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Journal articles on the topic 'Hilbert–Schmidt kernels'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

ZHANG, HAIZHANG, and LIANG ZHAO. "ON THE INCLUSION RELATION OF REPRODUCING KERNEL HILBERT SPACES." Analysis and Applications 11, no. 02 (2013): 1350014. http://dx.doi.org/10.1142/s0219530513500140.

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To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert–Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.

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2

Heo, Jaeseong. "Projectively invariant Hilbert–Schmidt kernels and convolution type operators." Studia Mathematica 213, no. 1 (2012): 61–79. http://dx.doi.org/10.4064/sm213-1-5.

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3

Ferguson, Sarah H., and Richard Rochberg. "Higher order Hilbert-Schmidt Hankel forms and tensors of analytical kernels." MATHEMATICA SCANDINAVICA 96, no. 1 (2005): 117. http://dx.doi.org/10.7146/math.scand.a-14948.

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The symbols of $n^{\hbox{th}}$-order Hankel forms defined on the product of certain reproducing kernel Hilbert spaces $H(k_{i})$, $i=1,2$, in the Hilbert-Schmidt class are shown to coincide with the orthogonal complement in $H(k_{1})\otimes H(k_{2})$ of the ideal of polynomials which vanish up to order $n$ along the diagonal. For tensor products of weighted Bergman and Dirichlet type spaces (including the Hardy space) we introduce a higher order restriction map which allows us to identify the relative quotient of the $n^{\hbox{th}}$-order ideal modulo the $(n+1)^{\hbox{st}}$-order one as a dir

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Kumar, Hemant, and R. C. Singh Chandel. "A THEORY OF MULTIDIMENSIONAL FREDHOLM INTEGRAL EQUATIONS HAVING SEPARABLE KERNELS: SOLVABLE IN A REGION SURROUNDING BY THE HYPERPLANES." jnanabha 54, no. 01 (2024): 169–79. http://dx.doi.org/10.58250/jnanabha.2024.54121.

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In this article, we present a theory of multidimensional Fredholm integral equations, having separable kernels, are solvable in a region surrounded by hyperplanes. In derivation of their solutions, we employ the generalized Hilbert-Schmidt theory involving eigenvalues and corresponding normalized eigen functions obtained by separable kernels in a region surrounded by the hyperplanes. Finally, we apply two variables Gegenbauer polynomials and derive a result on the inequality of the solution for the double symmetric Fredholm integral equation.

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5

Klimek, Malgorzata. "Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions." Symmetry 13, no. 12 (2021): 2265. http://dx.doi.org/10.3390/sym13122265.

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In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operato

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Cavoretto, Roberto, Gregory E. Fasshauer, and Michael McCourt. "An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels." Numerical Algorithms 68, no. 2 (2014): 393–422. http://dx.doi.org/10.1007/s11075-014-9850-z.

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7

Laumann, Felix, Julius von Kügelgen, Junhyung Park, Bernhard Schölkopf, and Mauricio Barahona. "Kernel-Based Independence Tests for Causal Structure Learning on Functional Data." Entropy 25, no. 12 (2023): 1597. http://dx.doi.org/10.3390/e25121597.

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Measurements of systems taken along a continuous functional dimension, such as time or space, are ubiquitous in many fields, from the physical and biological sciences to economics and engineering. Such measurements can be viewed as realisations of an underlying smooth process sampled over the continuum. However, traditional methods for independence testing and causal learning are not directly applicable to such data, as they do not take into account the dependence along the functional dimension. By using specifically designed kernels, we introduce statistical tests for bivariate, joint, and co

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8

Friesen, Martin, and Sven Karbach. "Stationary covariance regime for affine stochastic covariance models in Hilbert spaces." Finance and Stochastics 28, no. 4 (2024): 1077–116. http://dx.doi.org/10.1007/s00780-024-00543-3.

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AbstractThis paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive self-adjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order $p \in [1,2]$ p ∈ [ 1 , 2 ] , and give explicit

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9

Oehring, Charles. "Singular numbers of smooth kernels." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (1988): 511–14. http://dx.doi.org/10.1017/s0305004100065129.

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In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit

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10

Raman, S. Ganapathi, and R. Vittal Rao. "Extended Kac-Akhiezer formulae and the Fredholm determinant of finite section Hilbert-Schmidt kernels." Proceedings Mathematical Sciences 104, no. 3 (1994): 581–91. http://dx.doi.org/10.1007/bf02867122.

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11

B�ttcher, A., B. Silbermann, and H. Widom. "Determinants of truncated Wiener-Hopf operators with Hilbert-Schmidt kernels and piecewise continuous symbols." Archiv der Mathematik 63, no. 1 (1994): 60–71. http://dx.doi.org/10.1007/bf01196300.

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12

El Amri, Mohamed Reda, and Amandine Marrel. "MORE POWERFUL HSIC-BASED INDEPENDENCE TESTS, EXTENSION TO SPACE-FILLING DESIGNS AND FUNCTIONAL DATA." International Journal for Uncertainty Quantification 14, no. 2 (2024): 69–98. http://dx.doi.org/10.1615/int.j.uncertaintyquantification.v14.i2.40.

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The Hilbert-Schmidt independence criterion (HSIC) is a dependence measure based on reproducing kernel Hilbert spaces. This measure can be used for the global sensitivity analysis of numerical simulators whose objective is to identify the most influential inputs on the output(s) of the code. For this purpose, HSIC-based sensitivity measures and independence tests can be used for the ranking and screening of inputs, respectively. In this framework, this work proposes several improvements in the use of HSIC to increase their application spectrum and make the associated independence tests more pow

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13

Gerth, D. ,., and B. Hofmann,. "A NOTE ON OPEN QUESTIONS ASKED TO ANALYSIS AND NUMERICS CONCERNING THE HAUSDORFF MOMENT PROBLEM." Eurasian Journal of Mathematical and Computer Applications 10, no. 1 (2022): 40–50. http://dx.doi.org/10.32523/2306-6172-2022-10-1-40-50.

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Abstract We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function x ∈ L 2 (0, 1) from elements of the infinite dimensional sequence space ` 2 that characterize moments applied to the antiderivative of x. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator A, which is the composition of the compact simple integration operator mapping in L 2 (0, 1) and the non-compact Hausdorff moment operator B(H) mapping from L 2 (0, 1) to ` 2 .

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14

Martı́nez-Morales, José L. "The Kernel Theorem of Hilbert–Schmidt operators." Journal of Mathematical Analysis and Applications 279, no. 1 (2003): 71–77. http://dx.doi.org/10.1016/s0022-247x(02)00391-8.

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15

Gumah, Ghaleb. "Reproducing kernel Hilbert space method to solve fuzzy partial Volterra integro-differential equations." Filomat 38, no. 24 (2024): 8553–64. https://doi.org/10.2298/fil2424553g.

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In this paper, a reproducing kernel Hilbert space method for the numerical solution of fuzzy partial Volterra integro-differential equations has been presented. The reproducing Hilbert space, kernel function properties, Gram-Schmidt orthogonalization process and the bounded linear operator in the same space have been developed, which helps this method to demonstrate the convergence analysis. Moreover, we present some lemmas and theorems to prove the convergence of the reproducing kernel Hilbert space method. In this method, we give the approximate solution of the fuzzy partial Volterra integro

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16

Liu, Xi, Zengrong Zhan, and Guo Niu. "Hilbert–Schmidt Independence Criterion Regularization Kernel Framework on Symmetric Positive Definite Manifolds." Mathematical Problems in Engineering 2021 (October 11, 2021): 1–11. http://dx.doi.org/10.1155/2021/2402292.

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Image recognition tasks involve an increasingly high amount of symmetric positive definite (SPD) matrices data. SPD manifolds exhibit nonlinear geometry, and Euclidean machine learning methods cannot be directly applied to SPD manifolds. The kernel trick of SPD manifolds is based on the concept of projecting data onto a reproducing kernel Hilbert space. Unfortunately, existing kernel methods do not consider the connection of SPD matrices and linear projections. Thus, a framework that uses the correlation between SPD matrices and projections to model the kernel map is proposed herein. To realiz

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17

SUGIYAMA, Masashi, and Makoto YAMADA. "On Kernel Parameter Selection in Hilbert-Schmidt Independence Criterion." IEICE Transactions on Information and Systems E95.D, no. 10 (2012): 2564–67. http://dx.doi.org/10.1587/transinf.e95.d.2564.

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18

Wang, Tinghua, and Wei Li. "Kernel learning and optimization with Hilbert–Schmidt independence criterion." International Journal of Machine Learning and Cybernetics 9, no. 10 (2017): 1707–17. http://dx.doi.org/10.1007/s13042-017-0675-7.

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19

Zhang, Bingyuan, and Joe Suzuki. "Extending Hilbert–Schmidt Independence Criterion for Testing Conditional Independence." Entropy 25, no. 3 (2023): 425. http://dx.doi.org/10.3390/e25030425.

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The Conditional Independence (CI) test is a fundamental problem in statistics. Many nonparametric CI tests have been developed, but a common challenge exists: the current methods perform poorly with a high-dimensional conditioning set. In this paper, we considered a nonparametric CI test using a kernel-based test statistic, which can be viewed as an extension of the Hilbert–Schmidt Independence Criterion (HSIC). We propose a local bootstrap method to generate samples from the null distribution H0:X⫫Y∣Z. The experimental results showed that our proposed method led to a significant performance i

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20

Xiao, Min, and Yuhong Guo. "Semi-Supervised Kernel Matching for Domain Adaptation." Proceedings of the AAAI Conference on Artificial Intelligence 26, no. 1 (2021): 1183–89. http://dx.doi.org/10.1609/aaai.v26i1.8292.

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In this paper, we propose a semi-supervised kernel matching method to address domain adaptation problems where the source distribution substantially differs from the target distribution. Specifically, we learn a prediction function on the labeled source data while mapping the target data points to similar source data points by matching the target kernel matrix to a submatrix of the source kernel matrix based on a Hilbert Schmidt Independence Criterion. We formulate this simultaneous learning and mapping process as a non-convex integer optimization problem and present a local minimization proce

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21

Wang, Tinghua, Jie Lu, and Guangquan Zhang. "Two-Stage Fuzzy Multiple Kernel Learning Based on Hilbert–Schmidt Independence Criterion." IEEE Transactions on Fuzzy Systems 26, no. 6 (2018): 3703–14. http://dx.doi.org/10.1109/tfuzz.2018.2848224.

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22

Liu, Xi, Peng Yang, Zengrong Zhan, and Zhengming Ma. "Hilbert–Schmidt Independence Criterion Subspace Learning on Hybrid Region Covariance Descriptor for Image Classification." Mathematical Problems in Engineering 2021 (July 21, 2021): 1–15. http://dx.doi.org/10.1155/2021/6663710.

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The region covariance descriptor (RCD), which is known as a symmetric positive definite (SPD) matrix, is commonly used in image representation. As SPD manifolds have a non-Euclidean geometry, Euclidean machine learning methods are not directly applicable to them. In this work, an improved covariance descriptor called the hybrid region covariance descriptor (HRCD) is proposed. The HRCD incorporates the mean feature information into the RCD to improve the latter’s discriminative performance. To address the non-Euclidean properties of SPD manifolds, this study also proposes an algorithm called th

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23

Al-Smadi, Mohammed. "Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces." Filomat 33, no. 2 (2019): 583–97. http://dx.doi.org/10.2298/fil1902583a.

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Integral equations under uncertainty are utilized to describe different formulations of physical phenomena in nature. This paper aims to obtain analytical and approximate solutions for a class of integral equations under uncertainty. The scheme presented here is based upon the reproducing kernel theory and the fuzzy real-valued mappings. The solution methodology transforms the linear fuzzy integral equation to crisp linear system of integral equations. Several reproducing kernel spaces are defined to investigate the approximate solutions, convergence and the error estimate in terms of uniform

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24

Farzaneh Javan, Sedigheh, Saeid Abbasbandy, and M. Ali Fariborzi Araghi. "Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations." Mathematical Problems in Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/7498136.

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A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.

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25

Du, Ming-Jing, Yu-Lan Wang, and Temuer Chaolu. "Numerical Simulation Characteristics of Logging Response in Water Injection Well by Reproducing Kernel Method." Mathematical Problems in Engineering 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/272856.

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Reproducing kernel Hilbert space method (RKHSM) is an effective method. This paper, for the first time, uses the traditional RKHSM for solving the temperature field in two phase flows of multilayer water injection well. According to 2D oil-water temperature field mathematical model of two phase flows in cylindrical coordinates, selecting the properly initial and boundary conditions, by the process of Gram-Schmidt orthogonalization, the analytical solution was given by reproducing kernel functions in a series expansion form, and the approximate solution was expressed byn-term summation. The sat

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26

Yamada, Makoto, Wittawat Jitkrittum, Leonid Sigal, Eric P. Xing, and Masashi Sugiyama. "High-Dimensional Feature Selection by Feature-Wise Kernelized Lasso." Neural Computation 26, no. 1 (2014): 185–207. http://dx.doi.org/10.1162/neco_a_00537.

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The goal of supervised feature selection is to find a subset of input features that are responsible for predicting output values. The least absolute shrinkage and selection operator (Lasso) allows computationally efficient feature selection based on linear dependency between input features and output values. In this letter, we consider a feature-wise kernelized Lasso for capturing nonlinear input-output dependency. We first show that with particular choices of kernel functions, nonredundant features with strong statistical dependence on output values can be found in terms of kernel-based indep

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27

Damodaran, Bharath Bhushan, Nicolas Courty, and Sebastien Lefevre. "Sparse Hilbert Schmidt Independence Criterion and Surrogate-Kernel-Based Feature Selection for Hyperspectral Image Classification." IEEE Transactions on Geoscience and Remote Sensing 55, no. 4 (2017): 2385–98. http://dx.doi.org/10.1109/tgrs.2016.2642479.

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28

Chiappinelli, Raffaele. "Variational Methods for NLEV Approximation Near a Bifurcation Point." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–32. http://dx.doi.org/10.1155/2012/102489.

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We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear ellip

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29

ALKAUSKAS, GIEDRIUS. "THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION." Glasgow Mathematical Journal 52, no. 1 (2009): 41–64. http://dx.doi.org/10.1017/s0017089509990152.

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AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distrib

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30

Blower, Gordon, and Yang Chen. "On Determinant Expansions for Hankel Operators." Concrete Operators 7, no. 1 (2020): 13–44. http://dx.doi.org/10.1515/conop-2020-0002.

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AbstractLet w be a semiclassical weight that is generic in Magnus’s sense, and ({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ1Γϕ2) where Γϕ1 and Γϕ2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2mF2m-1. These re

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Januškevičienė, Olga. "Apie išsigimusios U-statistikos konvergavimo greitį." Lietuvos matematikos rinkinys 45 (December 18, 2005): 533–38. http://dx.doi.org/10.15388/lmr.2005.29319.

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Let X, X1, X2, ... be independent identically distributed random variables taking values in a measurable space (Ω, ℜ). Let h(x, y) be real valued measurable symmetric function of the arguments x, y ∈ ℜ.Assume that Eh(x, X)= 0, for all x. We consider U-statistics of type T = n−1 ∑1 ≤ i< k ≤ n h(Xi, Xk). Let qi, i ≥ 1 be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x, y) and q1 be the largest eigenvalue. Under the condition β3 := E|h(X, X1)|3 <∞, we prove that Δn = ρ(T, T0) ≤ cβ3q−11 n−1/7 +cq−11∑i ≥ 1 qin−1/4, where T0 is the limit statistic and ρ is a Kolmogor

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32

Grigoryan, Gevorg A. "Special factorization of a non-invertible integral Fredholm operator of the second kind with Hilbert-Schmidt kernel." Sbornik: Mathematics 198, no. 5 (2007): 627–37. http://dx.doi.org/10.1070/sm2007v198n05abeh003852.

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33

Zhang, Xin, Tinghua Wang, and Zhiyong Lai. "A Feature-Weighted Support Vector Regression Machine Based on Hilbert–Schmidt Independence Criterion Least Absolute Shrinkage and Selection Operator." Information 15, no. 10 (2024): 639. http://dx.doi.org/10.3390/info15100639.

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Support vector regression (SVR) is a powerful kernel-based regression prediction algorithm that performs excellently in various application scenarios. However, for real-world data, the general SVR often fails to achieve good predictive performance due to its inability to assess feature contribution accurately. Feature weighting is a suitable solution to address this issue, applying correlation measurement methods to obtain reasonable weights for features based on their contributions to the output. In this paper, based on the idea of a Hilbert–Schmidt independence criterion least absolute shrin

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Bachir, Ahmed, and Abdelkader Segres. "Asymmetric Putnam-Fuglede Theorem for (n,k)-Quasi-∗-Paranormal Operators." Symmetry 11, no. 1 (2019): 64. http://dx.doi.org/10.3390/sym11010064.

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T ∈ B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k and n, ∥ T ∗ ( T k x ) ∥ ( 1 + n ) ≤ ∥ T ( 1 + n ) ( T k x ) ∥ ∥ T k x ∥ n ; for all x ∈ H . In this paper, the asymmetric Putnam-Fuglede theorem for the pair ( A , B ) of power-bounded operators is proved when (i) A and B ∗ are n-∗-paranormal operators (ii) A is a ( n , k ) -quasi-∗-paranormal operator with reduced kernel and B ∗ is n-∗-paranormal operator. The class of ( n , k ) -quasi-∗-paranormal operators properly contains the classes of n-∗-paranormal operators, ( 1 , k ) -quasi-∗-paranormal

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Zhang, Hao, Yewei Xia, Yixin Ren, Jihong Guan, and Shuigeng Zhou. "Differentially Private Nonlinear Causal Discovery from Numerical Data." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 10 (2023): 12321–28. http://dx.doi.org/10.1609/aaai.v37i10.26452.

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Recently, several methods such as private ANM, EM-PC and Priv-PC have been proposed to perform differentially private causal discovery in various scenarios including bivariate, multivariate Gaussian and categorical cases. However, there is little effort on how to conduct private nonlinear causal discovery from numerical data. This work tries to challenge this problem. To this end, we propose a method to infer nonlinear causal relations from observed numerical data by using regression-based conditional independence test (RCIT) that consists of kernel ridge regression (KRR) and Hilbert-Schmidt i

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36

Hadi Gabber, Ali. "Approximate Solution of Fuzzy Caputo's-Katugampola differential equation with order 0<Beta<1." Wasit Journal for Pure sciences 4, no. 1 (2025): 12–28. https://doi.org/10.31185/wjps.593.

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In this paper that some of fuzzy second order Caputo's- Katugampola fractional which included also the first order Caputo's- Katugmpola fractional have been presented with analytic interesting result to explain the solution in fuzzy real numbers and distinguish space included the type of functions which suitable to the problem formulations which are under studied. all the fuzzy results are supported the numerical solutions that which used later on. The interesting illustrative examples for application some classes of fuzzy Caputo- Katugampla fractional order differential equations with and exp

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37

Yun, Hyeongu, Taegwan Kang, and Kyomin Jung. "Analyzing and Controlling Inter-Head Diversity in Multi-Head Attention." Applied Sciences 11, no. 4 (2021): 1548. http://dx.doi.org/10.3390/app11041548.

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Multi-head attention, a powerful strategy for Transformer, is assumed to utilize information from diverse representation subspaces. However, measuring diversity between heads’ representations or exploiting the diversity has been rarely studied. In this paper, we quantitatively analyze inter-head diversity of multi-head attention by applying recently developed similarity measures between two deep representations: Singular Vector Canonical Correlation Analysis (SVCCA) and Centered Kernel Alignment (CKA). By doing so, we empirically show that multi-head attention does diversify representation sub

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38

Nguyen, Dai Hai, Canh Hao Nguyen, and Hiroshi Mamitsuka. "ADAPTIVE: leArning DAta-dePendenT, concIse molecular VEctors for fast, accurate metabolite identification from tandem mass spectra." Bioinformatics 35, no. 14 (2019): i164—i172. http://dx.doi.org/10.1093/bioinformatics/btz319.

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Abstract Motivation Metabolite identification is an important task in metabolomics to enhance the knowledge of biological systems. There have been a number of machine learning-based methods proposed for this task, which predict a chemical structure of a given spectrum through an intermediate (chemical structure) representation called molecular fingerprints. They usually have two steps: (i) predicting fingerprints from spectra; (ii) searching chemical compounds (in database) corresponding to the predicted fingerprints. Fingerprints are feature vectors, which are usually very large to cover all

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Zhou, Haohao, Hao Wang, Yijie Ding, and Jijun Tang. "Multivariate information fusion for identifying antifungal peptides with Hilbert-Schmidt Independence Criterion." Current Bioinformatics 16 (July 27, 2021). http://dx.doi.org/10.2174/1574893616666210727161003.

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Background: Antifungal peptides (AFP) have been found to be effective against many fungal infections. Objective: However, it is difficult to identify AFP. Therefore, it is great practical significance to identify AFP via machine learning methods (with sequence information). Method: In this study, a Multi-Kernel Support Vector Machine (MKSVM) with Hilbert-Schmidt Independence Criterion (HSIC) is proposed. Proteins are encoded with five types of features (188-bit, AAC, ASDC, CKSAAP, DPC), and then construct kernels using Gaussian kernel function. HSIC are used to combine kernels and multi-kernel

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40

Gauthier, Bertrand. "Kernel embedding of measures and low-rank approximation of integral operators." Positivity 28, no. 2 (2024). http://dx.doi.org/10.1007/s11117-024-01041-8.

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AbstractWe describe a natural coisometry from the Hilbert space of all Hilbert-Schmidt operators on a separable reproducing kernel Hilbert space $$\hbox { (RKHS)}\, \mathcal {H}$$ (RKHS) H and onto the RKHS $$\mathcal {G}$$ G associated with the squared-modulus of the reproducing kernel of $$\mathcal {H}$$ H . Through this coisometry, trace-class integral operators defined by general measures and the reproducing kernel of $$\mathcal {H}$$ H are isometrically represented as potentials in $$\mathcal {G}$$ G , and the quadrature approximation of these operators is equivalent to the approximation

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41

Putinar, Mihai. "Moment Estimates of the Cloud of a Planar Measure." Acta Applicandae Mathematicae 176, no. 1 (2021). http://dx.doi.org/10.1007/s10440-021-00443-0.

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AbstractWith a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.

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Bertola, M., E. Blackstone, A. Katsevich, and A. Tovbis. "On singular limits of finite Hilbert transform operators on multi‐intervals." Mathematische Nachrichten, November 2, 2023. http://dx.doi.org/10.1002/mana.202300185.

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AbstractInthis paper, we study the small‐λ spectral asymptotics of an integral operator defined on two multi‐intervals J and E, when the multi‐intervals touch each other (but their interiors are disjoint). The operator is closely related to the multi‐interval finite Hilbert transform (FHT). This case can be viewed as a singular limit of self‐adjoint Hilbert–Schmidt integral operators with so‐called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when , and is of the Hilbert–Schmidt class, was studied in an earlier pape

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"An identity for reproducing kernels in a planar domain and Hilbert-Schmidt Hankel operators." Journal für die reine und angewandte Mathematik (Crelles Journal) 1990, no. 406 (1990): 179–99. http://dx.doi.org/10.1515/crll.1990.406.179.

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Moeller, Moritz, and Tino Ullrich. "$$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace." Sampling Theory, Signal Processing, and Data Analysis 19, no. 2 (2021). http://dx.doi.org/10.1007/s43670-021-00013-3.

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AbstractIn this paper we study $$L_2$$ L 2 -norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $$D \subset \mathbb {R}^d$$ D ⊂ R d based on random function samples. We only assume the finite trace of the kernel (Hilbert–Schmidt embedding into $$L_2$$ L 2 ) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomiall

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J., Emmanuel Johnson Valero Laparra Adrián Pérez-Suay Miguel D. Mahecha Gustau Camps-Valls. "Kernel methods and their derivatives: Concept and perspectives for the earth system sciences." October 29, 2020. https://doi.org/10.1371/ journal.pone.0235885.

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Kernel methods are powerful machine learning techniques which use generic non-linear functions to solve complex tasks. They have a solid mathematical foundation and exhibit excellent performance in practice. However, kernel machines are still considered black-box models as the kernel feature mapping cannot be accessed directly thus making the kernels difficult to interpret. The aim of this work is to show that it is indeed possible to interpret the functions learned by various kernel methods as they can be intuitive despite their complexity. Specifically, we show that derivatives of these func

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Boullé, Nicolas, and Alex Townsend. "Learning Elliptic Partial Differential Equations with Randomized Linear Algebra." Foundations of Computational Mathematics, January 18, 2022. http://dx.doi.org/10.1007/s10208-022-09556-w.

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AbstractGiven input–output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically rigorous scheme for learning the associated Green’s function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of $$\mathcal {O}(\varGamma _\epsilon ^{-1/2}\log ^3(1/\epsilon )\epsilon )$$ O ( Γ ϵ - 1 / 2 log 3 ( 1 / ϵ ) ϵ ) using at most $$\mathcal {O}(\epsilon ^{-6}\log ^4(1/\epsilon ))$$ O ( ϵ - 6 log 4 ( 1 / ϵ ) ) input–output training

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Szabłowski, Paweł J. "On a class of bivariate distributions built of q-ultraspherical polynomials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, December 2, 2024, 1–34. https://doi.org/10.1017/prm.2024.99.

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Our primary result concerns the positivity of specific kernels constructed using the q-ultraspherical polynomials. In other words, it concerns a two-parameter family of bivariate, compactly supported distributions. Moreover, this family has a property that all its conditional moments are polynomials in the conditioning random variable. The significance of this result is evident for individuals working on distribution theory, orthogonal polynomials, q-series theory, and the so-called quantum polynomials. Therefore, it may have a limited number of interested researchers. That is why, we put our

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Skrettingland, Eirik. "Equivalent Norms for Modulation Spaces from Positive Cohen’s Class Distributions." Journal of Fourier Analysis and Applications 28, no. 2 (2022). http://dx.doi.org/10.1007/s00041-022-09930-w.

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AbstractWe give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert–Schmidt operator. The main result is applied to Cohen’s class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen’s class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert–Schmidt operator must satisfy a nuclearity condition for these results to hold

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Fardi, Mojtaba, and Yasir Khan. "Numerical simulation of squeezing Cu–Water nanofluid flow by a kernel-based method." International Journal of Modeling, Simulation, and Scientific Computing, August 25, 2021, 2250005. http://dx.doi.org/10.1142/s1793962322500052.

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The main aim of this paper is to propose a kernel-based method for solving the problem of squeezing Cu–Water nanofluid flow between parallel disks. Our method is based on Gaussian Hilbert–Schmidt SVD (HS-SVD), which gives an alternate basis for the data-dependent subspace of “native” Hilbert space without ever forming kernel matrix. The well-conditioning linear system is one of the critical advantages of using the alternate basis obtained from HS-SVD. Numerical simulations are performed to illustrate the efficiency and applicability of the proposed method in the sense of accuracy. Numerical re

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Srashidinov, A. "TRANSITION METHOD FOR CONVOLUTION EQUATIONS FOR EXAMPLE." Bulletin of Issyk-Kul University, July 30, 2024. http://dx.doi.org/10.69722/1694-8211-2024-57-40-48.

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The essence of the so-called transition method for convolution equations [1] is that the integral equations of the first and second kind of Volterra convolution with the help of a special continuation of the kernel , solution and free term from the interval [0,1] to the interval [0,2] is reduced to equivalent to the Fredholm equations with a difference kernel of the first or second kind, which allows the well-known rich Hilbert-Schmitt theory of symmetric operators in a Hilbert space to be applied to the latter. All this is demonstrated on a private measurement. For the convenience of the read

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