As data analysis often incurs significant communication and computational costs,these tasks are increasingly outsourced to cloud computing platforms.However,this introduces privacy concerns,as sensitive data must be t...As data analysis often incurs significant communication and computational costs,these tasks are increasingly outsourced to cloud computing platforms.However,this introduces privacy concerns,as sensitive data must be transmitted to and processed by untrusted parties.To address this,fully homomorphic encryption(FHE)has emerged as a promising solution for privacy-preserving Machine-Learning-as-a-Service(MLaaS),enabling computation on encrypted data without revealing the plaintext.Nevertheless,FHE remains computationally expensive.As a result,approximate homomorphic encryption(AHE)schemes,such as CKKS,have attracted attention due to their efficiency.In our previous work,we proposed RP-OKC,a CKKS-based clustering scheme implemented via TenSEAL.However,errors inherent to CKKS operations—termed CKKS-errors—can affect the accuracy of the result after decryption.Since these errors can be mitigated through post-decryption rounding,we propose a data pre-scaling technique to increase the number of significant digits and reduce CKKS-errors.Furthermore,we introduce an Operation-Error-Estimation(OEE)table that quantifies upper-bound error estimates for various CKKS operations.This table enables error-aware decryption correction,ensuring alignment between encrypted and plaintext results.We validate our method on K-means clustering using the Kaggle Customer Segmentation dataset.Experimental results confirm that the proposed scheme enhances the accuracy and reliability of privacy-preserving data analysis in cloud environments.展开更多
Approximate Bayesian Computation (ABC) is a popular sampling method in applications involving intractable likelihood functions. Instead of evaluating the likelihood function, ABC approximates the posterior distributio...Approximate Bayesian Computation (ABC) is a popular sampling method in applications involving intractable likelihood functions. Instead of evaluating the likelihood function, ABC approximates the posterior distribution by a set of accepted samples which are simulated from a generating model. Simulated samples are accepted if the distances between the samples and the observation are smaller than some threshold. The distance is calculated in terms of summary statistics. This paper proposes Local Gradient Kernel Dimension Reduction (LGKDR) to construct low dimensional summary statistics for ABC. The proposed method identifies a sufficient subspace of the original summary statistics by implicitly considering all non-linear transforms therein, and a weighting kernel is used for the concentration of the projections. No strong assumptions are made on the marginal distributions, nor the regression models, permitting usage in a wide range of applications. Experiments are done with simple rejection ABC and sequential Monte Carlo ABC methods. Results are reported as competitive in the former and substantially better in the latter cases in which Monte Carlo errors are compressed as much as possible.展开更多
为了对大量电力用户的稀疏、不规律的日耗电量数据进行特征分析,并对用户进行分类,文章提出一种函数性数据聚类分析方法。首先,应用kernel方法将离散的电量数据还原成连续曲线;然后,受Sobolev空间距离的启发,定义了新的函数距离,用于k-m...为了对大量电力用户的稀疏、不规律的日耗电量数据进行特征分析,并对用户进行分类,文章提出一种函数性数据聚类分析方法。首先,应用kernel方法将离散的电量数据还原成连续曲线;然后,受Sobolev空间距离的启发,定义了新的函数距离,用于k-means算法进行聚类。以某城市10 000户居民538天的实际用电数据进行实验,得到了用户在不同距离和聚类个数下的聚类原型。实验结果显示,由于选取的用户主要是城市居民,其用电模式比较相似:大高峰时段主要在6—9月,小高峰时段主要在1—2月,日消耗波动较小。而不同用户类别的主要区别体现在用电量的范围上:低耗电用户整体低于13 k W?h/天,高耗电用户接近100 k W?h/天。展开更多
For the kernel K-mean cluster method is run in an implicit feature space, the initial and iterative cluster centers cannot be defined explicitly. Against the deficiency of the initial cluster centers selected in the o...For the kernel K-mean cluster method is run in an implicit feature space, the initial and iterative cluster centers cannot be defined explicitly. Against the deficiency of the initial cluster centers selected in the original space discretionarily in the existing methods, this paper proposes a new method for ensuring the clustering center that virtual clustering centers are defined in the feature space by the original classification as the initial cluster centers and the iteration clustering centers are ensured by the further virtual classification. The improved method is used for fault diagnosis of roller bearing that achieves a good cluster and diagnosis result, which demonstrates the effectiveness of the proposed method.展开更多
Wavelet, a powerful tool for signal processing, can be used to approximate the target func-tion. For enhancing the sparse property of wavelet approximation, a new algorithm was proposed by using wavelet kernel Support...Wavelet, a powerful tool for signal processing, can be used to approximate the target func-tion. For enhancing the sparse property of wavelet approximation, a new algorithm was proposed by using wavelet kernel Support Vector Machines (SVM), which can converge to minimum error with bet-ter sparsity. Here, wavelet functions would be firstly used to construct the admitted kernel for SVM according to Mercy theory; then new SVM with this kernel can be used to approximate the target fun-citon with better sparsity than wavelet approxiamtion itself. The results obtained by our simulation ex-periment show the feasibility and validity of wavelet kernel support vector machines.展开更多
This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, ...This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, is obtained, where n is the number of parameters needed in the approximation. By means of the approximation, a learning rate of regularized least square algorithm with the Lipschitz kernel on the sphere is also deduced.展开更多
In this paper, the normal approximation rate and the random weighting approximation rate of error distribution of the kernel estimator of conditional density function f(y|x) are studied. The results may be used to...In this paper, the normal approximation rate and the random weighting approximation rate of error distribution of the kernel estimator of conditional density function f(y|x) are studied. The results may be used to construct the confidence interval of f(y|x) .展开更多
LetΩbe homogeneous of degree zero,integrable on S^(d−1) and have vanishing moment of order one,a be a function on R^(d) such that ∇a∈L^(∞)(R^(d)).Let T*_(Ω,a) be the maximaloperator associated with the d-dimensional...LetΩbe homogeneous of degree zero,integrable on S^(d−1) and have vanishing moment of order one,a be a function on R^(d) such that ∇a∈L^(∞)(R^(d)).Let T*_(Ω,a) be the maximaloperator associated with the d-dimensional Calder´on commutator defined by T*_(Ωa)f(x):=sup_(ε>0)|∫_(|x-y|>ε)^Ω(x-y)/|x-y|^(d+1)(a(x)-a(y))f(y)dy.In this paper,the authors establish bilinear sparse domination for T*_(Ω,a) under the assumption Ω∈L∞(Sd−1).As applications,some quantitative weighted bounds for T*_(Ω,a) are obtained.展开更多
This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographi...This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographic velocity inversion. Based on the Bom/Rytov approximation of the frequency-domain wave equation, we derive the traveltime sensitivity kemels of the wave equation on the band-limited wave field and simultaneously obtain the traveltime residuals based on the Rytov approximation. In contrast to single-ray tomography, the modified velocity inversion method improves the inversion stability. Tests of the near- surface velocity model and field data prove that the proposed method has higher accuracy and Computational efficiency than ray theory tomography and full waveform inversion methods.展开更多
Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.
Multiple kernel clustering is an unsupervised data analysis method that has been used in various scenarios where data is easy to be collected but hard to be labeled.However,multiple kernel clustering for incomplete da...Multiple kernel clustering is an unsupervised data analysis method that has been used in various scenarios where data is easy to be collected but hard to be labeled.However,multiple kernel clustering for incomplete data is a critical yet challenging task.Although the existing absent multiple kernel clustering methods have achieved remarkable performance on this task,they may fail when data has a high value-missing rate,and they may easily fall into a local optimum.To address these problems,in this paper,we propose an absent multiple kernel clustering(AMKC)method on incomplete data.The AMKC method rst clusters the initialized incomplete data.Then,it constructs a new multiple-kernel-based data space,referred to as K-space,from multiple sources to learn kernel combination coefcients.Finally,it seamlessly integrates an incomplete-kernel-imputation objective,a multiple-kernel-learning objective,and a kernel-clustering objective in order to achieve absent multiple kernel clustering.The three stages in this process are carried out simultaneously until the convergence condition is met.Experiments on six datasets with various characteristics demonstrate that the kernel imputation and clustering performance of the proposed method is signicantly better than state-of-the-art competitors.Meanwhile,the proposed method gains fast convergence speed.展开更多
How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducin...How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.展开更多
In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwellian molecules. We first show that the global existence in time of the mild solution of the viscosity equation . ...In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwellian molecules. We first show that the global existence in time of the mild solution of the viscosity equation . We then study the asymptotic behaviour of the mild solution as the coefficients , and an estimate on is derived.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.
This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeab...This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeable sheet submerged in a moving fluid.To solve this equation,a numerical method is proposed based on a Laguerre functions with reproducing kernel Hilbert space method.Using the operational matrices of derivative,we reduced the problem to a set of algebraic equations.We also compare this work with some other numerical results and present a solution that proves to be highly accurate.展开更多
A new algorithm named kernel bisecting k-means and sample removal(KBK-SR) is proposed as sampling preprocessing for support vector machine(SVM) training to improve the efficiency.The proposed algorithm tends to quickl...A new algorithm named kernel bisecting k-means and sample removal(KBK-SR) is proposed as sampling preprocessing for support vector machine(SVM) training to improve the efficiency.The proposed algorithm tends to quickly produce balanced clusters of similar sizes in the kernel feature space,which makes it efficient and effective for reducing training samples.Theoretical analysis and experimental results on three UCI real data benchmarks both show that,with very short sampling time,the proposed algorithm dramatically accelerates SVM sampling and training while maintaining high test accuracy.展开更多
基金funded by National Science and Technology Council,Taiwan,grant numbers are 110-2401-H-002-094-MY2 and 112-2221-E-130-001.
文摘As data analysis often incurs significant communication and computational costs,these tasks are increasingly outsourced to cloud computing platforms.However,this introduces privacy concerns,as sensitive data must be transmitted to and processed by untrusted parties.To address this,fully homomorphic encryption(FHE)has emerged as a promising solution for privacy-preserving Machine-Learning-as-a-Service(MLaaS),enabling computation on encrypted data without revealing the plaintext.Nevertheless,FHE remains computationally expensive.As a result,approximate homomorphic encryption(AHE)schemes,such as CKKS,have attracted attention due to their efficiency.In our previous work,we proposed RP-OKC,a CKKS-based clustering scheme implemented via TenSEAL.However,errors inherent to CKKS operations—termed CKKS-errors—can affect the accuracy of the result after decryption.Since these errors can be mitigated through post-decryption rounding,we propose a data pre-scaling technique to increase the number of significant digits and reduce CKKS-errors.Furthermore,we introduce an Operation-Error-Estimation(OEE)table that quantifies upper-bound error estimates for various CKKS operations.This table enables error-aware decryption correction,ensuring alignment between encrypted and plaintext results.We validate our method on K-means clustering using the Kaggle Customer Segmentation dataset.Experimental results confirm that the proposed scheme enhances the accuracy and reliability of privacy-preserving data analysis in cloud environments.
文摘Approximate Bayesian Computation (ABC) is a popular sampling method in applications involving intractable likelihood functions. Instead of evaluating the likelihood function, ABC approximates the posterior distribution by a set of accepted samples which are simulated from a generating model. Simulated samples are accepted if the distances between the samples and the observation are smaller than some threshold. The distance is calculated in terms of summary statistics. This paper proposes Local Gradient Kernel Dimension Reduction (LGKDR) to construct low dimensional summary statistics for ABC. The proposed method identifies a sufficient subspace of the original summary statistics by implicitly considering all non-linear transforms therein, and a weighting kernel is used for the concentration of the projections. No strong assumptions are made on the marginal distributions, nor the regression models, permitting usage in a wide range of applications. Experiments are done with simple rejection ABC and sequential Monte Carlo ABC methods. Results are reported as competitive in the former and substantially better in the latter cases in which Monte Carlo errors are compressed as much as possible.
基金Projected Supported by the National High Technology Research and Development Program of China(863 Program)(2015AA050203)National Talents Training Base for Basic Research and Teaching of Natural Science of China(J1103105)~~
文摘为了对大量电力用户的稀疏、不规律的日耗电量数据进行特征分析,并对用户进行分类,文章提出一种函数性数据聚类分析方法。首先,应用kernel方法将离散的电量数据还原成连续曲线;然后,受Sobolev空间距离的启发,定义了新的函数距离,用于k-means算法进行聚类。以某城市10 000户居民538天的实际用电数据进行实验,得到了用户在不同距离和聚类个数下的聚类原型。实验结果显示,由于选取的用户主要是城市居民,其用电模式比较相似:大高峰时段主要在6—9月,小高峰时段主要在1—2月,日消耗波动较小。而不同用户类别的主要区别体现在用电量的范围上:低耗电用户整体低于13 k W?h/天,高耗电用户接近100 k W?h/天。
文摘For the kernel K-mean cluster method is run in an implicit feature space, the initial and iterative cluster centers cannot be defined explicitly. Against the deficiency of the initial cluster centers selected in the original space discretionarily in the existing methods, this paper proposes a new method for ensuring the clustering center that virtual clustering centers are defined in the feature space by the original classification as the initial cluster centers and the iteration clustering centers are ensured by the further virtual classification. The improved method is used for fault diagnosis of roller bearing that achieves a good cluster and diagnosis result, which demonstrates the effectiveness of the proposed method.
文摘Wavelet, a powerful tool for signal processing, can be used to approximate the target func-tion. For enhancing the sparse property of wavelet approximation, a new algorithm was proposed by using wavelet kernel Support Vector Machines (SVM), which can converge to minimum error with bet-ter sparsity. Here, wavelet functions would be firstly used to construct the admitted kernel for SVM according to Mercy theory; then new SVM with this kernel can be used to approximate the target fun-citon with better sparsity than wavelet approxiamtion itself. The results obtained by our simulation ex-periment show the feasibility and validity of wavelet kernel support vector machines.
基金Supported by the National Natural Science Foundation of China(61272023,91330118)
文摘This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, is obtained, where n is the number of parameters needed in the approximation. By means of the approximation, a learning rate of regularized least square algorithm with the Lipschitz kernel on the sphere is also deduced.
基金Supported by Natural Science Foundation of Beijing City and National Natural Science Foundation ofChina(2 2 30 4 1 0 0 1 30 1
文摘In this paper, the normal approximation rate and the random weighting approximation rate of error distribution of the kernel estimator of conditional density function f(y|x) are studied. The results may be used to construct the confidence interval of f(y|x) .
文摘LetΩbe homogeneous of degree zero,integrable on S^(d−1) and have vanishing moment of order one,a be a function on R^(d) such that ∇a∈L^(∞)(R^(d)).Let T*_(Ω,a) be the maximaloperator associated with the d-dimensional Calder´on commutator defined by T*_(Ωa)f(x):=sup_(ε>0)|∫_(|x-y|>ε)^Ω(x-y)/|x-y|^(d+1)(a(x)-a(y))f(y)dy.In this paper,the authors establish bilinear sparse domination for T*_(Ω,a) under the assumption Ω∈L∞(Sd−1).As applications,some quantitative weighted bounds for T*_(Ω,a) are obtained.
基金sponsored by the National Natural Science Foundation of China(No.41204086)the Self-governed Innovative Project of China University of Petroleum(No.13CX02041A)+2 种基金the Doctoral Fund of National Ministry of Education(No.20110133120001)the National 863 Project(2011AA060301)the Major National Science and Technology Program(No.2011ZX05006-002)
文摘This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographic velocity inversion. Based on the Bom/Rytov approximation of the frequency-domain wave equation, we derive the traveltime sensitivity kemels of the wave equation on the band-limited wave field and simultaneously obtain the traveltime residuals based on the Rytov approximation. In contrast to single-ray tomography, the modified velocity inversion method improves the inversion stability. Tests of the near- surface velocity model and field data prove that the proposed method has higher accuracy and Computational efficiency than ray theory tomography and full waveform inversion methods.
文摘Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.
基金funded by National Natural Science Foundation of China under Grant Nos.61972057 and U1836208Hunan Provincial Natural Science Foundation of China under Grant No.2019JJ50655+3 种基金Scientic Research Foundation of Hunan Provincial Education Department of China under Grant No.18B160Open Fund of Hunan Key Laboratory of Smart Roadway and Cooperative Vehicle Infrastructure Systems(Changsha University of Science and Technology)under Grant No.kfj180402the“Double First-class”International Cooperation and Development Scientic Research Project of Changsha University of Science and Technology under Grant No.2018IC25the Researchers Supporting Project No.(RSP-2020/102)King Saud University,Riyadh,Saudi Arabia.
文摘Multiple kernel clustering is an unsupervised data analysis method that has been used in various scenarios where data is easy to be collected but hard to be labeled.However,multiple kernel clustering for incomplete data is a critical yet challenging task.Although the existing absent multiple kernel clustering methods have achieved remarkable performance on this task,they may fail when data has a high value-missing rate,and they may easily fall into a local optimum.To address these problems,in this paper,we propose an absent multiple kernel clustering(AMKC)method on incomplete data.The AMKC method rst clusters the initialized incomplete data.Then,it constructs a new multiple-kernel-based data space,referred to as K-space,from multiple sources to learn kernel combination coefcients.Finally,it seamlessly integrates an incomplete-kernel-imputation objective,a multiple-kernel-learning objective,and a kernel-clustering objective in order to achieve absent multiple kernel clustering.The three stages in this process are carried out simultaneously until the convergence condition is met.Experiments on six datasets with various characteristics demonstrate that the kernel imputation and clustering performance of the proposed method is signicantly better than state-of-the-art competitors.Meanwhile,the proposed method gains fast convergence speed.
基金Project supported by the National Natural Science Foundation of China(No.10461005)
文摘How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.
文摘In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwellian molecules. We first show that the global existence in time of the mild solution of the viscosity equation . We then study the asymptotic behaviour of the mild solution as the coefficients , and an estimate on is derived.
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.
文摘This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeable sheet submerged in a moving fluid.To solve this equation,a numerical method is proposed based on a Laguerre functions with reproducing kernel Hilbert space method.Using the operational matrices of derivative,we reduced the problem to a set of algebraic equations.We also compare this work with some other numerical results and present a solution that proves to be highly accurate.
基金National Natural Science Foundation of China (No. 60975083)Key Grant Project,Ministry of Education,China(No. 104145)
文摘A new algorithm named kernel bisecting k-means and sample removal(KBK-SR) is proposed as sampling preprocessing for support vector machine(SVM) training to improve the efficiency.The proposed algorithm tends to quickly produce balanced clusters of similar sizes in the kernel feature space,which makes it efficient and effective for reducing training samples.Theoretical analysis and experimental results on three UCI real data benchmarks both show that,with very short sampling time,the proposed algorithm dramatically accelerates SVM sampling and training while maintaining high test accuracy.
