We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a ...
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We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a K-functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernelfunction spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernelfunctions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness.
In the incompressible material point method (iMPM), the momentum equations were solved at the background grid nodes while the divergence-free conditions were enforced at grid cell centers. The density of each particle...
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In the incompressible material point method (iMPM), the momentum equations were solved at the background grid nodes while the divergence-free conditions were enforced at grid cell centers. The density of each particle was assumed to be constant but the particles could distribute nonuniformly in space over time. Therefore, the fluid density would be nonuniform and violate the incompressible condition. In this paper, the original iMPM is improved by explicitly imposing the density-invariant condition. A new particle shifting scheme is proposed for particle density correction. Particles are shifted along their density gradient to guarantee that the density field of the fluid is constant and the momentum is conserved. The proposed method has been implemented in our MPM code, and validated by simulating a dam breaking inside a tank, another dam breaking with an obstacle and a sloshing problem.
kernel principal component analysis (KPCA) has become a popular technique for process monitoring in recent years. However, the performance largely depends on kernelfunction, yet methods to choose an appropriate kerne...
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ISBN:
(纸本)9781467397148
kernel principal component analysis (KPCA) has become a popular technique for process monitoring in recent years. However, the performance largely depends on kernelfunction, yet methods to choose an appropriate kernelfunction among infinite ones have only been sporadically touched in the research literatures. In this paper, a novel methodology to learn a data-dependent kernelfunction automatically from specific input data is proposed and the improved kernel principal component analysis is obtained through using the data-dependent kernelfunction in traditional KPCA. The learning procedure includes two parts: learning a kernel matrix and approximating a kernelfunction. The kernel matrix is learned via a manifold learning method named maximum variance unfolding (MVU) which considers underlying manifold structure to ensure that principal components are linear in kernel space. Then, a kernelfunction is approximated via generalized Nystrom formula. The effectiveness of the improved KPCA model is confirmed by a numerical simulation and the Tennessee Eastman (TE) process benchmark.
kernel principal component analysis(KPCA) has become a popular technique for process monitoring in recent ***,the performance largely depends on kernelfunction,yet methods to choose an appropriate kernelfunction amo...
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ISBN:
(纸本)9781467397155
kernel principal component analysis(KPCA) has become a popular technique for process monitoring in recent ***,the performance largely depends on kernelfunction,yet methods to choose an appropriate kernelfunction among infinite ones have only been sporadically touched in the research *** this paper,a novel methodology to learn a data-dependent kernelfunction automatically from specific input data is proposed and the improved kernel principal component analysis is obtained through using the data-dependent kernelfunction in traditional *** learning procedure includes two parts:learning a kernel matrix and approximating a kernel *** kernel matrix is learned via a manifold learning method named maximum variance unfolding(MVU) which considers underlying manifold structure to ensure that principal components are linear in kernel ***,a kernelfunction is approximated via generalized Nystr?m *** effectiveness of the improved KPCA model is confirmed by a numerical simulation and the Tennessee Eastman(TE) process benchmark.
An efficient approach of cancer classification using microarray expression data by vector-valued regularized kernel function approximation (VVRKFA) method is presented in a true computer aided diagnosis framework. A f...
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An efficient approach of cancer classification using microarray expression data by vector-valued regularized kernel function approximation (VVRKFA) method is presented in a true computer aided diagnosis framework. A fast dimensionality reduction method based on maximum relevance minimum redundancy (MRMR) criteria is used to select very few genes so that both the classification accuracy and computational speed are enhanced. The experimental results are compared with support vector machines (SVM). It is observed that VVRKFA has achieved at least equal or better classification accuracy. This method also has the advantage that the separability of the data set can be observed in the label space.
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