This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained resul...This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained results with those obtained by the Toeplitz matrix method (TMM) demonstrates that the proposed technique is powerful and simple.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
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 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
Through using weight function, we give a new Hilbert-type integral inequality with two independent parameters and two pair of conjugate exponents, which is a best extension of a Hilbert-type integral inequality with t...Through using weight function, we give a new Hilbert-type integral inequality with two independent parameters and two pair of conjugate exponents, which is a best extension of a Hilbert-type integral inequality with the homogeneous kernel of 0-degree. The equivalent form, the reverses and some particular results are considered.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
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
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
The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplifi...The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplified.On this basis the solutions and the solvable conditions in classH_(1)as well as the generalized Noether theorem for the complete equation are obtained.展开更多
文摘This paper proposes the combined Laplace-Adomian decomposition method (LADM) for solution two dimensional linear mixed integral equations of type Volterra-Fredholm with Hilbert kernel. Comparison of the obtained results with those obtained by the Toeplitz matrix method (TMM) demonstrates that the proposed technique is powerful and simple.
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 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.
基金Project supported by the Natural Science Foundation of the Institutions of Higher Learning of Guangdong Province (GrantNo.05Z026)the Natural Science Foundation of Guangdong Province (Grant No.7004344)
文摘Through using weight function, we give a new Hilbert-type integral inequality with two independent parameters and two pair of conjugate exponents, which is a best extension of a Hilbert-type integral inequality with the homogeneous kernel of 0-degree. The equivalent form, the reverses and some particular results are considered.
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.
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.
基金Supported by the National Natural Science Foundation of China(19971064)Ziqiang Invention Foundation of Wuhan University(201990336)
文摘The basic sets of solutions in classH(orH*)for the characteristic equation and its adjoint equation with Hilbert kernel are given respectively.Thus the expressions of solutions and its solvable conditions are simplified.On this basis the solutions and the solvable conditions in classH_(1)as well as the generalized Noether theorem for the complete equation are obtained.
