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
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
Kernel canonical correlation analysis(CCA) is a nonlinear extension of CCA,which aims at extract-ing the information shared by two random variables. It has wide applications in many fields,such as information retrieva...Kernel canonical correlation analysis(CCA) is a nonlinear extension of CCA,which aims at extract-ing the information shared by two random variables. It has wide applications in many fields,such as information retrieval. This paper gives the convergence rate analysis of kernel CCA under some approximation conditions and some suggestions on how to choose the regularization parameter. The result shows that the convergence rate only depends on two parameters:the rate of regularization parameter and the decay rate of eigenvalues of compact operator VY X,and it gives better understanding of kernel CCA.展开更多
We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space Kθ = H^2θθH^2 is studied. We...We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space Kθ = H^2θθH^2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H2-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.展开更多
When there is substantial heterogeneity of treatment effectiveness for comparative treatmentselection, it is crucial to identify individualised treatment rules for patients who have heterogeneous responses to treatmen...When there is substantial heterogeneity of treatment effectiveness for comparative treatmentselection, it is crucial to identify individualised treatment rules for patients who have heterogeneous responses to treatment. Existing approaches include directly modelling clinical outcomeby defining the optimal treatment rule according to the interactions between treatment andcovariates and outcome weighted approach that uses clinical outcome as weights to maximise atarget function whose value directly reflects correct treatment assignment. All existing articles ofestimating individualised treatment rules are all assuming just two treatment assignments. Herewe propose an outcome weighted learning approach that uses a vector hinge loss to extend estimating individualised treatment rules in multi-category treatments case. The consistency of theresulting estimator is shown. We also demonstrate the performance of our approach in simulationstudies and a real data analysis.展开更多
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.
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.
基金supported by National Natural Science Foundation of China (Grant Nos. 11001247, 11071276)
文摘Kernel canonical correlation analysis(CCA) is a nonlinear extension of CCA,which aims at extract-ing the information shared by two random variables. It has wide applications in many fields,such as information retrieval. This paper gives the convergence rate analysis of kernel CCA under some approximation conditions and some suggestions on how to choose the regularization parameter. The result shows that the convergence rate only depends on two parameters:the rate of regularization parameter and the decay rate of eigenvalues of compact operator VY X,and it gives better understanding of kernel CCA.
文摘We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space Kθ = H^2θθH^2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H2-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.
基金The author would like to thank Jun Shao and Menggang Yu for their help with preparing the manuscript.This work was supported by the Chinese 111 Project[grant number B14019](for Lou and Shao).
文摘When there is substantial heterogeneity of treatment effectiveness for comparative treatmentselection, it is crucial to identify individualised treatment rules for patients who have heterogeneous responses to treatment. Existing approaches include directly modelling clinical outcomeby defining the optimal treatment rule according to the interactions between treatment andcovariates and outcome weighted approach that uses clinical outcome as weights to maximise atarget function whose value directly reflects correct treatment assignment. All existing articles ofestimating individualised treatment rules are all assuming just two treatment assignments. Herewe propose an outcome weighted learning approach that uses a vector hinge loss to extend estimating individualised treatment rules in multi-category treatments case. The consistency of theresulting estimator is shown. We also demonstrate the performance of our approach in simulationstudies and a real data analysis.
