Based on exact penalty function, a new neural network for solving the L1-norm optimization problem is proposed. In comparison with Kennedy and Chua’s network(1988), it has better properties.Based on Bandler’s fault ...Based on exact penalty function, a new neural network for solving the L1-norm optimization problem is proposed. In comparison with Kennedy and Chua’s network(1988), it has better properties.Based on Bandler’s fault location method(1982), a new nonlinearly constrained L1-norm problem is developed. It can be solved with less computing time through only one optimization processing. The proposed neural network can be used to solve the analog diagnosis L1 problem. The validity of the proposed neural networks and the fault location L1 method are illustrated by extensive computer simulations.展开更多
Radial functions have become a useful tool in numerical mathematics. On the sphere they have to be identified with the zonal functions. We investigate zonal polynomials with mass concentration at the pole, in the sens...Radial functions have become a useful tool in numerical mathematics. On the sphere they have to be identified with the zonal functions. We investigate zonal polynomials with mass concentration at the pole, in the sense of their L1-norm is attaining the minimum value. Such polynomials satisfy a complicated system of nonlinear e-quations (algebraic if the space dimension is odd, only) and also a singular differential equation of third order. The exact order of decay of the minimum value with respect to the polynomial degree is determined. By our results we can prove that some nodal systems on the sphere, which are defined by a minimum-property, are providing fundamental matrices which are diagonal-dominant or bounded with respect to the ∞-norm, at least, as the polynomial degree tends to infinity.展开更多
Principal component analysis(PCA) is fundamental in many pattern recognition applications.Much research has been performed to minimize the reconstruction error in L1-norm based reconstruction error minimization(L1-PCA...Principal component analysis(PCA) is fundamental in many pattern recognition applications.Much research has been performed to minimize the reconstruction error in L1-norm based reconstruction error minimization(L1-PCA-REM) since conventional L2-norm based PCA(L2-PCA) is sensitive to outliers.Recently,the variance maximization formulation of PCA with L1-norm(L1-PCA-VM) has been proposed,where new greedy and nongreedy solutions are developed.Armed with the gradient ascent perspective for optimization,we show that the L1-PCA-VM formulation is problematic in learning principal components and that only a greedy solution can achieve robustness motivation,which are verified by experiments on synthetic and real-world datasets.展开更多
基金This work is sponsored by National Natural Science Foundation of China （No. 40874056）, Important National Science ＆ Technology Specific Projects 2008ZX05023-005-004, and the NCET Fund.Acknowledgements The authors are grateful to Liu Yang, and Zhu Sheng-wang for their constructive remarks on this manuscript.
基金Supported by Doctoral Special Fund of State Education Commissionthe National Natural Science Foundation of China,Grant No.59477001 and No.59707002
文摘Based on exact penalty function, a new neural network for solving the L1-norm optimization problem is proposed. In comparison with Kennedy and Chua’s network(1988), it has better properties.Based on Bandler’s fault location method(1982), a new nonlinearly constrained L1-norm problem is developed. It can be solved with less computing time through only one optimization processing. The proposed neural network can be used to solve the analog diagnosis L1 problem. The validity of the proposed neural networks and the fault location L1 method are illustrated by extensive computer simulations.
文摘Radial functions have become a useful tool in numerical mathematics. On the sphere they have to be identified with the zonal functions. We investigate zonal polynomials with mass concentration at the pole, in the sense of their L1-norm is attaining the minimum value. Such polynomials satisfy a complicated system of nonlinear e-quations (algebraic if the space dimension is odd, only) and also a singular differential equation of third order. The exact order of decay of the minimum value with respect to the polynomial degree is determined. By our results we can prove that some nodal systems on the sphere, which are defined by a minimum-property, are providing fundamental matrices which are diagonal-dominant or bounded with respect to the ∞-norm, at least, as the polynomial degree tends to infinity.
基金Project supported by the National Natural Science Foundation of China (Nos. 61071131 and 61271388)the Beijing Natural Science Foundation (No. 4122040)+1 种基金the Research Project of Tsinghua University (No. 2012Z01011)the United Technologies Research Center (UTRC)
文摘Principal component analysis(PCA) is fundamental in many pattern recognition applications.Much research has been performed to minimize the reconstruction error in L1-norm based reconstruction error minimization(L1-PCA-REM) since conventional L2-norm based PCA(L2-PCA) is sensitive to outliers.Recently,the variance maximization formulation of PCA with L1-norm(L1-PCA-VM) has been proposed,where new greedy and nongreedy solutions are developed.Armed with the gradient ascent perspective for optimization,we show that the L1-PCA-VM formulation is problematic in learning principal components and that only a greedy solution can achieve robustness motivation,which are verified by experiments on synthetic and real-world datasets.
