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
High-dimensional and sparse(HiDS)matrices commonly arise in various industrial applications,e.g.,recommender systems(RSs),social networks,and wireless sensor networks.Since they contain rich information,how to accurat...High-dimensional and sparse(HiDS)matrices commonly arise in various industrial applications,e.g.,recommender systems(RSs),social networks,and wireless sensor networks.Since they contain rich information,how to accurately represent them is of great significance.A latent factor(LF)model is one of the most popular and successful ways to address this issue.Current LF models mostly adopt L2-norm-oriented Loss to represent an HiDS matrix,i.e.,they sum the errors between observed data and predicted ones with L2-norm.Yet L2-norm is sensitive to outlier data.Unfortunately,outlier data usually exist in such matrices.For example,an HiDS matrix from RSs commonly contains many outlier ratings due to some heedless/malicious users.To address this issue,this work proposes a smooth L1-norm-oriented latent factor(SL-LF)model.Its main idea is to adopt smooth L1-norm rather than L2-norm to form its Loss,making it have both strong robustness and high accuracy in predicting the missing data of an HiDS matrix.Experimental results on eight HiDS matrices generated by industrial applications verify that the proposed SL-LF model not only is robust to the outlier data but also has significantly higher prediction accuracy than state-of-the-art models when they are used to predict the missing data of HiDS matrices.展开更多
Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m...Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m（m≥0）times continuously differentiable and its ruth derivative,g<sub>0</sub><sup>（m）</sup>,satisfies a H■lder condition of order γ（m +γ】1/2）.A piecewise polynomial L<sub>1</sub>- norm estimator of go is proposed.Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution,it is proved that the rates of convergence of the piecewise polynomial L<sub>1</sub>-norm estimator are o(n<sup>-2（m+γ）+1/m+γ-1/δ</sup>almost surely and o(n<sup>-2（m+γ）+1/m+γ-δ</sup>)in probability,which can arbitrarily approach the optimal rates of convergence for nonparametric regression,where δ is any number in （0, min（（m+γ-1/2）/3,γ））.展开更多
Based on the range space property （RSP）, the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted l1-norm minimization problem are studied in this paper. Different...Based on the range space property （RSP）, the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted l1-norm minimization problem are studied in this paper. Different from other conditions based on the spark property, the mutual coherence, the null space property （NSP） and the restricted isometry property （RIP）, the RSP- based conditions are easier to be verified. Moreover, the proposed conditions guarantee not only the strong equivalence, but also the equivalence between the two problems. First, according to the foundation of the strict complemenrarity theorem of linear programming, a sufficient and necessary condition, satisfying the RSP of the sensing matrix and the full column rank property of the corresponding sub-matrix, is presented for the unique nonnegative solution to the weighted l1-norm minimization problem. Then, based on this condition, the equivalence conditions between the two problems are proposed. Finally, this paper shows that the matrix with the RSP of order k can guarantee the strong equivalence of the two problems.展开更多
Consider the partly linear model K = X1& + go(Ti) + ei, where {(Ti, Xi)}T is a strictlystationary Sequence of random variable8, the ei’8 are i.i.d. random errorsl the K’s are realvalued responsest fo is a &v...Consider the partly linear model K = X1& + go(Ti) + ei, where {(Ti, Xi)}T is a strictlystationary Sequence of random variable8, the ei’8 are i.i.d. random errorsl the K’s are realvalued responsest fo is a &vector of parameters, X is a &vector of explanatory variables,Ti is another explanatory variable ranging over a nondegenerate compact interval. Bnd ona segmnt of observations (T1, Xi 1 Y1 ),’’’ f (Tn, X;, Yn), this article investigates the rates ofconvrgence of the M-estimators for Po and go obtained from the minimisation problemwhere H is a space of B-spline functions of order m + 1 and p(-) is a function chosen suitablyUnder some regularity conditions, it is shown that the estimator of go achieves the optimalglobal rate of convergence of estimators for nonparametric regression, and the estdriator offo is asymptotically normal. The M-estimators here include regression quantile estimators,Li-estimators, Lp-norm estimators, Huber’s type M-estimators and usual least squares estimators. Applications of the asymptotic theory to testing the hypothesis H0: A’β0 =β are alsodiscussed, where β is a given vector and A is a known d × do matrix with rank d0.展开更多
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.
基金supported in part by the National Natural Science Foundation of China(61702475,61772493,61902370,62002337)in part by the Natural Science Foundation of Chongqing,China(cstc2019jcyj-msxmX0578,cstc2019jcyjjqX0013)+1 种基金in part by the Chinese Academy of Sciences“Light of West China”Program,in part by the Pioneer Hundred Talents Program of Chinese Academy of Sciencesby Technology Innovation and Application Development Project of Chongqing,China(cstc2019jscx-fxydX0027)。
文摘High-dimensional and sparse(HiDS)matrices commonly arise in various industrial applications,e.g.,recommender systems(RSs),social networks,and wireless sensor networks.Since they contain rich information,how to accurately represent them is of great significance.A latent factor(LF)model is one of the most popular and successful ways to address this issue.Current LF models mostly adopt L2-norm-oriented Loss to represent an HiDS matrix,i.e.,they sum the errors between observed data and predicted ones with L2-norm.Yet L2-norm is sensitive to outlier data.Unfortunately,outlier data usually exist in such matrices.For example,an HiDS matrix from RSs commonly contains many outlier ratings due to some heedless/malicious users.To address this issue,this work proposes a smooth L1-norm-oriented latent factor(SL-LF)model.Its main idea is to adopt smooth L1-norm rather than L2-norm to form its Loss,making it have both strong robustness and high accuracy in predicting the missing data of an HiDS matrix.Experimental results on eight HiDS matrices generated by industrial applications verify that the proposed SL-LF model not only is robust to the outlier data but also has significantly higher prediction accuracy than state-of-the-art models when they are used to predict the missing data of HiDS matrices.
基金Supported by the National Natural Science Foundation of China.
文摘Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m（m≥0）times continuously differentiable and its ruth derivative,g<sub>0</sub><sup>（m）</sup>,satisfies a H■lder condition of order γ（m +γ】1/2）.A piecewise polynomial L<sub>1</sub>- norm estimator of go is proposed.Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution,it is proved that the rates of convergence of the piecewise polynomial L<sub>1</sub>-norm estimator are o(n<sup>-2（m+γ）+1/m+γ-1/δ</sup>almost surely and o(n<sup>-2（m+γ）+1/m+γ-δ</sup>)in probability,which can arbitrarily approach the optimal rates of convergence for nonparametric regression,where δ is any number in （0, min（（m+γ-1/2）/3,γ））.
基金Research supported by the National Natural Science Foundation of China under Grant 61672005
文摘Based on the range space property （RSP）, the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted l1-norm minimization problem are studied in this paper. Different from other conditions based on the spark property, the mutual coherence, the null space property （NSP） and the restricted isometry property （RIP）, the RSP- based conditions are easier to be verified. Moreover, the proposed conditions guarantee not only the strong equivalence, but also the equivalence between the two problems. First, according to the foundation of the strict complemenrarity theorem of linear programming, a sufficient and necessary condition, satisfying the RSP of the sensing matrix and the full column rank property of the corresponding sub-matrix, is presented for the unique nonnegative solution to the weighted l1-norm minimization problem. Then, based on this condition, the equivalence conditions between the two problems are proposed. Finally, this paper shows that the matrix with the RSP of order k can guarantee the strong equivalence of the two problems.
文摘Consider the partly linear model K = X1& + go(Ti) + ei, where {(Ti, Xi)}T is a strictlystationary Sequence of random variable8, the ei’8 are i.i.d. random errorsl the K’s are realvalued responsest fo is a &vector of parameters, X is a &vector of explanatory variables,Ti is another explanatory variable ranging over a nondegenerate compact interval. Bnd ona segmnt of observations (T1, Xi 1 Y1 ),’’’ f (Tn, X;, Yn), this article investigates the rates ofconvrgence of the M-estimators for Po and go obtained from the minimisation problemwhere H is a space of B-spline functions of order m + 1 and p(-) is a function chosen suitablyUnder some regularity conditions, it is shown that the estimator of go achieves the optimalglobal rate of convergence of estimators for nonparametric regression, and the estdriator offo is asymptotically normal. The M-estimators here include regression quantile estimators,Li-estimators, Lp-norm estimators, Huber’s type M-estimators and usual least squares estimators. Applications of the asymptotic theory to testing the hypothesis H0: A’β0 =β are alsodiscussed, where β is a given vector and A is a known d × do matrix with rank d0.
