We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R+n,{u（x） =2/（nωn）∫?R+n（xnf（v（y）））/（|x- y|n）dy, x ∈R+n,v（y） =2/（nωn）∫R+n（xng（u（x）））/（...We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R+n,{u（x） =2/（nωn）∫?R+n（xnf（v（y）））/（|x- y|n）dy, x ∈R+n,v（y） =2/（nωn）∫R+n（xng（u（x）））/（|x- y|n）dx, y ∈?R+n,where n 3, ωn is the volume of the unit ball in Rn. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Hang et al.（2008）.With natural structure conditions on f and g, we classify the positive solutions of the above system based on the method of moving spheres in integral form and the inequality mentioned above.展开更多
Semi-supervised learning is an emerging computational paradigm for machine learning,that aims to make better use of large amounts of inexpensive unlabeled data to improve the learning performance.While various methods...Semi-supervised learning is an emerging computational paradigm for machine learning,that aims to make better use of large amounts of inexpensive unlabeled data to improve the learning performance.While various methods have been proposed based on different intuitions,the crucial issue of generalization performance is still poorly understood.In this paper,we investigate the convergence property of the Laplacian regularized least squares regression,a semi-supervised learning algorithm based on manifold regularization.Moreover,the improvement of error bounds in terms of the number of labeled and unlabeled data is presented for the first time as far as we know.The convergence rate depends on the approximation property and the capacity of the reproducing kernel Hilbert space measured by covering numbers.Some new techniques are exploited for the analysis since an extra regularizer is introduced.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11571268)Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2014JM1021)
文摘We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R+n,{u（x） =2/（nωn）∫?R+n（xnf（v（y）））/（|x- y|n）dy, x ∈R+n,v（y） =2/（nωn）∫R+n（xng（u（x）））/（|x- y|n）dx, y ∈?R+n,where n 3, ωn is the volume of the unit ball in Rn. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Hang et al.（2008）.With natural structure conditions on f and g, we classify the positive solutions of the above system based on the method of moving spheres in integral form and the inequality mentioned above.
基金supported by National Natural Science Foundation of China (Grant Nos.11171014 and 11101024)National Basic Research Program of China (973 Project) (Grant No. 2010CB731900)
文摘Semi-supervised learning is an emerging computational paradigm for machine learning,that aims to make better use of large amounts of inexpensive unlabeled data to improve the learning performance.While various methods have been proposed based on different intuitions,the crucial issue of generalization performance is still poorly understood.In this paper,we investigate the convergence property of the Laplacian regularized least squares regression,a semi-supervised learning algorithm based on manifold regularization.Moreover,the improvement of error bounds in terms of the number of labeled and unlabeled data is presented for the first time as far as we know.The convergence rate depends on the approximation property and the capacity of the reproducing kernel Hilbert space measured by covering numbers.Some new techniques are exploited for the analysis since an extra regularizer is introduced.
