In the traditional unscented Kalman filter（UKF）,accuracy and robustness decline when uncertain disturbances exist in the practical system.To deal with the problem,a robust UKF algorithm based on an H-infinity norm i...In the traditional unscented Kalman filter（UKF）,accuracy and robustness decline when uncertain disturbances exist in the practical system.To deal with the problem,a robust UKF algorithm based on an H-infinity norm is proposed.In Krein space,a robust element is added in the simplified UKF so as to improve the algorithm.The filtering gain is adjusted by the robust element and in this way the performance of the robustness of the filtering algorithm is promoted.In the initial alignment process of the large heading misalignment angle of the strapdown inertial navigation system（SINS）,comparative studies are conducted on the robust UKF and the simplified UKF.The simulation results illustrate that compared with the simplified UKF,the robust UKF is more accurate,and the estimation error of heading misalignment decreases from 16.9＇ to 4.3＇.In short,the robust UKF can reduce the sensitivity to the system disturbances resulting in better performance.展开更多
Data hierarchy,as a hidden property of data structure,exists in a wide range of machine learning applications.A common practice to classify such hierarchical data is first to encode the data in the Euclidean space,and...Data hierarchy,as a hidden property of data structure,exists in a wide range of machine learning applications.A common practice to classify such hierarchical data is first to encode the data in the Euclidean space,and then train a Euclidean classifier.However,such a paradigm leads to a performance drop due to distortion of data embedding in the Euclidean space.To relieve this issue,hyperbolic geometry is investigated as an alternative space to encode the hierarchical data for its higher ability to capture the hierarchical structures.Those methods cannot explore the full potential of the hyperbolic geometry,in the sense that such methods define the hyperbolic operations in the tangent plane,causing the distortion of data embeddings.In this paper,we develop two novel kernel formulations in the hyperbolic space,with one being positive definite(PD)and another one being indefinite,to solve the classification tasks in hyperbolic space.The PD one is defined via mapping the hyperbolic data to the Drury-Arveson(DA)space,which is a special reproducing kernel Hilbert space(RKHS).To further increase the discrimination of the classifier,an indefinite kernel is further defined in the Krein spaces.Specifically,we design a 2-layer nested indefinite kernel which first maps hyperbolic data into the DA spaces,followed by a mapping from the DA spaces to the Krein spaces.Extensive experiments on real-world datasets demonstrate the superiority ofthe proposed kernels.展开更多
In this paper,we derive the optimal Cauchy–Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra,which heavily depend on the Clifford algebraic structure.The obtained inequalities furth...In this paper,we derive the optimal Cauchy–Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra,which heavily depend on the Clifford algebraic structure.The obtained inequalities further lead to very general uncertainty inequalities on these modules.Some new phenomena arise,due to the non-commutative nature,the Clifford-valued inner products and the Krein geometry.Taking into account applications,special attention is given to the Dirac operator and the Howe dual pair Pin(m)×osp(1|2).Moreover,it is surprisingly to find that the recent highly nontrivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy–Schwarz inequality.This new observation leads to refined uncertainty relations in terms of the Wigner–Yanase–Dyson skew information for mixed states and other generalizations.These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.展开更多
The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-def...The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…展开更多
This note is concerned with the H-infinity deconvolution filtering problem for linear time-varying discretetime systems described by state space models, The H-infinity deconvolution filter is derived by proposing a ne...This note is concerned with the H-infinity deconvolution filtering problem for linear time-varying discretetime systems described by state space models, The H-infinity deconvolution filter is derived by proposing a new approach in Krein space. With the new approach, it is clearly shown that the central deconvolution filter in an H-infinity setting is the same as the one in an H2 setting associated with one constructed stochastic state-space model. This insight allows us to calculate the complicated H-infinity deconvolution filter in an intuitive and simple way. The deconvolution filter is calculated by performing Riccati equation with the same order as that of the original system.展开更多
We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form: . We study some special type of factorization for plus-operators T, among them the following one: T ...We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form: . We study some special type of factorization for plus-operators T, among them the following one: T = BU, where B is a lower triangular plus-operator, U is a J-unitary operator. We apply the above factorization to the study of basical properties of relations (1), in particular, convexity and compactness of their images with respect to the weak operator topology. Obtained results we apply to the known Koenigs embedding problem, the Krein-Phillips problem of existing of invariant semidefinite subspaces for some families of plus-operators and to some other fields.展开更多
The operator sets, which are the subject of this paper, have been studied in many papers where, under different restrictions on the generating operators, convexity, compactness in the weak operator topology, and nonem...The operator sets, which are the subject of this paper, have been studied in many papers where, under different restrictions on the generating operators, convexity, compactness in the weak operator topology, and nonemptiness were proved for sets of different classes under study. Then the results obtained were used in these papers to solve several applied problems. Namely, they played the key role in establishing the dichotomy of nonautonomous dynamical systems, with either continuous or discrete time. In the present paper, we generalize and sharpen the already known criteria and obtain several new criteria for convexity, compactness, and nonemptiness of several special operator sets. Then, using the assertions obtained, we construct examples of sets of the form under study which are nonconvex, noncompact in the weak operator topology, as well as empty, and are generated by ＂smooth＂ operators of a special class. The existence problem for such sets remained open until the authors of this paper announced some of its results.展开更多
It has been shown that the negative-norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter spacetime. In this processes ultraviolet and infrared di- vergence...It has been shown that the negative-norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter spacetime. In this processes ultraviolet and infrared di- vergences have been automatically eliminated. A natural renormalization of the one-loop interacting quantum field in Minkowski spacetime (λφ 4) has been achieved through the consideration of the nega- tive-norm states defined in Krein space. It has been shown that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuation, results in quantum field the- ory without any divergences. Pursuing this approach, we express Wick’s theorem and calculate Mφller scattering in the one-loop approximation in generalized Krein space. The mathematical consequence of this method is the disappearance of the ultraviolet divergence in the one-loop approximation.展开更多
基金Supported by National Natural Science Foundation of China （60774071）, the Doctoral Program Foundation of Education Ministry of China （20050422036）, and Shandong Scientific and Research Grant.（2005BS01007.）
文摘这份报纸处理 H 差错评价的问题因为有 L2 标准的线性分离变化时间的系统的一个类围住未知输入。主要贡献是 H 差错评价的一条新 Krein 基于空间的途径的发展。H 差错评价的问题第一被等同到一种分级的二次的形式的最小。由在 Krein 空格介绍一个相应系统，然后，一个 H 差错评估者的存在上的一个足够、必要的条件被导出，它的参数矩阵的一个答案以矩阵 Riccati 方程被获得。最后，二个数字例子被给表明建议方法的效率。
基金The National Basic Research Program of China (973 Program) (No. 613121010202)
文摘In the traditional unscented Kalman filter（UKF）,accuracy and robustness decline when uncertain disturbances exist in the practical system.To deal with the problem,a robust UKF algorithm based on an H-infinity norm is proposed.In Krein space,a robust element is added in the simplified UKF so as to improve the algorithm.The filtering gain is adjusted by the robust element and in this way the performance of the robustness of the filtering algorithm is promoted.In the initial alignment process of the large heading misalignment angle of the strapdown inertial navigation system（SINS）,comparative studies are conducted on the robust UKF and the simplified UKF.The simulation results illustrate that compared with the simplified UKF,the robust UKF is more accurate,and the estimation error of heading misalignment decreases from 16.9＇ to 4.3＇.In short,the robust UKF can reduce the sensitivity to the system disturbances resulting in better performance.
基金supported by the National Natural Science Foundation of China(Grant No.62076062)the Fundamental Research Funds for the Central Universities(2242021k30056).
文摘Data hierarchy,as a hidden property of data structure,exists in a wide range of machine learning applications.A common practice to classify such hierarchical data is first to encode the data in the Euclidean space,and then train a Euclidean classifier.However,such a paradigm leads to a performance drop due to distortion of data embedding in the Euclidean space.To relieve this issue,hyperbolic geometry is investigated as an alternative space to encode the hierarchical data for its higher ability to capture the hierarchical structures.Those methods cannot explore the full potential of the hyperbolic geometry,in the sense that such methods define the hyperbolic operations in the tangent plane,causing the distortion of data embeddings.In this paper,we develop two novel kernel formulations in the hyperbolic space,with one being positive definite(PD)and another one being indefinite,to solve the classification tasks in hyperbolic space.The PD one is defined via mapping the hyperbolic data to the Drury-Arveson(DA)space,which is a special reproducing kernel Hilbert space(RKHS).To further increase the discrimination of the classifier,an indefinite kernel is further defined in the Krein spaces.Specifically,we design a 2-layer nested indefinite kernel which first maps hyperbolic data into the DA spaces,followed by a mapping from the DA spaces to the Krein spaces.Extensive experiments on real-world datasets demonstrate the superiority ofthe proposed kernels.
基金Supported by NSFC(Grant No.12101451)Tianjin Municipal Science and Technology Commission(Grant No.22JCQNJC00470)。
文摘In this paper,we derive the optimal Cauchy–Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra,which heavily depend on the Clifford algebraic structure.The obtained inequalities further lead to very general uncertainty inequalities on these modules.Some new phenomena arise,due to the non-commutative nature,the Clifford-valued inner products and the Krein geometry.Taking into account applications,special attention is given to the Dirac operator and the Howe dual pair Pin(m)×osp(1|2).Moreover,it is surprisingly to find that the recent highly nontrivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy–Schwarz inequality.This new observation leads to refined uncertainty relations in terms of the Wigner–Yanase–Dyson skew information for mixed states and other generalizations.These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.
基金Supported by the National Natural Science Foundation of China(10561005)the Doctor's Discipline Fund of the Ministry of Education of China(20040126008)
文摘The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…
基金supported by the National Natural Science Foundation of China (No.60574016,60804034)the Natural Science Foundation of Shandong Province (No.Y2007G34)+2 种基金the National Natural Science Foundation for Distinguished Youth Scholars of China (No.60825304)973 Program (No.2009cb320600)the first two authors are also supported by "Taishan Scholarship" Construction Engineering
文摘This note is concerned with the H-infinity deconvolution filtering problem for linear time-varying discretetime systems described by state space models, The H-infinity deconvolution filter is derived by proposing a new approach in Krein space. With the new approach, it is clearly shown that the central deconvolution filter in an H-infinity setting is the same as the one in an H2 setting associated with one constructed stochastic state-space model. This insight allows us to calculate the complicated H-infinity deconvolution filter in an intuitive and simple way. The deconvolution filter is calculated by performing Riccati equation with the same order as that of the original system.
文摘We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form: . We study some special type of factorization for plus-operators T, among them the following one: T = BU, where B is a lower triangular plus-operator, U is a J-unitary operator. We apply the above factorization to the study of basical properties of relations (1), in particular, convexity and compactness of their images with respect to the weak operator topology. Obtained results we apply to the known Koenigs embedding problem, the Krein-Phillips problem of existing of invariant semidefinite subspaces for some families of plus-operators and to some other fields.
文摘The operator sets, which are the subject of this paper, have been studied in many papers where, under different restrictions on the generating operators, convexity, compactness in the weak operator topology, and nonemptiness were proved for sets of different classes under study. Then the results obtained were used in these papers to solve several applied problems. Namely, they played the key role in establishing the dichotomy of nonautonomous dynamical systems, with either continuous or discrete time. In the present paper, we generalize and sharpen the already known criteria and obtain several new criteria for convexity, compactness, and nonemptiness of several special operator sets. Then, using the assertions obtained, we construct examples of sets of the form under study which are nonconvex, noncompact in the weak operator topology, as well as empty, and are generated by ＂smooth＂ operators of a special class. The existence problem for such sets remained open until the authors of this paper announced some of its results.
文摘It has been shown that the negative-norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter spacetime. In this processes ultraviolet and infrared di- vergences have been automatically eliminated. A natural renormalization of the one-loop interacting quantum field in Minkowski spacetime (λφ 4) has been achieved through the consideration of the nega- tive-norm states defined in Krein space. It has been shown that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuation, results in quantum field the- ory without any divergences. Pursuing this approach, we express Wick’s theorem and calculate Mφller scattering in the one-loop approximation in generalized Krein space. The mathematical consequence of this method is the disappearance of the ultraviolet divergence in the one-loop approximation.
