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
The Lt-norm method is one of the widely used matching filters for adaptive multiple subtraction. When the primaries and multiples are mixed together, the L1-norm method might damage the primaries, leading to poor late...The Lt-norm method is one of the widely used matching filters for adaptive multiple subtraction. When the primaries and multiples are mixed together, the L1-norm method might damage the primaries, leading to poor lateral continuity. In this paper, we propose a constrained L1-norm method for adaptive multiple subtraction by introducing the lateral continuity constraint for the estimated primaries. We measure the lateral continuity using prediction-error filters （PEF）. We illustrate our method with the synthetic Pluto dataset. The results show that the constrained L1-norm method can simultaneously attenuate the multiples and preserve the primaries.展开更多
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
In this work,we address the frequency estimation problem of a complex single-tone embedded in the heavy-tailed noise.With the use of the linear prediction(LP)property and l_(1)-norm minimization,a robust frequency est...In this work,we address the frequency estimation problem of a complex single-tone embedded in the heavy-tailed noise.With the use of the linear prediction(LP)property and l_(1)-norm minimization,a robust frequency estimator is developed.Since the proposed method employs the weighted l_(1)-norm on the LP errors,it can be regarded as an extension of the l_(1)-generalized weighted linear predictor.Computer simulations are conducted in the environment of α-stable noise,indicating the superiority of the proposed algorithm,in terms of its robust to outliers and nearly optimal estimation performance.展开更多
In this work, we consider a homotopic principle for solving large-scale and dense l1underdetermined problems and its applications in image processing and classification. We solve the face recognition problem where the...In this work, we consider a homotopic principle for solving large-scale and dense l1underdetermined problems and its applications in image processing and classification. We solve the face recognition problem where the input image contains corrupted and/or lost pixels. The approach involves two steps: first, the incomplete or corrupted image is subject to an inpainting process, and secondly, the restored image is used to carry out the classification or recognition task. Addressing these two steps involves solving large scale l1minimization problems. To that end, we propose to solve a sequence of linear equality constrained multiquadric problems that depends on a regularization parameter that converges to zero. The procedure generates a central path that converges to a point on the solution set of the l1underdetermined problem. In order to solve each subproblem, a conjugate gradient algorithm is formulated. When noise is present in the model, inexact directions are taken so that an approximate solution is computed faster. This prevents the ill conditioning produced when the conjugate gradient is required to iterate until a zero residual is attained.展开更多
A joint two-dimensional(2D)direction-of-arrival(DOA)and radial Doppler frequency estimation method for the L-shaped array is proposed in this paper based on the compressive sensing(CS)framework.Revised from the conven...A joint two-dimensional(2D)direction-of-arrival(DOA)and radial Doppler frequency estimation method for the L-shaped array is proposed in this paper based on the compressive sensing(CS)framework.Revised from the conventional CS-based methods where the joint spatial-temporal parameters are characterized in one large scale matrix,three smaller scale matrices with independent azimuth,elevation and Doppler frequency are introduced adopting a separable observation model.Afterwards,the estimation is achieved by L1-norm minimization and the Bayesian CS algorithm.In addition,under the L-shaped array topology,the azimuth and elevation are separated yet coupled to the same radial Doppler frequency.Hence,the pair matching problem is solved with the aid of the radial Doppler frequency.Finally,numerical simulations corroborate the feasibility and validity of the proposed algorithm.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
This paper presents a new method for image separation through employing a combined dictionary consisting of wavelets and complex shearlets. Because the combined dictionary sparsely represents points and curvilinear si...This paper presents a new method for image separation through employing a combined dictionary consisting of wavelets and complex shearlets. Because the combined dictionary sparsely represents points and curvilinear singularities respectively, the image can be decomposed into pointlike and curvelike parts as accurate as possible. The proposed method based on the geo- metric separation theory introduced by Donoho in 2005 shows that accurate geometric separation of the morphologically distinct fea- tures of points and curves can be achieved by l1 minimization. The experimental results show that the proposed method can not only be effective but also greatly reduce the computing time.展开更多
The maximum of k numerical functions defined on , , by , ??is used here in Statistical classification. Previously, it has been used in Statistical Discrimination [1] and in Clustering [2]. We present first some theore...The maximum of k numerical functions defined on , , by , ??is used here in Statistical classification. Previously, it has been used in Statistical Discrimination [1] and in Clustering [2]. We present first some theoretical results on this function, and then its application in classification using a computer program we have developed. This approach leads to clear decisions, even in cases where the extension to several classes of Fisher’s linear discriminant function fails to be effective.展开更多
In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for t...In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for the original problem and design an extragradient thresholding algorithm (ETA) to solve the regularized model. Furthermore, we prove that any cluster point of the sequence generated by ETA is a solution of MCP. Finally, numerical experiments show that the ETA algorithm can effectively solve the l1 regularized projection minimization model and obtain the sparse solution of the mixed complementarity problem.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
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.
文摘The Lt-norm method is one of the widely used matching filters for adaptive multiple subtraction. When the primaries and multiples are mixed together, the L1-norm method might damage the primaries, leading to poor lateral continuity. In this paper, we propose a constrained L1-norm method for adaptive multiple subtraction by introducing the lateral continuity constraint for the estimated primaries. We measure the lateral continuity using prediction-error filters （PEF）. We illustrate our method with the synthetic Pluto dataset. The results show that the constrained L1-norm method can simultaneously attenuate the multiples and preserve the primaries.
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In this work,we address the frequency estimation problem of a complex single-tone embedded in the heavy-tailed noise.With the use of the linear prediction(LP)property and l_(1)-norm minimization,a robust frequency estimator is developed.Since the proposed method employs the weighted l_(1)-norm on the LP errors,it can be regarded as an extension of the l_(1)-generalized weighted linear predictor.Computer simulations are conducted in the environment of α-stable noise,indicating the superiority of the proposed algorithm,in terms of its robust to outliers and nearly optimal estimation performance.
文摘In this work, we consider a homotopic principle for solving large-scale and dense l1underdetermined problems and its applications in image processing and classification. We solve the face recognition problem where the input image contains corrupted and/or lost pixels. The approach involves two steps: first, the incomplete or corrupted image is subject to an inpainting process, and secondly, the restored image is used to carry out the classification or recognition task. Addressing these two steps involves solving large scale l1minimization problems. To that end, we propose to solve a sequence of linear equality constrained multiquadric problems that depends on a regularization parameter that converges to zero. The procedure generates a central path that converges to a point on the solution set of the l1underdetermined problem. In order to solve each subproblem, a conjugate gradient algorithm is formulated. When noise is present in the model, inexact directions are taken so that an approximate solution is computed faster. This prevents the ill conditioning produced when the conjugate gradient is required to iterate until a zero residual is attained.
文摘A joint two-dimensional(2D)direction-of-arrival(DOA)and radial Doppler frequency estimation method for the L-shaped array is proposed in this paper based on the compressive sensing(CS)framework.Revised from the conventional CS-based methods where the joint spatial-temporal parameters are characterized in one large scale matrix,three smaller scale matrices with independent azimuth,elevation and Doppler frequency are introduced adopting a separable observation model.Afterwards,the estimation is achieved by L1-norm minimization and the Bayesian CS algorithm.In addition,under the L-shaped array topology,the azimuth and elevation are separated yet coupled to the same radial Doppler frequency.Hence,the pair matching problem is solved with the aid of the radial Doppler frequency.Finally,numerical simulations corroborate the feasibility and validity of the proposed algorithm.
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 Aviation Science Foundation(201120M5007)the Natural Science Foundation of Beijing(4102050)
文摘This paper presents a new method for image separation through employing a combined dictionary consisting of wavelets and complex shearlets. Because the combined dictionary sparsely represents points and curvilinear singularities respectively, the image can be decomposed into pointlike and curvelike parts as accurate as possible. The proposed method based on the geo- metric separation theory introduced by Donoho in 2005 shows that accurate geometric separation of the morphologically distinct fea- tures of points and curves can be achieved by l1 minimization. The experimental results show that the proposed method can not only be effective but also greatly reduce the computing time.
文摘The maximum of k numerical functions defined on , , by , ??is used here in Statistical classification. Previously, it has been used in Statistical Discrimination [1] and in Clustering [2]. We present first some theoretical results on this function, and then its application in classification using a computer program we have developed. This approach leads to clear decisions, even in cases where the extension to several classes of Fisher’s linear discriminant function fails to be effective.
文摘In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for the original problem and design an extragradient thresholding algorithm (ETA) to solve the regularized model. Furthermore, we prove that any cluster point of the sequence generated by ETA is a solution of MCP. Finally, numerical experiments show that the ETA algorithm can effectively solve the l1 regularized projection minimization model and obtain the sparse solution of the mixed complementarity problem.
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.
