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REVIEW 3 major objections 4 minor 56 references

Beyond the Gegenbauer Paradigm: q-Orthogonal Kernels for Machine Learning

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper argues that kernels built from discrete q-Hermite I polynomials need no scaling to stay bounded on [-1,1], making them valid Mercer kernels that match Gegenbauer accuracy while training faster.

desk verdict A useful new kernel with honest experiments, but the no-scaling claim rests on a false bound and the paper's own clipping; fix Proposition 1 and the framing. read the letter →

arxiv 2608.03482 v1 pith:M62AJ6VO submitted 2026-08-04 cs.LG stat.ML

Álvaro Sánchez-Paniagua Ríos , Juan P. Llerena , Alberto Lastra , Nuria Torrado , Edmundo J. Huertas This is my paper
classification cs.LGstat.ML MSC 33D4568T05
keywords supportvectormachineskernelmethodsorthogonalpolynomialkernelsq-HermitepolynomialsMercerq-calculusnumericalstabilitydesign
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 claims that a kernel built from discrete q-Hermite I polynomials, a q-deformed relative of Hermite polynomials indexed by a deformation parameter q in (0,1), is a valid Mercer kernel—symmetric, continuous, and positive semidefinite—whose terms stay bounded on [-1,1] by themselves. If true, that boundedness removes the need for the explicit scaling function that the Gegenbauer kernel construction requires to stop kernel values from vanishing (annihilation) or blowing up (explosion). The paper proves the weight function lies strictly between 0 and 1, proves positive semidefiniteness by writing the kernel as a sum of squared integrals, and reports experiments on 20 datasets where the new kernel ranks third in accuracy behind Gegenbauer and polynomial kernels, beats Gegenbauer on 9 of 20 datasets, and trains faster than Gegenbauer with the speed advantage growing with dimensionality. The payoff, if the claim holds, is a numerically stable SVM kernel whose single parameter q tunes behavior between a classical-Hermite-like regime and strongly deformed regimes.

What carries the argument

The central object is the discrete q-Hermite I polynomial family H_n(x;q) with parameter a = -1, generated by the three-term recurrence xH_n = H_{n+1} + q^{n-1}(1 - q^n)H_{n-1} and supported on [-1,1]. Two properties carry the argument: a claimed uniform bound on these polynomials over [-1,1] (a Fibonacci-type growth constant times (q;q)_∞/(q;q)_n q^{n/2}), and the bivariate q-Pochhammer weight w_{-1,q}(x,z), which is confined to (0,1] through the identity (qx,-qx;q)_∞ = (q²x²;q²)_∞, a product of factors 1 - q^{2m}x². Together they make the kernel expansion bounded term by term, which is what lets the construction skip the scaling function that earlier Gegenbauer-based kernels require. The k

What would settle it

Compute sup_{x in [-1,1]} |H_n(x;q)| directly from the recurrence for q = 0.99 and q = 0.999 and n up to a few hundred; if these suprema grow with n instead of staying bounded, the uniform-boundedness premise fails and the no-scaling claim collapses. A one-line spot check: the proof asserts sup|H_1| = 2 on [-1,1], but H_1(x) = x has supremum 1, so the stated bound cannot be correct as written.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the discrete q-Hermite I polynomials specialized at a = -1 are uniformly bounded on [-1,1], and their associated weight function w_{-1,q}(x,z) = (qx,-qx;q)_∞ (qz,-qz;q)_∞ takes values strictly inside (0,1]. Together these two facts guarantee that the kernel K_Q-HERMITE(x,z) = ∏_j ∑_i H_i(x_j;q)H_i(z_j;q) w_{-1,q}(x_j,z_j) neither annihilates nor explodes, eliminating the scaling function the Gegenbauer formulation needs. The kernel is shown to be a Mercer kernel: its scalar form is a sum of squares of integrals ∫ H_i(x;q)(qx,-qx;q)_∞ g(x) dx, so it is positive semidefinite, and products of positive semidefinite kernels extend the construction t

Load-bearing premise

The load-bearing premise is that the discrete q-Hermite I polynomials are uniformly bounded on [-1,1] for every q in (0,1); the proof of that bound contains a reversed inequality and a wrong supremum for the first polynomial, so the premise is not established as printed for q close to 1, and without it the kernel's no-scaling stability claim unravels.

Editorial extensions

If this is right

  • SVM users gain a valid Mercer kernel that needs no scaling function, removing the annihilation and explosion failure modes that the Gegenbauer construction had to patch with a scaling map and a weight offset.
  • The single parameter q tunes the polynomial family continuously from a classical-Hermite-like regime (q approaching 1) to strongly deformed regimes, giving one formula a tunable kernel family.
  • The training-time advantage over the Gegenbauer kernel grows with feature count (the paper reports r = 0.479, p = 0.0325), so high-dimensional tabular problems are where the kernel pays off most.
  • The weight function can be truncated to a small number of q-Pochhammer factors at negligible cost, keeping the kernel cheap to evaluate while preserving its stability properties.
  • As a Mercer kernel with an explicit feature map, the construction is offered by the authors as a natural basis for quantum support-vector machines, where q-structures already arise.

Reading between the lines

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

  • The no-scaling claim rests on the uniform bound in Proposition 1, and as printed that proof contains a reversed inequality (it states C(n) ≤ sup|H_n| where the comparison yields the opposite direction) and sets sup|H_1| = 2 although H_1(x) = x has supremum 1 on [-1,1]; my inference is that uniform boundedness is not actually established for q near 1, and the stabilization claim needs a corrected p
  • The paper does not analyze how q affects generalization; a natural testable hypothesis, my own, is that q acts like a bandwidth or smoothness parameter analogous to the RBF kernel's γ, with a monotone effect on margin and accuracy that a simple sweep over q could reveal.
  • Because the kernel factorizes per dimension and its feature map is explicit, the mapped features could be plugged into linear models or used without forming the Gram matrix in moderate dimension—a use the paper does not explore.
  • The paper reports a dissociation between support-vector proportion and accuracy (e.g., high accuracy with high PSV on ionosphere and fertility); my inference is that support-vector counts should not be treated as a reliable complexity or performance proxy for this kernel family, despite their use in earlier orthogonal-polynomial kernel studies.
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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

3 major / 4 minor

Summary. The paper proposes a new SVM kernel, K_Q-HERMITE (Eq. 14), built from discrete q-Hermite I polynomials with a=-1 and q in (0,1). It claims Mercer validity, inherent boundedness that makes explicit scaling unnecessary, and reports experiments on 20 datasets showing accuracy competitive with Gegenbauer and faster training times. The Mercer/PSD argument in Section 4 is correct, and the experimental pipeline is substantial. However, the theoretical anchor for the no-scaling claim, Proposition 1, is false as stated, and the weight's lack of a useful lower bound plus the paper's own clipping undermines the advertised advantage. These are fixable by revising the proofs and claims; the kernel construction and empirical comparison remain of interest.

Significance. If the kernel construction and experimental results are taken at face value, the paper contributes a new orthogonal-polynomial kernel with a tunable deformation parameter q and a clean factorization that makes positive semidefiniteness immediate. The implementation is reproducible, and the comparison methodology is more careful than many kernel papers (35 repetitions, 10-fold CV during optimization, Bayesian hyperparameter search, open-source pipeline). The measured training-time advantage over Gegenbauer is a genuine empirical finding. But the advertised absence of scaling is load-bearing and is not established: Proposition 1 is false, and Section 3.2.1 reintroduces clipping. With a corrected argument and a more modest statement of the stability claim, the contribution would be publishable; with the current claims it is not.

major comments (3)
  1. [Section 3.2, Proposition 1 (Eq. 15)] The stated uniform bound is false. For n=1, the LHS is sup_{[-1,1]} |H_1(x;q)| = sup_{[-1,1]} |x| = 1, while the RHS is 2*sqrt(q)*(q;q)_inf/(q;q)_1 = 2*sqrt(q)*prod_{m>=2}(1-q^m), which is <1 for every q in (0,1) and is astronomically small for q=.99. The proof's line 'C(n) <= sup |H_n|' has the inequality reversed: the recurrence gives sup |H_n| <= C(n), not >=. Moreover M_1=2 is wrong for H_1(x)=x, whose sup is 1. The final normalization identity does not repair the bound. Since the no-explosion claim in Sections 3.2.1, the abstract, and the conclusions rests on this proposition, this is a load-bearing error.
  2. [Section 3.2.1, Proposition 2 and clipping] Proposition 2 establishes only 0<w<=1; it gives no positive lower bound. At x=z=1 and q=.99, w_{-1,q}(1,1) = ((q^2;q^2)_inf)^2 is about 10^-44, so raw kernel entries are exponentially small. The paper then clips w<0.1 to 0.1, which is exactly the epsilon-regularization used for the Gegenbauer kernel in Eq. (3). This contradicts the sentence 'such a correction is not required' and the abstract's claim that no explicit scaling mechanisms are needed. Even if clipping is described as purely practical, it is part of the implementation, and the no-scaling advantage is not supported by the stated results.
  3. [Section 3.2.1 vs. Algorithm 1] There is a mismatch between the mathematical definition and the implementation. Section 3.2.1 says that whenever w<0.1, it is clipped to 0.1, but Eq. (14) and Algorithm 1 contain no clipping step. If the experiments use the raw weight, boundary Gram entries can be of order 10^-44, raising numerical concerns. If they use the clipped weight, then the evaluated kernel is not the kernel of Eq. (14) and the no-scaling statement is void. The paper should state exactly which kernel is used in the experiments and report sensitivity to the clipping threshold and to the Pochhammer truncation bound theta.
minor comments (4)
  1. [Section 3.2] Proposition 2 appears twice with two different proofs; consolidate into one statement and one proof.
  2. [Algorithm 1] Typo: 'ilustrates' should be 'illustrates'. The algorithm also does not show the clipping described in Section 3.2.1.
  3. [Eq. (11) and references] The weight function is cited to [34], which is a q-difference-equations paper, not a kernel paper. Since the author lists overlap, please identify the exact formula being borrowed and state explicitly that no kernel result is imported from [34].
  4. [Section 6.1] The paper acknowledges that epsilon=0.1 and theta=10 are not optimized and that no sensitivity analysis is given. A short experiment varying these two parameters would strengthen the numerical-stability claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kernel construction and Mercer proof are self-contained; the only overlapping-author citation is provenance for a formula, not a load-bearing argument.

full rationale

The paper's central derivation is Eq. (14): K_Q-HERMITE(x,z)=∏_j Σ_i H_i(x_j;q)H_i(z_j;q) w_q(x_j,z_j), with w_q(x,z)=(qx,-qx;q)_∞(qz,-qz;q)_∞. The Mercer proof in Section 4 expands the double integral and, because w_q(x,z)=w_q(x)w_q(z), rewrites it as Σ_i (∫ H_i(x)w_q(x)g(x)dx)^2 ≥ 0. This is a direct, self-contained positive-semidefiniteness argument; it does not presuppose the conclusion or depend on the cited [34] as a theorem. [34] is cited only as the source of the weight formula in Section 3.1.1; the overlap of authors is incidental and the formula is a definition with standard q-Pochhammer identities available in [32,33]. The boundedness claim for the weight (Proposition 2) is essentially immediate from the definition. Proposition 1's proof appears to contain mathematical errors (M_1=2 for H_1(x)=x and the inequality direction 'C(n)≤sup|H_n|'), but those are correctness problems, not circularity: the no-explosion conclusion would be unsupported, not identical to its input. The paper's Section 6.1 explicitly concedes that ε=0.1 clipping and Pochhammer truncation are unoptimized design decisions, and Section 3.2.1 actually applies the clipping, softening the 'no explicit scaling' claim; again this is a limitation/contradiction, not a circular reduction. The q hyperparameter is tuned per dataset by Bayesian optimization, but the reported accuracies are benchmark outcomes, not predictions forced by a fitted value. No fitted parameter is renamed a prediction, no uniqueness theorem is imported, and no ansatz is smuggled in as an external result. Therefore no circular step is present.

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

The central kernel depends on the q-deformation parameter q, the degree n, and two implementation constants (clipping threshold and Pochhammer truncation). These are all fitted or chosen rather than derived. No new physical entities are introduced; the mathematical objects (q-Hermite polynomials, weight function) come from prior literature including the authors' own [34].

free parameters (4)
  • q (deformation parameter) = range [0.01, 0.99], selected per dataset by TPE Bayesian optimization
    Controls the q-deformation of the discrete q-Hermite I polynomials; the kernel's shape and numerical behavior are tuned to each dataset through this parameter.
  • n (polynomial degree) = integers 1 to 6, selected per dataset
    Truncation degree in the kernel sum; higher degrees risk overfitting, so the range is bounded by design.
  • weight clipping threshold epsilon = 0.1
    Any kernel weight below 0.1 is clipped to 0.1, contradicting the abstract's 'no explicit scaling' claim; acknowledged as a design decision in Section 6.1.
  • Pochhammer truncation bound theta = 10
    Infinite product for w_q is truncated at theta=10 in implementation; affects numerical resolution of the weight (Section 6.1).
assumptions (4)
  • standard math q-Pochhammer infinite products converge absolutely for 0<q<1 on [-1,1]
    Used in Proposition 2 to prove the weight is finite and in (0,1].
  • standard math Discrete q-Hermite I polynomials satisfy recurrence (10) with gamma_n = q^{n-1}(1-q^n)
    Definition of the polynomial family; from the q-Askey scheme literature [23, 35].
  • domain assumption The weight function w_{-1,q}(x)=(qx,-qx;q)_infinity is the correct orthogonality weight on [-1,1] for these polynomials
    The weight is taken from [34] and the continuous interval [-1,1] is treated as the support, which is nonstandard for Al-Salam-Carlitz I polynomials (discrete lattice orthogonality). The Mercer proof does not require orthogonality, but the 'orthogonal kernel' framing depends on this assumption.
  • standard math Mercer's theorem and the product rule for positive semidefinite kernels
    Used in Section 4 to establish validity of the multivariate kernel.

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Cite this review

Pith. "Pith review of Beyond the Gegenbauer Paradigm: q-Orthogonal Kernels for Machine Learning." pith.science (2026). https://pith.science/paper/M62AJ6VO

@misc{pith2026260803482,
  author       = {Pith},
  title        = {Pith review of: Beyond the Gegenbauer Paradigm: q-Orthogonal Kernels for Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M62AJ6VO}},
  note         = {Machine review of arXiv:2608.03482}
}
abstract

The performance of Support Vector Machines (SVMs) critically depends on the kernel function choice, which enables implicit mapping of data into high-dimensional feature spaces. While classical kernels like Radial Basis Function (RBF) remain popular, orthogonal polynomial kernels offer mathematically interpretable alternatives that can incorporate structured prior knowledge. This work extends the orthogonal polynomial kernel paradigm by introducing a novel family based on discrete $q$-Hermite I polynomials, a class of $q$-orthogonal polynomials that generalize classical Hermite polynomials through a deformation parameter $q$. We formally define the q-Hermite kernel and establish its validity under Mercer's theorem. The kernel's inherent boundedness properties naturally prevent annihilation and explosion effects without requiring explicit scaling mechanisms. Extensive experiments across 20 benchmark datasets demonstrate that the proposed kernel achieves competitive performance compared to both classical kernels and other orthogonal polynomial kernels, while offering advantages in numerical stability and computational simplicity. Our results confirm that $q$-orthogonal polynomials constitute a promising direction for kernel design, bridging mathematical elegance with practical machine learning applications, that provides conceptual and algorithmic resources that may be further extended to emerging quantum computing paradigms. To facilitate full reproducibility, we provide the complete implementation and experimental pipeline in an open-access GitHub repository at https://github.com/Kokechacho/SVMs-QSVMs.

Figures

Figures reproduced from arXiv: 2608.03482 by the authors.

Figure 1
Figure 1. discrete q-Hermite I polynomials (A) Polynomials without the weight function for q = 0.5. (B) Polynomials incorporating the weight function for q = 0.5. 3.1.1 The weight and scaling functions: kernel construction The weight function used in our case is taken from [34]: wa,q(x) = (qx, a−1 qx; q)∞ (11) To obtain the bivariate version, we simply multiply the two univariate functions, as speci￾fied in [37], yielding: wa… view at source ↗
Figure 2
Figure 2. Bivariate weight function w−1,q(x, z) comparison for the q-Hermite polynomials. (A) Weight function for q = 0.05. (B) Weight function for q = 0.99. In the approach by [21], a scaling function is now required to constrain the amplitudes of Gegenbauer polynomials, which diverge beyond the interval [−1, 1] for certain parameter values. In contrast, as demonstrated in the following section, our method eliminates this ne… view at source ↗
Figure 3
Figure 3. Comparative analysis of Q-Hermite and Gegenbauer kernel performance across mul [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Training-time ratio log2 (GEGEN/Q-HERMITE) across 20 UCI datasets as a func￾tion of feature dimensionality. Positive values (red dots) indicate a speed advantage for KqH for that dataset (while blue dots represent datasets where KGegen is faster). A linear green fit re…

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Works this paper leans on

56 extracted references · 49 canonical work pages

  1. [1]

    Survey on svm and their application in image classification.International Journal of Information Technology, 13(5):1–11, 2021

    Mayank Arya Chandra and SS Bedi. Survey on svm and their application in image classification.International Journal of Information Technology, 13(5):1–11, 2021

    work page 2021

  2. [2]

    Applicationsofsupportvectormachine(svm)learningincancergenomics

    Shujun Huang, Nianguang Cai, Pedro Penzuti Pacheco, Shavira Narrandes, Yang Wang, andWayneXu. Applicationsofsupportvectormachine(svm)learningincancergenomics. Cancer genomics & proteomics, 15(1):41–51, 2018

    work page 2018

  3. [3]

    James Mercer. Xvi. functions of positive and negative type, and their connection the theory of integral equations.Philosophical transactions of the royal society of London. Series A, containing papers of a mathematical or physical character, 209(441-458):415– 446, 1909

    work page 1909

  4. [4]

    A review of optimization methodologies in support vector machines.Neurocomputing, 74(17):3609–3618, 2011

    John Shawe-Taylor and Shiliang Sun. A review of optimization methodologies in support vector machines.Neurocomputing, 74(17):3609–3618, 2011

    work page 2011

  5. [5]

    A novel active learning method using svm for text classification.International Journal of Au- tomation and Computing, 15(3):290–298, 2018

    Mohamed Goudjil, Mouloud Koudil, Mouldi Bedda, and Noureddine Ghoggali. A novel active learning method using svm for text classification.International Journal of Au- tomation and Computing, 15(3):290–298, 2018

    work page 2018

  6. [6]

    Face recognition by support vector machines

    Guodong Guo, Stan Z Li, and Kapluk Chan. Face recognition by support vector machines. InProceedings fourth IEEE international conference on automatic face and gesture recog- nition (cat. no. PR00580), pages 196–201. IEEE, 2000

    work page 2000

  7. [7]

    Biological applications of support vector machines.Briefings in bioin- formatics, 5(4):328–338, 2004

    Zheng Rong Yang. Biological applications of support vector machines.Briefings in bioin- formatics, 5(4):328–338, 2004

    work page 2004

  8. [8]

    Quantum support vector machine for classification task: A review

    Muhamad Akrom. Quantum support vector machine for classification task: A review. Journal of Multiscale Materials Informatics, 1(2):1–8, 2024

    work page 2024

Show all 56 references
  1. [9]

    The com- plexity of quantum support vector machines, 2022

    Gian Gentinetta, Arne Thomsen, David Sutter, and Stefan Woerner. The com- plexity of quantum support vector machines, 2022. doi: 10.48550.arXiv preprint ARXIV.2203.00031

    2022 arXiv

  2. [10]

    Performance analysis of classical and quantum support vector machines for diagnosis of chronic kidney disease.Informatics and Health, 2025

    Muhammad Minoar Hossain, Arslan Munir, Md Ahsan Habib, Md Sadiq Iqbal, Md Mosaddik Hasan, and Mohammad Motiur Rahman. Performance analysis of classical and quantum support vector machines for diagnosis of chronic kidney disease.Informatics and Health, 2025

    2025

  3. [11]

    Prac- tical application improvement to quantum svm: theory to practice.arXiv preprint arXiv:2012.07725, 2020

    Jae-Eun Park, Brian Quanz, Steve Wood, Heather Higgins, and Ray Harishankar. Prac- tical application improvement to quantum svm: theory to practice.arXiv preprint arXiv:2012.07725, 2020. 23

    2012 arXiv

  4. [12]

    Ensemble and optimization algorithm in support vector machines for classification of wheat genotypes.Scientific Reports, 14(1):22728, 2024

    Mujahid Khan, BK Hooda, Arpit Gaur, Vikram Singh, Yogesh Jindal, Hemender Tanwar, Sushma Sharma, Sonia Sheoran, Dinesh Kumar Vishwakarma, Mohammad Khalid, et al. Ensemble and optimization algorithm in support vector machines for classification of wheat genotypes.Scientific Rep...

    2024

  5. [13]

    Radial basis function kernel optimization for support vector machine classifiers

    Karl Thurnhofer-Hemsi, Ezequiel López-Rubio, Miguel A Molina-Cabello, and Kayvan Najarian. Radial basis function kernel optimization for support vector machine classifiers. arXiv preprint arXiv:2007.08233, 2020

    2007 arXiv

  6. [14]

    Support vector machines and kernels for computational biology.PLoS computa- tional biology, 4(10):e1000173, 2008

    Asa Ben-Hur, Cheng Soon Ong, Sören Sonnenburg, Bernhard Schölkopf, and Gunnar Rätsch. Support vector machines and kernels for computational biology.PLoS computa- tional biology, 4(10):e1000173, 2008

    2008

  7. [15]

    Exploring kernel machines and support vector machines: Principles, techniques, and future directions

    Ke-Lin Du, Bingchun Jiang, Jiabin Lu, Jingyu Hua, and MNS Swamy. Exploring kernel machines and support vector machines: Principles, techniques, and future directions. Mathematics, 12(24):3935, 2024

    2024

  8. [16]

    Model selection for support vector machines

    Olivier Chapelle and Vladimir Vapnik. Model selection for support vector machines. Advances in neural information processing systems, 12, 1999

    1999

  9. [17]

    An overview on the advancements of support vector machine models in healthcare applications: a review

    Rosita Guido, Stefania Ferrisi, Danilo Lofaro, and Domenico Conforti. An overview on the advancements of support vector machine models in healthcare applications: a review. Information, 15(4):235, 2024

    2024

  10. [18]

    An orthogonal polynomial kernel-based machine learning model for differential-algebraic equations.arXiv preprint arXiv:2401.14382, 2024

    Tayebeh Taheri, Alireza Afzal Aghaei, and Kourosh Parand. An orthogonal polynomial kernel-based machine learning model for differential-algebraic equations.arXiv preprint arXiv:2401.14382, 2024

    2024 arXiv

  11. [19]

    Rational jacobi kernel functions: A novel massively parallelizable orthogonal kernel for support vector machines

    Mahdi Movahedian Moghaddam, Alireza Afzal Aghaei, and Kourosh Parand. Rational jacobi kernel functions: A novel massively parallelizable orthogonal kernel for support vector machines. In2024 Third International Conference on Distributed Computing and High Performance Computing...

    2024

  12. [20]

    New hermite orthogonal poly- nomial kernel and combined kernels in support vector machine classifier.Pattern Recog- nition, 60:921–935, 2016

    Vahid Hooshmand Moghaddam and Javad Hamidzadeh. New hermite orthogonal poly- nomial kernel and combined kernels in support vector machine classifier.Pattern Recog- nition, 60:921–935, 2016

    2016

  13. [21]

    A novel formulation of orthogonal polynomial kernel functions for svm classifiers: The gegenbauer family.Pattern Recognition, 84:211–225, 2018

    LuisCarlosPadierna, MartínCarpio, AlfonsoRojas-Domínguez, HéctorPuga, andHéctor Fraire. A novel formulation of orthogonal polynomial kernel functions for svm classifiers: The gegenbauer family.Pattern Recognition, 84:211–225, 2018

    2018

  14. [22]

    A review of q-difference equations for al-salam–carlitz polynomials and applications to u (n+ 1) type generating functions and ramanujan’s integrals.Mathematics, 11(7):1655, 2023

    Jian Cao, Jin-Yan Huang, Mohammed Fadel, and Sama Arjika. A review of q-difference equations for al-salam–carlitz polynomials and applications to u (n+ 1) type generating functions and ramanujan’s integrals.Mathematics, 11(7):1655, 2023

    2023

  15. [23]

    Hypergeometric orthogonal polynomials

    Roelof Koekoek, Peter A Lesky, and René F Swarttouw. Hypergeometric orthogonal polynomials. InHypergeometric Orthogonal Polynomials and Their q-Analogues, pages 183–253. Springer, 2010

    2010

  16. [24]

    Bivariate continuous q-hermite poly- nomials and deformed quantum serre relations.Journal of Algebra and Its Applications, 20(01):2140016, 2021

    W Riley Casper, Stefan Kolb, and Milen Yakimov. Bivariate continuous q-hermite poly- nomials and deformed quantum serre relations.Journal of Algebra and Its Applications, 20(01):2140016, 2021

    2021

  17. [25]

    Deformed gaussian operators on weighted q-fock spaces.Journal of Stochastic Analysis, 1(4):6, 2020

    Nobuhiro Asai and Hiroaki Yoshida. Deformed gaussian operators on weighted q-fock spaces.Journal of Stochastic Analysis, 1(4):6, 2020. 24

    2020

  18. [26]

    Supportvectormachinewithorthogonal chebyshev kernel

    NingYe, RuixiangSun, YinganLiu, andLinCao. Supportvectormachinewithorthogonal chebyshev kernel. In18th International Conference on Pattern Recognition (ICPR’06), volume 2, pages 752–755. IEEE, 2006

    2006

  19. [27]

    A set of new chebyshev kernel functions for support vector machine pattern classification.Pattern Recognition, 44(7):1435–1447, 2011

    Sedat Ozer, Chi H Chen, and Hakan A Cirpan. A set of new chebyshev kernel functions for support vector machine pattern classification.Pattern Recognition, 44(7):1435–1447, 2011

    2011

  20. [28]

    An adaptive support vector regression based on a new sequence of unified orthogonal polynomials

    Jinwei Zhao, Guirong Yan, Boqin Feng, Wentao Mao, and Junqing Bai. An adaptive support vector regression based on a new sequence of unified orthogonal polynomials. Pattern recognition, 46(3):899–913, 2013

    2013

  21. [29]

    Some sets of orthogonal polynomial kernel functions

    Meng Tian and Wenjian Wang. Some sets of orthogonal polynomial kernel functions. Applied Soft Computing, 61:742–756, 2017

    2017

  22. [30]

    Number 2

    George E Andrews.The theory of partitions. Number 2. Cambridge university press, 1998

    1998

  23. [31]

    Springer Science & Business Media, 2012

    Christian Kassel.Quantum groups, volume 155. Springer Science & Business Media, 2012

    2012

  24. [32]

    Cambridge university press, 2011

    George Gasper and Mizan Rahman.Basic hypergeometric series, volume 96. Cambridge university press, 2011

    2011

  25. [33]

    Springer Science & Business Media, 2012

    Thomas Ernst.A comprehensive treatment of q-calculus. Springer Science & Business Media, 2012

    2012

  26. [34]

    Onsecond order q-difference equations satisfied by al-salam–carlitz i-sobolev type polynomials of higher order.Mathematics, 8(8):1300, 2020

    CarlosHermoso, EdmundoJHuertas, AlbertoLastra, andAnierSoria-Lorente. Onsecond order q-difference equations satisfied by al-salam–carlitz i-sobolev type polynomials of higher order.Mathematics, 8(8):1300, 2020

    2020

  27. [35]

    On combinatorics of al-salam carlitz polynomials.European Journal of Combinatorics, 18(3):295–302, 1997

    Dongsu Kim. On combinatorics of al-salam carlitz polynomials.European Journal of Combinatorics, 18(3):295–302, 1997

    1997

  28. [36]

    Multivariable al–salam & carlitz polynomials associated with the type a q–dunkl kernel.Mathematische Nachrichten, 212(1):5–35, 2000

    TH Baker and PJ Forrester. Multivariable al–salam & carlitz polynomials associated with the type a q–dunkl kernel.Mathematische Nachrichten, 212(1):5–35, 2000

    2000

  29. [37]

    On discrete orthogonal polynomials of several variables.Advances in Applied Mathematics, 33(3):615–632, 2004

    Yuan Xu. On discrete orthogonal polynomials of several variables.Advances in Applied Mathematics, 33(3):615–632, 2004

    2004

  30. [38]

    Constructing support vector machine kernels from orthogonal polynomials for face and speaker verification

    Feng Zhou, Zhigang Fang, and Jie Xu. Constructing support vector machine kernels from orthogonal polynomials for face and speaker verification. InFourth International Conference on Image and Graphics (ICIG 2007), pages 627–632. IEEE, 2007

    2007

  31. [39]

    Bohb: Robust and efficient hyperpa- rameter optimization at scale

    Stefan Falkner, Aaron Klein, and Frank Hutter. Bohb: Robust and efficient hyperpa- rameter optimization at scale. InInternational conference on machine learning, pages 1437–1446. PMLR, 2018

    2018

  32. [40]

    OptunaHub: A platform for black-box optimization.arXiv preprint arXiv:2510.02798, 2025

    Yoshihiko Ozaki, Shuhei Watanabe, and Toshihiko Yanase. OptunaHub: A platform for black-box optimization.arXiv preprint arXiv:2510.02798, 2025

    2025 arXiv

  33. [41]

    Algorithms for hyper- parameter optimization.Advances in neural information processing systems, 24, 2011

    James Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. Algorithms for hyper- parameter optimization.Advances in neural information processing systems, 24, 2011

    2011

  34. [42]

    Hyperparameter optimization

    Matthias Feurer and Frank Hutter. Hyperparameter optimization. InAutomated machine learning: Methods, systems, challenges, pages 3–33. Springer International Publishing Cham, 2019. 25

    2019

  35. [43]

    The entire regularization path for the support vector machine.Journal of Machine Learning Research, 5(Oct):1391– 1415, 2004

    Trevor Hastie, Saharon Rosset, Robert Tibshirani, and Ji Zhu. The entire regularization path for the support vector machine.Journal of Machine Learning Research, 5(Oct):1391– 1415, 2004

    2004

  36. [44]

    Shawkat Ali and Kate Smith-Miles. On optimal degree selection for polynomial kernel with support vector machines: Theoretical and empirical investigations.International Journal of Knowledge-Based and Intelligent Engineering Systems, 11(1):1–18, 2007

    2007

  37. [45]

    Efficient kernel selection via spectral analysis

    Jian Li, Yong Liu, Hailun Lin, Yinliang Yue, and Weiping Wang. Efficient kernel selection via spectral analysis. InIJCAI, pages 2124–2130, 2017

    2017

  38. [46]

    Infinite kernel learning: generalization bounds and algorithms

    Yong Liu, Shizhong Liao, Hailun Lin, Yinliang Yue, and Weiping Wang. Infinite kernel learning: generalization bounds and algorithms. InProceedings of the AAAI Conference on artificial intelligence, volume 31, 2017

    2017

  39. [47]

    Jie Liu and Yubo Xu. T-friedman test: A new statistical test for multiple comparison with an adjustable conservativeness measure.International Journal of Computational Intelligence Systems, 15(1):29, 2022

    2022

  40. [48]

    Performance evaluation in machine learning: the good, the bad, the ugly, and the way forward

    Peter Flach. Performance evaluation in machine learning: the good, the bad, the ugly, and the way forward. InProceedings of the AAAI conference on artificial intelligence, volume 33, pages 9808–9814, 2019

    2019

  41. [49]

    Statistical comparisons of classifiers over multiple data sets.Journal of Machine learning research, 7(Jan):1–30, 2006

    Janez Demšar. Statistical comparisons of classifiers over multiple data sets.Journal of Machine learning research, 7(Jan):1–30, 2006

    2006

  42. [50]

    Salvador García, Alberto Fernández, Julián Luengo, and Francisco Herrera. Advanced nonparametric tests for multiple comparisons in the design of experiments in computa- tionalintelligenceanddatamining: Experimentalanalysisofpower.Information sciences, 180(10):2044–2064, 2010

  43. [51]

    Time for a change: a tutorial for comparing multiple classifiers through bayesian analysis.Journal of Machine Learning Research, 18(77):1–36, 2017

    Alessio Benavoli, Giorgio Corani, Janez Demšar, and Marco Zaffalon. Time for a change: a tutorial for comparing multiple classifiers through bayesian analysis.Journal of Machine Learning Research, 18(77):1–36, 2017

    2017

  44. [52]

    Accounting for variance in machine learning benchmarks.Proceedings of Machine Learning and Systems, 3:747–769, 2021

    Xavier Bouthillier, Pierre Delaunay, Mirko Bronzi, Assya Trofimov, Brennan Nichyporuk, Justin Szeto, Nazanin Mohammadi Sepahvand, Edward Raff, Kanika Madan, Vikram Voleti, et al. Accounting for variance in machine learning benchmarks.Proceedings of Machine Learning and Systems...

    2021

  45. [53]

    The asa statement on p-values: context, process, and purpose, 2016

    Ronald L Wasserstein and Nicole A Lazar. The asa statement on p-values: context, process, and purpose, 2016

    2016

  46. [54]

    Dataset meta-level and statistical features affect machine learning performance.Scientific Reports, 14(1):1670, 2024

    Shahadat Uddin and Haohui Lu. Dataset meta-level and statistical features affect machine learning performance.Scientific Reports, 14(1):1670, 2024

    2024

  47. [55]

    Impact of dataset size on classi- fication performance: an empirical evaluation in the medical domain.Applied sciences, 11(2):796, 2021

    Alhanoof Althnian, Duaa AlSaeed, Heyam Al-Baity, Amani Samha, Alanoud Bin Dris, Najla Alzakari, Afnan Abou Elwafa, and Heba Kurdi. Impact of dataset size on classi- fication performance: an empirical evaluation in the medical domain.Applied sciences, 11(2):796, 2021

    2021

  48. [56]

    Avoiding common machine learning pitfalls.Patterns, 5(10), 2024

    Michael A Lones. Avoiding common machine learning pitfalls.Patterns, 5(10), 2024. 26 Table 8: Average ranks from Friedman tests (the lower values the better). Kernels are sorted by average rank for each metric. Metric Average Rank Linear Hermite RBF Q-Hermite Polynomial Gegenb...

    2024

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