This study aimed to investigate the pollution characteristics, source apportionment, and health risks associated with trace metal(loid)s(TMs) in the major agricultural producing areas in Chongqing, China. We analyzed ...This study aimed to investigate the pollution characteristics, source apportionment, and health risks associated with trace metal(loid)s(TMs) in the major agricultural producing areas in Chongqing, China. We analyzed the source apportionment and assessed the health risk of TMs in agricultural soils by using positive matrix factorization(PMF) model and health risk assessment(HRA) model based on Monte Carlo simulation. Meanwhile, we combined PMF and HRA models to explore the health risks of TMs in agricultural soils by different pollution sources to determine the priority control factors. Results showed that the average contents of cadmium(Cd), arsenic (As), lead(Pb), chromium(Cr), copper(Cu), nickel(Ni), and zinc(Zn) in the soil were found to be 0.26, 5.93, 27.14, 61.32, 23.81, 32.45, and 78.65 mg/kg, respectively. Spatial analysis and source apportionment analysis revealed that urban and industrial sources, agricultural sources, and natural sources accounted for 33.0%, 27.7%, and 39.3% of TM accumulation in the soil, respectively. In the HRA model based on Monte Carlo simulation, noncarcinogenic risks were deemed negligible(hazard index <1), the carcinogenic risks were at acceptable level(10^(-6)<total carcinogenic risk ≤ 10^(-4)), with higher risks observed for children compared to adults. The relationship between TMs, their sources, and health risks indicated that urban and industrial sources were primarily associated with As, contributing to 75.1% of carcinogenic risks and 55.7% of non-carcinogenic risks, making them the primary control factors. Meanwhile, agricultural sources were primarily linked to Cd and Pb, contributing to 13.1% of carcinogenic risks and 21.8% of non-carcinogenic risks, designating them as secondary control factors.展开更多
This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative ...This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .展开更多
Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square...Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.展开更多
The problem of high-precision indoor positioning in the 5G era has attracted more and more attention.A fingerprint location method based on matrix completion(MC-FPL)is proposed for 5G ultradense networks to overcome t...The problem of high-precision indoor positioning in the 5G era has attracted more and more attention.A fingerprint location method based on matrix completion(MC-FPL)is proposed for 5G ultradense networks to overcome the high costs of traditional fingerprint database construction and matching algorithms.First,a partial fingerprint database constructed and the accelerated proximal gradient algorithm is used to fill the partial fingerprint database to construct a full fingerprint database.Second,a fingerprint database division method based on the strongest received signal strength indicator is proposed,which divides the original fingerprint database into several sub-fingerprint databases.Finally,a classification weighted K-nearest neighbor fingerprint matching algorithm is proposed.The estimated coordinates of the point to be located can be obtained by fingerprint matching in a sub-fingerprint database.The simulation results show that the MC-FPL algorithm can reduce the complexity of database construction and fingerprint matching and has higher positioning accuracy compared with the traditional fingerprint algorithm.展开更多
This paper discusses the solutions of the linear matrix equation BT X B=Don some linear manifolds.Some necessary and sufficient conditions for the existenceof the solution and the expression of the general solution ar...This paper discusses the solutions of the linear matrix equation BT X B=Don some linear manifolds.Some necessary and sufficient conditions for the existenceof the solution and the expression of the general solution are given.And also someoptimal approximation solutions are discussed.展开更多
Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm},We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A...Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm},We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best"approximate to λixi, i = 1, 2,..., m in Frobenius norm and (Ⅱ) the Y in set ( )which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem Ⅰ and Problem Ⅱ is given andthe general expression of solutions for Problem Ⅰ is derived. Some sufficient conditionsunder which Problem Ⅰ and Problem Ⅱ have an explicit solution is provided. A numer-ical algorithm of the solution for Problem Ⅱ has been presented.展开更多
In this paper, we construct one of the forms of totally positive Toeplitz matrices from upper or lower bidiagonal totally nonnegative matrix. In addition, some properties related to this matrix involving its factoriza...In this paper, we construct one of the forms of totally positive Toeplitz matrices from upper or lower bidiagonal totally nonnegative matrix. In addition, some properties related to this matrix involving its factorization are presented.展开更多
In this paper, we consider the positive semidefinite solution of the matrix equation (AT X A, BT X B) - (C, D). A necessary and sufficient condition for the existence of such solution is derived using the generalized ...In this paper, we consider the positive semidefinite solution of the matrix equation (AT X A, BT X B) - (C, D). A necessary and sufficient condition for the existence of such solution is derived using the generalized singular value decomposition.The general forms of positive semidefinite solution are given.展开更多
In this paper, the totally non-positive matrix is introduced. The totally non-positive completion asks which partial totally non-positive matrices have a completion to a totally non-positive matrix. This problem has. ...In this paper, the totally non-positive matrix is introduced. The totally non-positive completion asks which partial totally non-positive matrices have a completion to a totally non-positive matrix. This problem has. in general, a negative answer. Therefore, our question is for what kind of labeled graphs G each partial totally non-positive matrix whose associated graph is G has a totally non-positive completion? If G is not a monotonically labeled graph or monotonically labeled cycle, we give necessary and sufficient conditions that guarantee the existence of the desired completion.展开更多
For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 30...For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).展开更多
The knowledge of subnuclear localization in eukaryotic cells is indispensable for under-standing the biological function of nucleus, genome regulation and drug discovery. In this study, a new feature representation wa...The knowledge of subnuclear localization in eukaryotic cells is indispensable for under-standing the biological function of nucleus, genome regulation and drug discovery. In this study, a new feature representation was pro-posed by combining position specific scoring matrix (PSSM) and auto covariance (AC). The AC variables describe the neighboring effect between two amino acids, so that they incorpo-rate the sequence-order information;PSSM de-scribes the information of biological evolution of proteins. Based on this new descriptor, a support vector machine (SVM) classifier was built to predict subnuclear localization. To evaluate the power of our predictor, the benchmark dataset that contains 714 proteins localized in nine subnuclear compartments was utilized. The total jackknife cross validation ac-curacy of our method is 76.5%, that is higher than those of the Nuc-PLoc (67.4%), the OET- KNN (55.6%), AAC based SVM (48.9%) and ProtLoc (36.6%). The prediction software used in this article and the details of the SVM parameters are freely available at http://chemlab.scu.edu.cn/ predict_SubNL/index.htm and the dataset used in our study is from Shen and Chou’s work by downloading at http://chou.med.harvard.edu/ bioinf/Nuc-PLoc/Data.htm.展开更多
A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from ...A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from {π_1,π_2} is positive. A necessary and sufficient condition is obtained for a pair {π_1,π_2} to have property(C),and an easily checked necessary and sufficient condition for the pairs of the form {π~p,π~q} to have property(C) is given, whereπ is the permutation defined by π(i) = i + 1 mod n and 1≤ p < q≤ n.展开更多
We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put...We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.展开更多
As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevinsky[7],in this paper,we give a test method of pos...As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevinsky[7],in this paper,we give a test method of positive definite matrices via the planar networks and the so-called mixing-type sub-cluster algebras respectively,introduced here originally.This work firstly gives a combinatorial realization of all matrices through planar network,and then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks.On the other hand,mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.展开更多
Identification of the drug-binding residues on the surface of proteins is a vital step in drug discovery and it is important for understanding protein function. Most previous researches are based on the structural inf...Identification of the drug-binding residues on the surface of proteins is a vital step in drug discovery and it is important for understanding protein function. Most previous researches are based on the structural information of proteins, but the structures of most proteins are not available. So in this article, a sequence-based method was proposed by combining the support vector machine (SVM)-based ensemble learning and the improved position specific scoring matrix (PSSM). In order to take the local environment information of a drug-binding site into account, an improved PSSM profile scaled by the sliding window and smoothing window was used to improve the prediction result. In addition, a new SVM-based ensemble learning method was developed to deal with the imbalanced data classification problem that commonly exists in the binding site predictions. When performed on the dataset of 985 drug-binding residues, the method achieved a very promising prediction result with the area under the curve (AUC) of 0.9264. Furthermore, an independent dataset of 349 drug- binding residues was used to evaluate the pre- diction model and the prediction accuracy is 84.68%. These results suggest that our method is effective for predicting the drug-binding sites in proteins. The code and all datasets used in this article are freely available at http://cic.scu.edu.cn/bioinformatics/Ensem_DBS.zip.展开更多
Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which i...Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.展开更多
基金supported by Project of Chongqing Science and Technology Bureau (cstc2022jxjl0005)。
文摘This study aimed to investigate the pollution characteristics, source apportionment, and health risks associated with trace metal(loid)s(TMs) in the major agricultural producing areas in Chongqing, China. We analyzed the source apportionment and assessed the health risk of TMs in agricultural soils by using positive matrix factorization(PMF) model and health risk assessment(HRA) model based on Monte Carlo simulation. Meanwhile, we combined PMF and HRA models to explore the health risks of TMs in agricultural soils by different pollution sources to determine the priority control factors. Results showed that the average contents of cadmium(Cd), arsenic (As), lead(Pb), chromium(Cr), copper(Cu), nickel(Ni), and zinc(Zn) in the soil were found to be 0.26, 5.93, 27.14, 61.32, 23.81, 32.45, and 78.65 mg/kg, respectively. Spatial analysis and source apportionment analysis revealed that urban and industrial sources, agricultural sources, and natural sources accounted for 33.0%, 27.7%, and 39.3% of TM accumulation in the soil, respectively. In the HRA model based on Monte Carlo simulation, noncarcinogenic risks were deemed negligible(hazard index <1), the carcinogenic risks were at acceptable level(10^(-6)<total carcinogenic risk ≤ 10^(-4)), with higher risks observed for children compared to adults. The relationship between TMs, their sources, and health risks indicated that urban and industrial sources were primarily associated with As, contributing to 75.1% of carcinogenic risks and 55.7% of non-carcinogenic risks, making them the primary control factors. Meanwhile, agricultural sources were primarily linked to Cd and Pb, contributing to 13.1% of carcinogenic risks and 21.8% of non-carcinogenic risks, designating them as secondary control factors.
文摘This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .
文摘Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.
基金supported in part by Sub Project of National Key Research and Development plan in 2020.NO.2020YFC1511704Beijing Information Science and Technology University.NO.2020KYNH212,NO.2021CGZH302+1 种基金Beijing Science and Technology Project(Grant No.Z211100004421009)in part by the National Natural Science Foundation of China(Grant No.61971048)。
文摘The problem of high-precision indoor positioning in the 5G era has attracted more and more attention.A fingerprint location method based on matrix completion(MC-FPL)is proposed for 5G ultradense networks to overcome the high costs of traditional fingerprint database construction and matching algorithms.First,a partial fingerprint database constructed and the accelerated proximal gradient algorithm is used to fill the partial fingerprint database to construct a full fingerprint database.Second,a fingerprint database division method based on the strongest received signal strength indicator is proposed,which divides the original fingerprint database into several sub-fingerprint databases.Finally,a classification weighted K-nearest neighbor fingerprint matching algorithm is proposed.The estimated coordinates of the point to be located can be obtained by fingerprint matching in a sub-fingerprint database.The simulation results show that the MC-FPL algorithm can reduce the complexity of database construction and fingerprint matching and has higher positioning accuracy compared with the traditional fingerprint algorithm.
基金This work was supposed by the National Nature Science Foundation of China
文摘This paper discusses the solutions of the linear matrix equation BT X B=Don some linear manifolds.Some necessary and sufficient conditions for the existenceof the solution and the expression of the general solution are given.And also someoptimal approximation solutions are discussed.
基金Suported by National Nature Science Foundation of China
文摘Let A and C denote real n × n matrices. Given real n-vectors x1,……… ,xm,m≤n,and a set of numbers (L)={λ1,λ2…λm},We descrbe(Ⅰ)the set( ) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best"approximate to λixi, i = 1, 2,..., m in Frobenius norm and (Ⅱ) the Y in set ( )which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem Ⅰ and Problem Ⅱ is given andthe general expression of solutions for Problem Ⅰ is derived. Some sufficient conditionsunder which Problem Ⅰ and Problem Ⅱ have an explicit solution is provided. A numer-ical algorithm of the solution for Problem Ⅱ has been presented.
文摘In this paper, we construct one of the forms of totally positive Toeplitz matrices from upper or lower bidiagonal totally nonnegative matrix. In addition, some properties related to this matrix involving its factorization are presented.
文摘We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.
基金Partially supported by the National Natural Science Foundation of China(No10071035) and the Doctor Foundation of Hunan Normal University.
文摘In this paper, we consider the positive semidefinite solution of the matrix equation (AT X A, BT X B) - (C, D). A necessary and sufficient condition for the existence of such solution is derived using the generalized singular value decomposition.The general forms of positive semidefinite solution are given.
基金The work was supported by the National Science Foundation of China (10571146).
文摘In this paper, the totally non-positive matrix is introduced. The totally non-positive completion asks which partial totally non-positive matrices have a completion to a totally non-positive matrix. This problem has. in general, a negative answer. Therefore, our question is for what kind of labeled graphs G each partial totally non-positive matrix whose associated graph is G has a totally non-positive completion? If G is not a monotonically labeled graph or monotonically labeled cycle, we give necessary and sufficient conditions that guarantee the existence of the desired completion.
文摘For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).
文摘The knowledge of subnuclear localization in eukaryotic cells is indispensable for under-standing the biological function of nucleus, genome regulation and drug discovery. In this study, a new feature representation was pro-posed by combining position specific scoring matrix (PSSM) and auto covariance (AC). The AC variables describe the neighboring effect between two amino acids, so that they incorpo-rate the sequence-order information;PSSM de-scribes the information of biological evolution of proteins. Based on this new descriptor, a support vector machine (SVM) classifier was built to predict subnuclear localization. To evaluate the power of our predictor, the benchmark dataset that contains 714 proteins localized in nine subnuclear compartments was utilized. The total jackknife cross validation ac-curacy of our method is 76.5%, that is higher than those of the Nuc-PLoc (67.4%), the OET- KNN (55.6%), AAC based SVM (48.9%) and ProtLoc (36.6%). The prediction software used in this article and the details of the SVM parameters are freely available at http://chemlab.scu.edu.cn/ predict_SubNL/index.htm and the dataset used in our study is from Shen and Chou’s work by downloading at http://chou.med.harvard.edu/ bioinf/Nuc-PLoc/Data.htm.
基金partially supported by National Natural Science Foundation of China(11671294)
文摘A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from {π_1,π_2} is positive. A necessary and sufficient condition is obtained for a pair {π_1,π_2} to have property(C),and an easily checked necessary and sufficient condition for the pairs of the form {π~p,π~q} to have property(C) is given, whereπ is the permutation defined by π(i) = i + 1 mod n and 1≤ p < q≤ n.
基金This work is supported by NSF of Shanxi province,20011041.
文摘We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.
基金Supported by the National Natural Science Foundation of China(11671350,11571173,11801043)Natural Science Foundation for Youths of Jiangsu Province(BK20181031).
文摘As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevinsky[7],in this paper,we give a test method of positive definite matrices via the planar networks and the so-called mixing-type sub-cluster algebras respectively,introduced here originally.This work firstly gives a combinatorial realization of all matrices through planar network,and then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks.On the other hand,mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.
文摘Identification of the drug-binding residues on the surface of proteins is a vital step in drug discovery and it is important for understanding protein function. Most previous researches are based on the structural information of proteins, but the structures of most proteins are not available. So in this article, a sequence-based method was proposed by combining the support vector machine (SVM)-based ensemble learning and the improved position specific scoring matrix (PSSM). In order to take the local environment information of a drug-binding site into account, an improved PSSM profile scaled by the sliding window and smoothing window was used to improve the prediction result. In addition, a new SVM-based ensemble learning method was developed to deal with the imbalanced data classification problem that commonly exists in the binding site predictions. When performed on the dataset of 985 drug-binding residues, the method achieved a very promising prediction result with the area under the curve (AUC) of 0.9264. Furthermore, an independent dataset of 349 drug- binding residues was used to evaluate the pre- diction model and the prediction accuracy is 84.68%. These results suggest that our method is effective for predicting the drug-binding sites in proteins. The code and all datasets used in this article are freely available at http://cic.scu.edu.cn/bioinformatics/Ensem_DBS.zip.
文摘Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.