In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then c...In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then construct a sequence of approximating operators with the help of Jacobi orthogonal polynomials, with which we establish a kind of Jackson inequality to describe the error estimate. Finally, The results are used to discuss an approximation problem arising from learning theory.展开更多
The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the s...The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.展开更多
This research proposes a method called enhanced collaborative andgeometric multi-kernel learning (E-CGMKL) that can enhance the CGMKLalgorithm which deals with multi-class classification problems with non-lineardata d...This research proposes a method called enhanced collaborative andgeometric multi-kernel learning (E-CGMKL) that can enhance the CGMKLalgorithm which deals with multi-class classification problems with non-lineardata distributions. CGMKL combines multiple kernel learning with softmaxfunction using the framework of multi empirical kernel learning (MEKL) inwhich empirical kernel mapping (EKM) provides explicit feature constructionin the high dimensional kernel space. CGMKL ensures the consistent outputof samples across kernel spaces and minimizes the within-class distance tohighlight geometric features of multiple classes. However, the kernels constructed by CGMKL do not have any explicit relationship among them andtry to construct high dimensional feature representations independently fromeach other. This could be disadvantageous for learning on datasets with complex hidden structures. To overcome this limitation, E-CGMKL constructskernel spaces from hidden layers of trained deep neural networks (DNN).Due to the nature of the DNN architecture, these kernel spaces not onlyprovide multiple feature representations but also inherit the compositionalhierarchy of the hidden layers, which might be beneficial for enhancing thepredictive performance of the CGMKL algorithm on complex data withnatural hierarchical structures, for example, image data. Furthermore, ourproposed scheme handles image data by constructing kernel spaces from aconvolutional neural network (CNN). Considering the effectiveness of CNNarchitecture on image data, these kernel spaces provide a major advantageover the CGMKL algorithm which does not exploit the CNN architecture forconstructing kernel spaces from image data. Additionally, outputs of hiddenlayers directly provide features for kernel spaces and unlike CGMKL, do notrequire an approximate MEKL framework. E-CGMKL combines the consistency and geometry preserving aspects of CGMKL with the compositionalhierarchy of kernel spaces extracted from DNN hidden layers to enhance the predictive performance of CGMKL significantly. The experimental results onvarious data sets demonstrate the superior performance of the E-CGMKLalgorithm compared to other competing methods including the benchmarkCGMKL.展开更多
Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods ...Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods in neural networks,such as using complex network architectures and introducing sparse techniques,always suffer from the difficulty of estimating hyperparameters and the lack of physical interpretability.To address this issue,this paper proposes a novel interpretable denoising layer based on reproducing kernel Hilbert space(RKHS)as the first layer for standard neural networks,with the aim to combine the advantages of both traditional signal processing technology with physical interpretation and network modeling strategy with parameter adaption.By investigating the influencing mechanism of parameters on the regularization procedure in RKHS,the key parameter that dynamically controls the signal smoothness with low computational cost is selected as the only trainable parameter of the proposed layer.Besides,the forward and backward propagation algorithms of the designed layer are formulated to ensure that the selected parameter can be automatically updated together with other parameters in the neural network.Moreover,exponential and piecewise functions are introduced in the weight updating process to keep the trainable weight within a reasonable range and avoid the ill-conditioned problem.Experiment studies verify the effectiveness and compatibility of the proposed layer design method in intelligent fault diagnosis of machinery in noisy environments.展开更多
The kernel function method in support vector machine(SVM)is an excellent tool for nonlinear classification.How to design a kernel function is difficult for an SVM nonlinear classification problem,even for the polynomi...The kernel function method in support vector machine(SVM)is an excellent tool for nonlinear classification.How to design a kernel function is difficult for an SVM nonlinear classification problem,even for the polynomial kernel function.In this paper,we propose a new kind of polynomial kernel functions,called semi-tensor product kernel(STP-kernel),for an SVM nonlinear classification problem by semi-tensor product of matrix(STP)theory.We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel.In addition,we have shown the existence of the reproducing kernel Hilbert space(RKHS)associated with the STP-kernel function.Compared to the existing methods,it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.展开更多
It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in C[a, b] or L2 [a, b]. In this paper, the representation of the solution f...It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in C[a, b] or L2 [a, b]. In this paper, the representation of the solution for Fredholm integral equation of the first kind is given if it has a unique solution. The stability of the solution is proved in the reproducing kernel space, namely, the measurement errors of the experimental data cannot result in unbounded errors of the true solution. The computation of approximate solution is also stable with respect to ｜｜· ｜｜c or ｜｜L2· A numerical experiment shows that the method given in this paper is stable in the reproducing kernel space.展开更多
In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel fu...In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.展开更多
The application of high-performance imaging sensors in space-based space surveillance systems makes it possible to recognize space objects and estimate their poses using vision-based methods. In this paper, we propose...The application of high-performance imaging sensors in space-based space surveillance systems makes it possible to recognize space objects and estimate their poses using vision-based methods. In this paper, we proposed a kernel regression-based method for joint multi-view space object recognition and pose estimation. We built a new simulated satellite image dataset named BUAA-SID 1.5 to test our method using different image representations. We evaluated our method for recognition-only tasks, pose estimation-only tasks, and joint recognition and pose estimation tasks. Experimental results show that our method outperforms the state-of-the-arts in space object recognition, and can recognize space objects and estimate their poses effectively and robustly against noise and lighting conditions.展开更多
Many websites use verification codes to prevent users from using the machine automatically to register,login,malicious vote or irrigate but it brought great burden to the enterprises involved in internet marketing as ...Many websites use verification codes to prevent users from using the machine automatically to register,login,malicious vote or irrigate but it brought great burden to the enterprises involved in internet marketing as entering the verification code manually.Improving the verification code security system needs the identification method as the corresponding testing system.We propose an anisotropic heat kernel equation group which can generate a heat source scale space during the kernel evolution based on infinite heat source axiom,design a multi-step anisotropic verification code identification algorithm which includes core procedure of building anisotropic heat kernel,settingwave energy information parameters,combing outverification codccharacters and corresponding peripheral procedure of gray scaling,binarizing,denoising,normalizing,segmenting and identifying,give out the detail criterion and parameter set.Actual test show the anisotropic heat kernel identification algorithm can be used on many kinds of verification code including text characters,mathematical,Chinese,voice,3D,programming,video,advertising,it has a higher rate of 25%and 50%than neural network and context matching algorithm separately for Yahoo site,49%and 60%for Captcha site,20%and 52%for Baidu site,60%and 65%for 3DTakers site,40%,and 51%.for MDP site.展开更多
In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the sec...In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations.Then,the uniform convergence of the numerical solution is proved,and the time consuming Schmidt orthogonalization process is avoided.The algorithm is employed successfully on some numerical examples.展开更多
Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the itera...Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.展开更多
Complementary-label learning(CLL)aims at finding a classifier via samples with complementary labels.Such data is considered to contain less information than ordinary-label samples.The transition matrix between the tru...Complementary-label learning(CLL)aims at finding a classifier via samples with complementary labels.Such data is considered to contain less information than ordinary-label samples.The transition matrix between the true label and the complementary label,and some loss functions have been developed to handle this problem.In this paper,we show that CLL can be transformed into ordinary classification under some mild conditions,which indicates that the complementary labels can supply enough information in most cases.As an example,an extensive misclassification error analysis was performed for the Kernel Ridge Regression(KRR)method applied to multiple complementary-label learning(MCLL),which demonstrates its superior performance compared to existing approaches.展开更多
Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the mod...Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the model bias comes from a reproducing kernel Hilbert space. Two different design criteria are proposed by applying the minimax approach for estimating the parameters of the regression response and extrapolating the regression response to points outside of the design space. A simulated annealing algorithm is applied to construct the minimax designs. These minimax designs are compared with the classical D-optimal designs and all-bias extrapolation designs. Numerical results indicate that the simulated annealing algorithm is feasible and the minimax designs are robust against bias caused by model misspecification.展开更多
The conditional kernel correlation is proposed to measure the relationship between two random variables under covariates for multivariate data.Relying on the framework of reproducing kernel Hilbert spaces,we give the ...The conditional kernel correlation is proposed to measure the relationship between two random variables under covariates for multivariate data.Relying on the framework of reproducing kernel Hilbert spaces,we give the definitions of the conditional kernel covariance and conditional kernel correlation.We also provide their respective sample estimators and give the asymptotic properties,which help us construct a conditional independence test.According to the numerical results,the proposed test is more effective compared to the existing one under the considered scenarios.A real data is further analyzed to illustrate the efficacy of the proposed method.展开更多
By combining the wavelet decomposition with kernel method, a practical approach of universal multiscale wavelet kernels constructed in reproducing kernel Hilbert space （RKHS） is discussed, and an identification sche...By combining the wavelet decomposition with kernel method, a practical approach of universal multiscale wavelet kernels constructed in reproducing kernel Hilbert space （RKHS） is discussed, and an identification scheme using wavelet support vector machines （WSVM） estimator is proposed for nordinear dynamic systems. The good approximating properties of wavelet kernel function enhance the generalization ability of the proposed method, and the comparison of some numerical experimental results between the novel approach and some existing methods is encouraging.展开更多
In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[...In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.展开更多
In this paper we make use of a special procedure on the repro ducing kernel space to give an expansion theorem for the function with two unkno wns and a surface approximation formula. The error of the surface posses...In this paper we make use of a special procedure on the repro ducing kernel space to give an expansion theorem for the function with two unkno wns and a surface approximation formula. The error of the surface possesses mono tonically decreasing and uniformly convergent characteristics in the sense of t he norm on the space.展开更多
In this paper, we apply the new algorithm of reproducing kernel method to give the approximate solution to some functional-differential equations. The numerical results demonstrate the accuracy of the proposed algorithm.
In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness cri...In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B_1^(0.1)(R^n)into itself.展开更多
We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner...We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.展开更多
基金The research is supported by the National Natural Science Foundation under Grant No. 10471130 and the Zhejiang Province Science Foundation under Grant No. Y604003. Acknowledgements The author thanks the referees for giving valuable comments on this paper which make him rewrite this paper in a better form.
文摘In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then construct a sequence of approximating operators with the help of Jacobi orthogonal polynomials, with which we establish a kind of Jackson inequality to describe the error estimate. Finally, The results are used to discuss an approximation problem arising from learning theory.
基金the NSFC(60473034)the Science Foundation of Zhejiang Province(Y604003).
文摘The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.
文摘This research proposes a method called enhanced collaborative andgeometric multi-kernel learning (E-CGMKL) that can enhance the CGMKLalgorithm which deals with multi-class classification problems with non-lineardata distributions. CGMKL combines multiple kernel learning with softmaxfunction using the framework of multi empirical kernel learning (MEKL) inwhich empirical kernel mapping (EKM) provides explicit feature constructionin the high dimensional kernel space. CGMKL ensures the consistent outputof samples across kernel spaces and minimizes the within-class distance tohighlight geometric features of multiple classes. However, the kernels constructed by CGMKL do not have any explicit relationship among them andtry to construct high dimensional feature representations independently fromeach other. This could be disadvantageous for learning on datasets with complex hidden structures. To overcome this limitation, E-CGMKL constructskernel spaces from hidden layers of trained deep neural networks (DNN).Due to the nature of the DNN architecture, these kernel spaces not onlyprovide multiple feature representations but also inherit the compositionalhierarchy of the hidden layers, which might be beneficial for enhancing thepredictive performance of the CGMKL algorithm on complex data withnatural hierarchical structures, for example, image data. Furthermore, ourproposed scheme handles image data by constructing kernel spaces from aconvolutional neural network (CNN). Considering the effectiveness of CNNarchitecture on image data, these kernel spaces provide a major advantageover the CGMKL algorithm which does not exploit the CNN architecture forconstructing kernel spaces from image data. Additionally, outputs of hiddenlayers directly provide features for kernel spaces and unlike CGMKL, do notrequire an approximate MEKL framework. E-CGMKL combines the consistency and geometry preserving aspects of CGMKL with the compositionalhierarchy of kernel spaces extracted from DNN hidden layers to enhance the predictive performance of CGMKL significantly. The experimental results onvarious data sets demonstrate the superior performance of the E-CGMKLalgorithm compared to other competing methods including the benchmarkCGMKL.
基金Supported by National Natural Science Foundation of China(Grant Nos.12072188,11632011,11702171,11572189,51121063)Shanghai Municipal Natural Science Foundation of China(Grant No.20ZR1425200).
文摘Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods in neural networks,such as using complex network architectures and introducing sparse techniques,always suffer from the difficulty of estimating hyperparameters and the lack of physical interpretability.To address this issue,this paper proposes a novel interpretable denoising layer based on reproducing kernel Hilbert space(RKHS)as the first layer for standard neural networks,with the aim to combine the advantages of both traditional signal processing technology with physical interpretation and network modeling strategy with parameter adaption.By investigating the influencing mechanism of parameters on the regularization procedure in RKHS,the key parameter that dynamically controls the signal smoothness with low computational cost is selected as the only trainable parameter of the proposed layer.Besides,the forward and backward propagation algorithms of the designed layer are formulated to ensure that the selected parameter can be automatically updated together with other parameters in the neural network.Moreover,exponential and piecewise functions are introduced in the weight updating process to keep the trainable weight within a reasonable range and avoid the ill-conditioned problem.Experiment studies verify the effectiveness and compatibility of the proposed layer design method in intelligent fault diagnosis of machinery in noisy environments.
基金supported by the National Natural Science Foundation of China(61573288)the Key Programs in Shaanxi Province of China(2021JZ-12)and the Yulin Science and Technology Bureau project(2019-89-2).
文摘The kernel function method in support vector machine(SVM)is an excellent tool for nonlinear classification.How to design a kernel function is difficult for an SVM nonlinear classification problem,even for the polynomial kernel function.In this paper,we propose a new kind of polynomial kernel functions,called semi-tensor product kernel(STP-kernel),for an SVM nonlinear classification problem by semi-tensor product of matrix(STP)theory.We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel.In addition,we have shown the existence of the reproducing kernel Hilbert space(RKHS)associated with the STP-kernel function.Compared to the existing methods,it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.
文摘It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in C[a, b] or L2 [a, b]. In this paper, the representation of the solution for Fredholm integral equation of the first kind is given if it has a unique solution. The stability of the solution is proved in the reproducing kernel space, namely, the measurement errors of the experimental data cannot result in unbounded errors of the true solution. The computation of approximate solution is also stable with respect to ｜｜· ｜｜c or ｜｜L2· A numerical experiment shows that the method given in this paper is stable in the reproducing kernel space.
文摘In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.
基金co-supported by the National Natural Science Foundation of China (Grant Nos. 61371134, 61071137)the National Basic Research Program of China (No. 2010CB327900)
文摘The application of high-performance imaging sensors in space-based space surveillance systems makes it possible to recognize space objects and estimate their poses using vision-based methods. In this paper, we proposed a kernel regression-based method for joint multi-view space object recognition and pose estimation. We built a new simulated satellite image dataset named BUAA-SID 1.5 to test our method using different image representations. We evaluated our method for recognition-only tasks, pose estimation-only tasks, and joint recognition and pose estimation tasks. Experimental results show that our method outperforms the state-of-the-arts in space object recognition, and can recognize space objects and estimate their poses effectively and robustly against noise and lighting conditions.
基金The national natural science foundation(61273290,61373147)Xiamen Scientific Plan Project(2014S0048,3502Z20123037)+1 种基金Fujian Scientific Plan Project(2013HZ0004-1)FuJian provincial education office A-class project(-JA13238)
文摘Many websites use verification codes to prevent users from using the machine automatically to register,login,malicious vote or irrigate but it brought great burden to the enterprises involved in internet marketing as entering the verification code manually.Improving the verification code security system needs the identification method as the corresponding testing system.We propose an anisotropic heat kernel equation group which can generate a heat source scale space during the kernel evolution based on infinite heat source axiom,design a multi-step anisotropic verification code identification algorithm which includes core procedure of building anisotropic heat kernel,settingwave energy information parameters,combing outverification codccharacters and corresponding peripheral procedure of gray scaling,binarizing,denoising,normalizing,segmenting and identifying,give out the detail criterion and parameter set.Actual test show the anisotropic heat kernel identification algorithm can be used on many kinds of verification code including text characters,mathematical,Chinese,voice,3D,programming,video,advertising,it has a higher rate of 25%and 50%than neural network and context matching algorithm separately for Yahoo site,49%and 60%for Captcha site,20%and 52%for Baidu site,60%and 65%for 3DTakers site,40%,and 51%.for MDP site.
基金This work is supported by a Young Innovative Talents Program in Universities and Colleges of Guangdong Province(2018KQNCX338)two Scientific Research-Innovation Team Projects at Zhuhai Campus,Beijing Institute of Technology(XK-2018-15,XK-2019-10).
文摘In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations.Then,the uniform convergence of the numerical solution is proved,and the time consuming Schmidt orthogonalization process is avoided.The algorithm is employed successfully on some numerical examples.
文摘Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.
基金Supported by the Indigenous Innovation’s Capability Development Program of Huizhou University(HZU202003,HZU202020)Natural Science Foundation of Guangdong Province(2022A1515011463)+2 种基金the Project of Educational Commission of Guangdong Province(2023ZDZX1025)National Natural Science Foundation of China(12271473)Guangdong Province’s 2023 Education Science Planning Project(Higher Education Special Project)(2023GXJK505)。
文摘Complementary-label learning(CLL)aims at finding a classifier via samples with complementary labels.Such data is considered to contain less information than ordinary-label samples.The transition matrix between the true label and the complementary label,and some loss functions have been developed to handle this problem.In this paper,we show that CLL can be transformed into ordinary classification under some mild conditions,which indicates that the complementary labels can supply enough information in most cases.As an example,an extensive misclassification error analysis was performed for the Kernel Ridge Regression(KRR)method applied to multiple complementary-label learning(MCLL),which demonstrates its superior performance compared to existing approaches.
基金Supported by National Natural Science Foundation of China(11471216,11301332)E-Institutes of Shanghai Municipal Education Commission(E03004)+1 种基金Central Finance Project(YC-XK-13105)Shanghai Municipal Science and Technology Research Project(14DZ1201902)
文摘Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the model bias comes from a reproducing kernel Hilbert space. Two different design criteria are proposed by applying the minimax approach for estimating the parameters of the regression response and extrapolating the regression response to points outside of the design space. A simulated annealing algorithm is applied to construct the minimax designs. These minimax designs are compared with the classical D-optimal designs and all-bias extrapolation designs. Numerical results indicate that the simulated annealing algorithm is feasible and the minimax designs are robust against bias caused by model misspecification.
基金partially supported by Knowledge Innovation Program of Hubei Province(No.2019CFB810)partially supported by NSFC(No.12325110)the CAS Project for Young Scientists in Basic Research(No.YSBR-034)。
文摘The conditional kernel correlation is proposed to measure the relationship between two random variables under covariates for multivariate data.Relying on the framework of reproducing kernel Hilbert spaces,we give the definitions of the conditional kernel covariance and conditional kernel correlation.We also provide their respective sample estimators and give the asymptotic properties,which help us construct a conditional independence test.According to the numerical results,the proposed test is more effective compared to the existing one under the considered scenarios.A real data is further analyzed to illustrate the efficacy of the proposed method.
基金the National 973 Key Fundamental Research Project of China (Grant No.2002CB312200)
文摘By combining the wavelet decomposition with kernel method, a practical approach of universal multiscale wavelet kernels constructed in reproducing kernel Hilbert space （RKHS） is discussed, and an identification scheme using wavelet support vector machines （WSVM） estimator is proposed for nordinear dynamic systems. The good approximating properties of wavelet kernel function enhance the generalization ability of the proposed method, and the comparison of some numerical experimental results between the novel approach and some existing methods is encouraging.
基金University of Macao Multi-Year Research Grant Ref.No MYRG2016-00053-FST and MYRG2018-00168-FSTthe Science and Technology Development Fund,Macao SAR FDCT/0123/2018/A3.
文摘In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.
文摘In this paper we make use of a special procedure on the repro ducing kernel space to give an expansion theorem for the function with two unkno wns and a surface approximation formula. The error of the surface possesses mono tonically decreasing and uniformly convergent characteristics in the sense of t he norm on the space.
文摘In this paper, we apply the new algorithm of reproducing kernel method to give the approximate solution to some functional-differential equations. The numerical results demonstrate the accuracy of the proposed algorithm.
基金This research was partially supported by NNSF NEC in P. R. China
文摘In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B_1^(0.1)(R^n)into itself.
文摘We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.
