Several kernel-based methods for multi-task learning have been proposed, which leverage relations among tasks as regularization to enhance the overall learning accuracies. These methods assume that the tasks share the...Several kernel-based methods for multi-task learning have been proposed, which leverage relations among tasks as regularization to enhance the overall learning accuracies. These methods assume that the tasks share the same kernel, which could limit their applications because in practice different tasks may need different kernels. The main challenge of introducing multiple kernels into multiple tasks is that models from different reproducing kernel Hilbert spaces （RKHSs） are not comparable, making it difficult to exploit relations among tasks. This paper addresses the challenge by formalizing the problem in the square integrable space （SIS）. Specially, it proposes a kernel-based method which makes use of a regularization term defined in SIS to represent task relations. We prove a new representer theorem for the proposed approach in SIS. We further derive a practical method for solving the learning problem and conduct consistency analysis of the method. We discuss the relationship between our method and an existing method. We also give an SVM （support vector machine）- based implementation of our method for multi-label classification. Experiments on an artificial example and two real-world datasets show that the proposed method performs better than the existing method.展开更多
In this paper,a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term.The hyperbolic part d(x)∂u/∂t+c(x,t)·∇u...In this paper,a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term.The hyperbolic part d(x)∂u/∂t+c(x,t)·∇u is handled by the characteristic method and the diffusion term∇·(a(x,t)∇u+b(x,t)∇ut)is approximated by the new expanded mixed method,whose gradient belongs to the simple square integrable(L^(2)(Ω))^(2)space instead of the classical H(div;Ω)space.For a priori error estimates,some important lemmas based on the new expanded mixed projection are introduced.An optimal priori error estimates in L^(2)-norm for the scalar unknown u and a priori error estimates in(L^(2))^(2)-norm for its gradientλ,and its fluxσ(the coefficients times the negative gradient)are derived.In particular,an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.展开更多
文摘Several kernel-based methods for multi-task learning have been proposed, which leverage relations among tasks as regularization to enhance the overall learning accuracies. These methods assume that the tasks share the same kernel, which could limit their applications because in practice different tasks may need different kernels. The main challenge of introducing multiple kernels into multiple tasks is that models from different reproducing kernel Hilbert spaces （RKHSs） are not comparable, making it difficult to exploit relations among tasks. This paper addresses the challenge by formalizing the problem in the square integrable space （SIS）. Specially, it proposes a kernel-based method which makes use of a regularization term defined in SIS to represent task relations. We prove a new representer theorem for the proposed approach in SIS. We further derive a practical method for solving the learning problem and conduct consistency analysis of the method. We discuss the relationship between our method and an existing method. We also give an SVM （support vector machine）- based implementation of our method for multi-label classification. Experiments on an artificial example and two real-world datasets show that the proposed method performs better than the existing method.
基金supported by the National Natural Science Fund of China(11061021)the Scientific Research Projection of Higher Schools of Inner Mongolia(NJZZ12011,NJZY13199)+1 种基金the Natural Science Fund of Inner Mongolia Province(2012MS0108,2012MS0106)the Program of Higher-level talents of Inner Mongolia University(125119,30105-125132).
文摘In this paper,a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term.The hyperbolic part d(x)∂u/∂t+c(x,t)·∇u is handled by the characteristic method and the diffusion term∇·(a(x,t)∇u+b(x,t)∇ut)is approximated by the new expanded mixed method,whose gradient belongs to the simple square integrable(L^(2)(Ω))^(2)space instead of the classical H(div;Ω)space.For a priori error estimates,some important lemmas based on the new expanded mixed projection are introduced.An optimal priori error estimates in L^(2)-norm for the scalar unknown u and a priori error estimates in(L^(2))^(2)-norm for its gradientλ,and its fluxσ(the coefficients times the negative gradient)are derived.In particular,an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.
