corollary 2, §3.3] The proof uses the bound |(a+ε)/(b−ε) − a/b| ≤ 3ε/b without derivation, and the step from cosine difference O((ρ/σ)^2) to angle difference O(ρ/σ) requires a lower bound on the relevant sine term; otherwise angles near 0 or π behave differently. This corollary is secondary, but the proof should be made precise.
  5. [§4.1.2, Algorithm 1] The threshold τ for grouping equal eigenvalues is fixed at 10^-6 with a note that 10^-9 and 10^-12 give the same results. Since the theoretical assumption is that eigenvalues are distinct, it would be useful to report how often the algorithm actually grouped multiple eigenvalues for the top-k analysis, and whether this changes the interpretation of Fig. 8 and Tables 3–4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorems 2 and 3 derive from a direct Taylor expansion of the Gaussian kernel and from Kato perturbation theory; the self-citations to [47] are background and not load-bearing.

full rationale

The central derivation is self-contained. Theorem 2 is obtained by expanding the Gaussian Gram matrix entries e^{-|x_i-x_j|^2/(2 d_X^2 sigma^2)} in powers of 1/(d_X sigma), keeping the quadratic term to identify (1/sigma_X^2) K_E, and then using Lemma 2, which is proved in the paper, to identify the double-centered Euclidean Gram matrix with the double-centered linear Gram matrix. The eccentricity rho = diam(X)/d_X appears algebraically through the bound alpha_{p,q} <= diam(X) and sigma_X = d_X sigma; it is not a fitted parameter and is not chosen to make the theorem true. Theorem 3 follows from the same expansion, written as K_G(sigma) = epsilon^2 (K_E + O(epsilon^2)), plus standard analytic perturbation theory. The distinct-eigenvalue condition is an explicit assumption of Theorem 3; the abstract's broader wording is a correctness/scope concern, not a circularity. The self-citations to [47] for the eccentricity measure, for Theorem 1 (CKA asymptotics), and for a proof technique are not load-bearing: Theorem 1 is motivation, the measure is redefined in Eq. 4, and the proof of Lemma 2 contains the full computation. Experimental slopes and r^2 values are after-the-fact summaries of measurements against the derived bound, not parameters inserted into the derivation. I therefore find no step in which a prediction is equivalent by construction to a fitted input or to a self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The theorems introduce no fitted free parameters: ρ, d_X, and η are computed directly from pairwise distances, and the experimental threshold τ=10^{-6} is a numerical convention. The axioms are standard mathematical background plus the explicitly stated simplicity assumption on K_lin's spectrum and non-degeneracy of the distance distribution. No new entities (particles, forces, dimensions) are postulated.

assumptions (6)
  • standard math Moore–Aronszajn theorem guarantees an RKHS feature map for any positive semidefinite kernel
    Used in §1 and §2.1 to justify replacing kernel evaluations by inner products in a Hilbert space.
  • standard math Double-centering a Gram matrix corresponds to centering feature vectors and preserves pairwise distances
    Proved as Lemma 1; elementary linear algebra, used throughout §3.
  • domain assumption Gaussian kernel Gram matrices are analytic in ε=1/σ_X and the matrix perturbation series converges
    Used in the proof of Theorem 3; true for finite data but not uniform in N or in data eccentricity.
  • domain assumption The centered linear Gram matrix K_lin has no repeated eigenvalues
    Explicitly assumed in Theorem 3; needed for individual eigenvector convergence; not mentioned in the abstract.
  • domain assumption d_X (median pairwise distance) is nonzero and η=diam/min-nonzero-distance is finite
    Needed for ρ and the relative near-isometry bound; degenerate datasets with duplicates violate it.
  • standard math Kato analytic perturbation theory for symmetric matrices (Chapter II, §1)
    Invoked in the proof of Theorem 3 to obtain analytic eigenvalue and eigenprojection convergence.

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Cite this review

Pith. "Pith review of Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA." pith.science (2026). https://pith.science/paper/74UGUQJM

@misc{pith2026260721823,
  author       = {Pith},
  title        = {Pith review of: Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/74UGUQJM}},
  note         = {Machine review of arXiv:2607.21823}
}
abstract

We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $\sigma \rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\frac{\rho}{\sigma} \right )^2$, where $\rho$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $\rho$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.

Figures

Figures reproduced from arXiv: 2607.21823 by the authors.

Figure 1
Figure 1. Nonlinearly co-varying attributes with null covariance. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Sample eigenprojection dissonance results for dif [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. RKHS metric convergence, six sample data sets; [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Large-bandwidth RKHS metric dissonance (Eq. 17) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 7
Figure 7. Figure 7: Convergence onset also mirrors ρ; see [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 6
Figure 6. Figure 6: Eigenprojection convergence, top eigenvalue only, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Large-σ normalized eigenprojection dissonance (Eq. 19) vs. log2 data eccentricity ρ (Eq. 4), top eigenvalue only. Results shown for σ = 27 , and 30 data sets. Dissonance increases with data eccentricity, ρ. Slope of least-squares linear fit is 2.05; coefficient of dete…
Figure 8
Figure 8. Figure 8: Eigenprojection convergence: top two eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reference graph

Works this paper leans on

64 extracted references · 1 canonical work pages

  1. [50]

    Spectral properties of kernel matrices in the flat limit,

    S. Barthelm´e and K. Usevich, “Spectral properties of kernel matrices in the flat limit,”SIAM Journal on Matrix Analysis and Applications, vol. 42, no. 1, pp. 17–57, 2021

    2021

  2. [51]

    Asymptotic behaviors of support vector machines with Gaussian kernel,

    S. S. Keerthi and C.-J. Lin, “Asymptotic behaviors of support vector machines with Gaussian kernel,”Neur. Comp., vol. 15, no. 7, pp. 1667–1689, 2003

    2003

  3. [47]

    Kernel PCA and de-noising in feature spaces,

    S. Mika, B. Sch ¨olkopf, A. Smola, K.-R. M ¨uller, M. Scholz, and G. R ¨atsch, “Kernel PCA and de-noising in feature spaces,” in Advances in Neural Information Processing Systems, M. Kearns, S. Solla, and D. Cohn, Eds., vol. 11. MIT Press, 1998. [Online]. Available: https://proceedings.neurips.cc/paper/1998/ file/226d1f15ecd35f784d2a20c3ecf56d7f-Paper.pdf

    1998

  4. [1]

    Theory of reproducing kernels,

    N. Aronszajn, “Theory of reproducing kernels,”Transactions of the American Mathematical Society, vol. 68, no. 3, pp. 337–404, 1950. [Online]. Available: http://dx.doi.org/10.2307/1990404

    doi:10.2307/1990404 1950

  5. [2]

    The kernel trick for distances,

    B. Sch ¨olkopf, “The kernel trick for distances,”Advances in neural information processing systems, vol. 13, 2000

    2000

  6. [3]

    Universal kernels

    C. A. Micchelli, Y. Xu, and H. Zhang, “Universal kernels.”Journal of Machine Learning Research, vol. 7, no. 12, 2006

    2006

  7. [4]

    Universality, characteristic kernels and RKHS embedding of measures,

    B. K. Sriperumbudur, K. Fukumizu, and G. R. Lanckriet, “Universality, characteristic kernels and RKHS embedding of measures,”J. Mach. Learn. Res., vol. 12, no. 70, pp. 2389– 2410, 2011. [Online]. Available: http://jmlr.org/papers/v12/ sriperumbudur11a.html

    2011

  8. [5]

    Some results on Tchebycheffian spline functions,

    G. Kimeldorf and G. Wahba, “Some results on Tchebycheffian spline functions,”Journal of Mathematical Analysis and Applications, vol. 33, no. 1, pp. 82–95, 1971

    1971

Show all 64 references
  1. [6]

    Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV,

    G. Wahbaet al., “Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV,”Advances in Kernel Methods-Support Vector Learning, vol. 6, pp. 69–87, 1999

    1999

  2. [7]

    A generalized representer theorem,

    B. Sch ¨olkopf, R. Herbrich, and A. J. Smola, “A generalized representer theorem,” inComputational Learning Theory: 14th Annual Conference on Computational Learning Theory, COLT 2001 and 5th European Conference on Computational Learning Theory, EuroCOLT 2001 Amsterdam, The Net...

    2001

  3. [8]

    Support-vector networks,

    C. Cortes and V . Vapnik, “Support-vector networks,”Machine learning, vol. 20, pp. 273–297, 1995

    1995

  4. [9]

    Comparing support vector machines with Gaussian kernels to radial basis function classifiers,

    B. Sch¨olkopf, K.-K. Sung, C. J. Burges, F. Girosi, P . Niyogi, T. Poggio, and V . Vapnik, “Comparing support vector machines with Gaussian kernels to radial basis function classifiers,”IEEE Transactions on Signal Processing, vol. 45, no. 11, pp. 2758–2765, 1997

    1997

  5. [10]

    A practical approach to model selection for support vector machines with a Gaussian kernel,

    M. Varewyck and J.-P . Martens, “A practical approach to model selection for support vector machines with a Gaussian kernel,”IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 41, no. 2, pp. 330–340, 2010

    2010

  6. [11]

    An approximation of the Gaussian RBF kernel for efficient classification with SVMs,

    M. Ring and B. M. Eskofier, “An approximation of the Gaussian RBF kernel for efficient classification with SVMs,”Pattern Recognition Letters, vol. 84, pp. 107–113, 2016

    2016

  7. [12]

    An on-chip-trainable Gaussian-kernel analog support vector machine,

    K. Kang and T. Shibata, “An on-chip-trainable Gaussian-kernel analog support vector machine,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, no. 7, pp. 1513–1524, 2009. 10

    2009

  8. [13]

    Parameter selection of Gaussian kernel for one-class SVM,

    Y. Xiao, H. Wang, and W. Xu, “Parameter selection of Gaussian kernel for one-class SVM,”IEEE Transactions on Cybernetics, vol. 45, no. 5, pp. 941–953, 2014

    2014

  9. [14]

    Efficient model selection for support vector machine with Gaussian kernel function,

    Y. Tang, W. Guo, and J. Gao, “Efficient model selection for support vector machine with Gaussian kernel function,” in2009 IEEE Symposium on Computational Intelligence and Data Mining. IEEE, 2009, pp. 40–45

    2009

  10. [15]

    Random Fourier features for kernel ridge regression: Approximation bounds and statistical guarantees,

    H. Avron, M. Kapralov, C. Musco, C. Musco, A. Velingker, and A. Zandieh, “Random Fourier features for kernel ridge regression: Approximation bounds and statistical guarantees,” inInternational Conference on Machine Learning. PMLR, 2017, pp. 253–262

    2017

  11. [16]

    Scalable and memory-efficient kernel ridge regression,

    G. Ch ´avez, Y. Liu, P . Ghysels, X. S. Li, and E. Rebrova, “Scalable and memory-efficient kernel ridge regression,” in2020 IEEE International Parallel and Distributed Processing Symposium (IPDPS). IEEE, 2020, pp. 956–965

    2020

  12. [17]

    Predicting nuclear masses with the kernel ridge regression,

    X. Wu and P . Zhao, “Predicting nuclear masses with the kernel ridge regression,”Physical Review C, vol. 101, no. 5, p. 051301, 2020

    2020

  13. [18]

    Kernel ridge regression-based chirplet transform for non-stationary signal anal- ysis and its application in machine fault detection under varying speed conditions,

    C. Ding, M. Zhao, J. Lin, K. Liang, and J. Jiao, “Kernel ridge regression-based chirplet transform for non-stationary signal anal- ysis and its application in machine fault detection under varying speed conditions,”Measurement, vol. 192, p. 110871, 2022

    2022

  14. [19]

    Divide and conquer kernel ridge regression,

    Y. Zhang, J. Duchi, and M. Wainwright, “Divide and conquer kernel ridge regression,” inProceedings of the 26th Annual Conference on Learning Theory, ser. Proceedings of Machine Learning Research, S. Shalev-Shwartz and I. Steinwart, Eds., vol. 30. Princeton, NJ, USA: PMLR, 12–1...

    2013

  15. [20]

    Kernel ridge regression for general noise model with its application,

    S. Zhang, Q. Hu, Z. Xie, and J. Mi, “Kernel ridge regression for general noise model with its application,”Neurocomputing, vol. 149, pp. 836–846, 2015

    2015

  16. [21]

    Model selection in kernel ridge regression,

    P . Exterkate, “Model selection in kernel ridge regression,”Computa- tional Statistics & Data Analysis, vol. 68, pp. 1–16, 2013

    2013

  17. [22]

    Conformalized kernel ridge regression,

    E. Burnaev and I. Nazarov, “Conformalized kernel ridge regression,” in2016 15th IEEE International Conference on Machine Learning and Applications (ICMLA). IEEE, 2016, pp. 45–52

    2016

  18. [23]

    Two-stage learning kernel algorithms,

    C. Cortes, M. Mohri, and A. Rostamizadeh, “Two-stage learning kernel algorithms,” inProc. 27th Intl. Conf. on Machine Learning, ser. ICML’10. Madison, WI, USA: Omnipress, 2010, p. 239–246

    2010

  19. [24]

    Algorithms for learning kernels based on centered alignment,

    ——, “Algorithms for learning kernels based on centered alignment,”J. Mach. Learn. Res., vol. 13, no. 28, pp. 795–828, 2012. [Online]. Available: http://jmlr.org/papers/v13/cortes12a.html

    2012

  20. [25]

    Similarity of neural network representations revisited,

    S. Kornblith, M. Norouzi, H. Lee, and G. E. Hinton, “Similarity of neural network representations revisited,” inProc. 36th Intl. Conf. Mach. Learn., ICML 2019, 9-15 June 2019, Long Beach, California, USA, ser. Proc. Mach. Learn. Res., K. Chaudhuri and R. Salakhutdinov, Eds., v...

    2019

  21. [26]

    Do Wide and Deep Networks Learn the Same Things? Uncovering How Neural Network Representations Vary with Width and Depth,

    T. Nguyen, M. Raghu, and S. Kornblith, “Do Wide and Deep Networks Learn the Same Things? Uncovering How Neural Network Representations Vary with Width and Depth,”CoRR, vol. abs/2010.15327, 2020 (ICLR 2021). [Online]. Available: https://arxiv.org/abs/2010.15327

    2010 arXiv

  22. [27]

    Emerging cross-lingual structure in pretrained language models,

    A. Conneau, S. Wu, H. Li, L. Zettlemoyer, and V . Stoyanov, “Emerging cross-lingual structure in pretrained language models,” inProc. 58th Ann. Meeting Assoc. Comp. Ling.Online: Assoc. Comp. Ling., Jul. 2020, pp. 6022–6034. [Online]. Available: https://www.aclweb.org/anthology...

    2020

  23. [28]

    Universality and individuality in neural dynamics across large populations of recurrent networks,

    N. Maheswaranathan, A. Williams, M. Golub, S. Ganguli, and D. Sussillo, “Universality and individuality in neural dynamics across large populations of recurrent networks,” inAdv. Neur. Inf. Proc. Syst., H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alch´e- Buc, E. Fox, and R...

  24. [29]

    Identification of drug-side effect association via multiple information integration with centered kernel alignment,

    Y. Ding, J. Tang, and F. Guo, “Identification of drug-side effect association via multiple information integration with centered kernel alignment,”Neurocomputing, vol. 325, pp. 211–224,

  25. [30]

    Kernel-based relevance analysis with enhanced interpretability for detection of brain activity patterns,

    A. M. Alvarez-Meza, A. Orozco-Gutierrez, and G. Castellanos- Dominguez, “Kernel-based relevance analysis with enhanced interpretability for detection of brain activity patterns,”Frontiers in Neuroscience, vol. 11, p. 550, 2017. [Online]. Available: https://www.frontiersin.org/...

    2017

  26. [31]

    Available: https://www.sciencedirect.com/science/ article/pii/S0925231218312165

    [Online]. Available: https://www.sciencedirect.com/science/ article/pii/S0925231218312165

  27. [32]

    Pyrcca: regularized kernel canonical correlation analysis in python and its applications to neuroimaging,

    N. Y. Bilenko and J. L. Gallant, “Pyrcca: regularized kernel canonical correlation analysis in python and its applications to neuroimaging,” Frontiers in neuroinformatics, vol. 10, p. 49, 2016

    2016

  28. [33]

    Entangled watermarks as a defense against model extraction,

    H. Jia, C. A. Choquette-Choo, V . Chandrasekaran, and N. Papernot, “Entangled watermarks as a defense against model extraction,” in 30th USENIX Security Symposium (USENIX Security 21). USENIX Association, Aug. 2021. [Online]. Available: https://www.usenix. org/conference/useni...

    2021

  29. [34]

    Graph multiview canonical correlation analysis,

    J. Chen, G. Wang, and G. B. Giannakis, “Graph multiview canonical correlation analysis,”IEEE Transactions on Signal Processing, vol. 67, no. 11, pp. 2826–2838, 2019

    2019

  30. [35]

    Independence test and canonical correlation analysis based on the alignment between ker- nel matrices for multivariate functional data,

    T. G´orecki, M. Krzy´sko, and W. Woły´nski, “Independence test and canonical correlation analysis based on the alignment between ker- nel matrices for multivariate functional data,”Artificial Intelligence Review, vol. 53, pp. 475–499, 2020

    2020

  31. [36]

    Real-time data driven de- formation using kernel canonical correlation analysis,

    W.-W. Feng, B.-U. Kim, and Y. Yu, “Real-time data driven de- formation using kernel canonical correlation analysis,” inACM SIGGRAPH 2008 papers, 2008, pp. 1–9

    2008

  32. [37]

    Statistical consistency of kernel canonical correlation analysis

    K. Fukumizu, F. R. Bach, and A. Gretton, “Statistical consistency of kernel canonical correlation analysis.”Journal of Machine Learning Research, vol. 8, no. 2, 2007

    2007

  33. [38]

    Kernel principal component analysis,

    B. Sch ¨olkopf, A. Smola, and K.-R. M ¨uller, “Kernel principal component analysis,” inArtificial Neural Networks — ICANN’97, W. Gerstner, A. Germond, M. Hasler, and J.-D. Nicoud, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997, pp. 583–588

    1997

  34. [39]

    Facial expression recognition using kernel canonical correlation analysis (KCCA),

    W. Zheng, X. Zhou, C. Zou, and L. Zhao, “Facial expression recognition using kernel canonical correlation analysis (KCCA),” IEEE Transactions on Neural Networks, vol. 17, no. 1, pp. 233–238, 2006

    2006

  35. [40]

    Model selection for Gaussian kernel PCA denoising,

    K. W. Jorgensen and L. K. Hansen, “Model selection for Gaussian kernel PCA denoising,”IEEE transactions on neural networks and learning systems, vol. 23, no. 1, pp. 163–168, 2011

    2011

  36. [41]

    Nonlinear component analysis as a kernel eigenvalue problem,

    B. Sch ¨olkopf, A. Smola, and K.-R. M ¨uller, “Nonlinear component analysis as a kernel eigenvalue problem,”Neural Computation, vol. 10, no. 5, pp. 1299–1319, 1998

    1998

  37. [42]

    Convex formulation for kernel PCA and its use in semisupervised learning,

    C. M. Ala´ız, M. Fanuel, and J. A. K. Suykens, “Convex formulation for kernel PCA and its use in semisupervised learning,”IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 8, pp. 3863–3869, 2018

    2018

  38. [43]

    Streaming kernel principal component analysis,

    M. Ghashami, D. J. Perry, and J. Phillips, “Streaming kernel principal component analysis,” inProceedings of the 19th International Conference on Artificial Intelligence and Statistics, ser. Proceedings of Machine Learning Research, A. Gretton and C. C. Robert, Eds., vol. 51. ...

    2016

  39. [44]

    Emerging evolutionary algorithm integrated with kernel principal component analysis for modeling the performance of a water treatment plant,

    S. Abba, Q. B. Pham, A. Usman, N. T. T. Linh, D. Aliyu, Q. Nguyen, and Q.-V . Bach, “Emerging evolutionary algorithm integrated with kernel principal component analysis for modeling the performance of a water treatment plant,”Journal of Water Process Engineering, vol. 33, p. 1...

    2020

  40. [45]

    Kernel PCA for novelty detection,

    H. Hoffmann, “Kernel PCA for novelty detection,”Pattern Recognition, vol. 40, no. 3, pp. 863–874, 2007. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0031320306003414

    2007

  41. [46]

    Kernel PCA for feature extraction and de-noising in nonlinear regression,

    R. Rosipal, M. Girolami, L. J. Trejo, and A. Cichocki, “Kernel PCA for feature extraction and de-noising in nonlinear regression,” Neural Computing & Applications, vol. 10, no. 3, pp. 231–243, 2001

    2001

  42. [48]

    Kato,Perturbation Theory for Linear Operators, ser

    T. Kato,Perturbation Theory for Linear Operators, ser. Classics in Mathematics. Springer Berlin Heidelberg, 1995

    1995

  43. [49]

    Gaussian RBF centered kernel alignment (CKA) in the large-bandwidth limit,

    S. A. Alvarez, “Gaussian RBF centered kernel alignment (CKA) in the large-bandwidth limit,”IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1–8, Oct. 2022, early access

    2022

  44. [52]

    Mesures d’ind´ependance dans des rkhs en limite plate,

    P .-O. Amblard, S. Barthelme, N. Tremblay, and K. Usevich, “Mesures d’ind´ependance dans des rkhs en limite plate,” inGRETSI 2022, 2022

    2022

  45. [53]

    Gaussian process regression in the flat limit,

    S. Barthelm ´e, P .-O. Amblard, N. Tremblay, and K. Usevich, “Gaussian process regression in the flat limit,”arXiv preprint arXiv:2201.01074, 2022. 11

    2022 arXiv

  46. [54]

    SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,

    P . Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P . Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Fe...

    2020

  47. [55]

    OpenML: Networked science in machine learning,

    J. Vanschoren, J. N. van Rijn, B. Bischl, and L. Torgo, “OpenML: Networked science in machine learning,”SIGKDD Explorations, vol. 15, no. 2, pp. 49–60, 2013. [Online]. Available: http://doi.acm.org/10.1145/2641190.2641198

    2013

  48. [56]

    Principal angles between subspaces and their tangents,

    A. V . Knyazev and P . Zhu, “Principal angles between subspaces and their tangents,”arXiv preprint arXiv:1209.0523, 2012

    2012 arXiv

  49. [57]

    Principal angles between subspaces in an A-based scalar product: Algorithms and perturbation estimates,

    A. V . Knyazev and M. E. Argentati, “Principal angles between subspaces in an A-based scalar product: Algorithms and perturbation estimates,”SIAM Journal on Scientific Computing, vol. 23, no. 6, pp. 2008–2040, 2002. [Online]. Available: https://doi.org/10.1137/S1064827500377332

    2008 doi

  50. [58]

    Scikit-learn: Machine learning in Python,

    F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P . Prettenhofer, R. Weiss, V . Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,”J. Machine Learning Res...

    2011

  51. [59]

    CodeCarbon, https://doi.org/10.5281/zenodo.21445297

    doi

  52. [60]

    Matplotlib: A 2D graphics environment,

    J. D. Hunter, “Matplotlib: A 2D graphics environment,”Computing in Science & Engineering, vol. 9, no. 3, pp. 90–95, 2007

    2007

  53. [61]

    Array programming with NumPy,

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P . Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R´ıo, M. Wiebe, P . Peterson, P . G´erard-Marchant, K. Sheppar...

    2020 doi

  54. [62]

    Data Structures for Statistical Computing in Python,

    Wes McKinney, “Data Structures for Statistical Computing in Python,” inProceedings of the 9th Python in Science Conference, St ´efan van der Walt and Jarrod Millman, Eds., 2010, pp. 56 – 61

    2010

  55. [63]

    pandas-dev/pandas: Pandas,

    The pandas development team, “pandas-dev/pandas: Pandas,” Feb. 2020. [Online]. Available: https://doi.org/10.5281/zenodo. 3509134

    2020 doi

  56. [2019]

    Available: https://proceedings.neurips.cc/paper/ 2019/file/5f5d472067f77b5c88f69f1bcfda1e08-Paper.pdf

    [Online]. Available: https://proceedings.neurips.cc/paper/ 2019/file/5f5d472067f77b5c88f69f1bcfda1e08-Paper.pdf

    2019

Pith tools

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