The Experts below are selected from a list of 180 Experts worldwide ranked by ideXlab platform

Haibin Chen - One of the best experts on this subject based on the ideXlab platform.

  • Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
    Journal of Industrial & Management Optimization, 2015
    Co-Authors: Haibin Chen
    Abstract:

    Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their Generating Vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its Generating Vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its Generating Vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.

  • Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
    arXiv: Spectral Theory, 2014
    Co-Authors: Haibin Chen
    Abstract:

    Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their Generating Vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its Generating Vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its Generating Vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. Furthermore, we prove that the Hadamard product of two positive semi-definite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.

Yonina C. Eldar - One of the best experts on this subject based on the ideXlab platform.

  • A semidefinite programming approach to optimal unambiguous discrimination of quantum states
    IEEE Transactions on Information Theory, 2003
    Co-Authors: Yonina C. Eldar
    Abstract:

    We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single Generating Vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple Generating Vectors, where the Generating Vectors satisfy a certain (weighted) norm constraint.

  • Geometrically uniform frames
    IEEE Transactions on Information Theory, 2003
    Co-Authors: Yonina C. Eldar, Helmut Bölcskei
    Abstract:

    We introduce a new class of finite-dimensional frames with strong symmetry properties, called geometrically uniform (GU) frames, that are defined over a finite Abelian group of unitary matrices and are generated by a single Generating Vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by a finite Abelian group of unitary matrices using multiple Generating Vectors. The dual frame Vectors and canonical tight frame Vectors associated with GU frames are shown to be GU and, therefore, also generated by a single Generating Vector, which can be computed very efficiently using a Fourier transform (FT) defined over the Generating group of the frame. Similarly, the dual frame Vectors and canonical tight frame Vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame Vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the Abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the Generating Vector.

  • Geometrically Uniform Frames
    arXiv: Functional Analysis, 2001
    Co-Authors: Yonina C. Eldar, Helmut Bölcskei
    Abstract:

    We introduce a new class of frames with strong symmetry properties called geometrically uniform frames (GU), that are defined over an abelian group of unitary matrices and are generated by a single Generating Vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by an abelian group of unitary matrices using multiple Generating Vectors. The dual frame Vectors and canonical tight frame Vectors associated with GU frames are shown to be GU and therefore generated by a single Generating Vector, which can be computed very efficiently using a Fourier transform defined over the Generating group of the frame. Similarly, the dual frame Vectors and canonical tight frame Vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame Vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the Generating Vector.

Helmut Bölcskei - One of the best experts on this subject based on the ideXlab platform.

  • Geometrically uniform frames
    IEEE Transactions on Information Theory, 2003
    Co-Authors: Yonina C. Eldar, Helmut Bölcskei
    Abstract:

    We introduce a new class of finite-dimensional frames with strong symmetry properties, called geometrically uniform (GU) frames, that are defined over a finite Abelian group of unitary matrices and are generated by a single Generating Vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by a finite Abelian group of unitary matrices using multiple Generating Vectors. The dual frame Vectors and canonical tight frame Vectors associated with GU frames are shown to be GU and, therefore, also generated by a single Generating Vector, which can be computed very efficiently using a Fourier transform (FT) defined over the Generating group of the frame. Similarly, the dual frame Vectors and canonical tight frame Vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame Vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the Abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the Generating Vector.

  • Geometrically Uniform Frames
    arXiv: Functional Analysis, 2001
    Co-Authors: Yonina C. Eldar, Helmut Bölcskei
    Abstract:

    We introduce a new class of frames with strong symmetry properties called geometrically uniform frames (GU), that are defined over an abelian group of unitary matrices and are generated by a single Generating Vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by an abelian group of unitary matrices using multiple Generating Vectors. The dual frame Vectors and canonical tight frame Vectors associated with GU frames are shown to be GU and therefore generated by a single Generating Vector, which can be computed very efficiently using a Fourier transform defined over the Generating group of the frame. Similarly, the dual frame Vectors and canonical tight frame Vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame Vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the Generating Vector.

Peter Kritzer - One of the best experts on this subject based on the ideXlab platform.

  • Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness
    Journal of Complexity, 2021
    Co-Authors: Adrian Ebert, Peter Kritzer, Dirk Nuyens, Onyekachi Osisiogu
    Abstract:

    Abstract Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an s -dimensional integral is fully specified by its Generating Vector z ∈ Z s and its number of points  N . While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions for good Generating Vectors for dimensions s ≥ 3 . This is why one usually resorts to computer search algorithms. Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate. Moreover, contrary to most other algorithms, we do not need to know the smoothness of our integrands in advance, the Generating Vector will still recover the convergence rate associated with the smoothness of the particular integrand, and, under appropriate conditions on the weights, the error bounds can be stated without dependence on s . The search algorithms presented in this paper are two variants of the well-known component-by-component (CBC) construction, one of which is combined with a digit-by-digit (DBD) construction. We present numerical results for both algorithms using fast construction algorithms in the case of product weights. They confirm our theoretical findings.

  • Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness.
    arXiv: Numerical Analysis, 2020
    Co-Authors: Adrian Ebert, Peter Kritzer, Dirk Nuyens, Onyekachi Osisiogu
    Abstract:

    Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an $s$-dimensional integral is fully specified by its Generating Vector $\mathbf{z} \in \mathbb{Z}^s$ and its number of points $N$. While there are many results on the existence of "good" rank-1 lattice rules, there are no explicit constructions for good Generating Vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate. Moreover, contrary to most other algorithms, we do not need to know the smoothness of our integrands in advance, the Generating Vector will still recover the convergence rate associated with the smoothness of the particular integrand, and, under appropriate conditions on the weights, the error bounds can be stated without dependence on $s$. The search algorithms presented in this paper are two variants of the well-known component-by-component (CBC) construction, one of which is combined with a digit-by-digit (DBD) construction. We present numerical results for both algorithms using fast construction algorithms in the case of product weights. They confirm our theoretical findings.

  • On a projection-corrected component-by-component construction
    Journal of Complexity, 2016
    Co-Authors: Josef Dick, Peter Kritzer
    Abstract:

    © 2015 Elsevier Inc. The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the Generating Vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm

  • On a projection-corrected component-by-component construction
    arXiv: Numerical Analysis, 2015
    Co-Authors: Josef Dick, Peter Kritzer
    Abstract:

    The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the Generating Vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm.

Qiling Deng - One of the best experts on this subject based on the ideXlab platform.

  • Vector optical field generation based on birefringent phase plate
    Optics express, 2017
    Co-Authors: Jiazhou Wang, Jian Chen, Axiu Cao, Hui Pang, Man Zhang, Wang Guangyi, Lifang Shi, Qiling Deng
    Abstract:

    Vector optical field has recently gained interest in a variety of application fields due to its novel characteristics. Conventional approaches of Generating Vector optical fields have difficulties in forming highly continuous polarization and suffer from the issue of high energy utilization rates. In order to address these issues, in this study a single optical path was proposed to generate Vector optical fields where the birefringent phase plate modulated a linear polarized light into a Vector optical field, which was then demodulated to a non-uniform linear polarization distribution of the Vector optical field by the polarization demodulation module. Both a theoretical model and numerical simulations of the Vector optical field generator were developed, illustrating the relationship between the polarization distribution of the target Vector optical field and the depth distribution of the birefringent phase plate. Furthermore, the birefringent phase plate with predefined surface distributions was fabricated by grayscale exposure and ion etching. The generated Vector optical field was experimentally characterized, capable of producing continuous polarization with high light energy utilization ratio, consistent with simulations. This new approach may have the potential of being widely used in future studies of Generating well-controlled Vector optical fields.