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In this work, we present a method of decomposition of arbitrary unitary matrix U is an element of U(2(k)) into a product of single-qubit negator and controlled-root NOT gates. Since the product results with negator matrix, which c...
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In this work, we present a method of decomposition of arbitrary unitary matrix U is an element of U(2(k)) into a product of single-qubit negator and controlled-root NOT gates. Since the product results with negator matrix, which can be treated as complex analogue of bistochastic matrix, our method can be seen as complex analogue of Sinkhorn-Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively, and resulting circuit consists of O(4(k)) entangling gates, which is proved to be optimal. An example of such transformation is presented.
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We propose a new fast randomized algorithm for interpolative decomposition of matrices which utilizes CountSketch. We then extend this approach to the tensor interpolative decomposition problem introduced by Biagioni et al. (J. Co...
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We propose a new fast randomized algorithm for interpolative decomposition of matrices which utilizes CountSketch. We then extend this approach to the tensor interpolative decomposition problem introduced by Biagioni et al. (J. Comput. Phys. 281(C), 116-134 (2015)). Theoretical performance guarantees are provided for both the matrix and tensor settings. Numerical experiments on both synthetic and real data demonstrate that our algorithms maintain the accuracy of competing methods, while running in less time, achieving at least an order of magnitude speedup on large matrices and tensors.
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Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition...
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Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A~*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A-BXB~*-CYC~* and then solve two conjectures on the maximal and minimal possible ranks of A-BXB~*-CYC~* with respect to X=±X~* and Y=±Y~*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB~* + CYC~*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation.
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Matrix decompositions are used for many data mining purposes. One of these purposes is to find a concise but interpretable representation of a given data matrix. Different decomposition formulations have been proposed for this tas...
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Matrix decompositions are used for many data mining purposes. One of these purposes is to find a concise but interpretable representation of a given data matrix. Different decomposition formulations have been proposed for this task, many of which assume a certain property of the input data ( e. g., nonnegativity) and aim at preserving that property in the decomposition. In this paper we propose new decomposition formulations for binary matrices, namely the Boolean CX and CUR decompositions. They are natural combinations of two previously presented decomposition formulations. We consider also two subproblems of these decompositions and present a rigorous theoretical study of the subproblems. We give algorithms for the decompositions and for the subproblems, and study their performance via extensive experimental evaluation. We show that even simple algorithms can give accurate and intuitive decompositions of real data, thus demonstrating the power and usefulness of the proposed decompositions.
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摘要 :
Matrix decompositions are used for many data mining purposes. One of these purposes is to find a concise but interpretable representation of a given data matrix. Different decomposition formulations have been proposed for this tas...
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Matrix decompositions are used for many data mining purposes. One of these purposes is to find a concise but interpretable representation of a given data matrix. Different decomposition formulations have been proposed for this task, many of which assume a certain property of the input data (e.g., nonnegativity) and aim at preserving that property in the decomposition. In this paper we propose new decomposition formulations for binary matrices, namely the Boolean CX and CUR decompositions. They are natural combinations of two previously presented decomposition formulations. We consider also two subproblems of these decompositions and present a rigorous theoretical study of the subproblems. We give algorithms for the decompositions and for the subproblems, and study their performance via extensive experimental evaluation. We show that even simple algorithms can give accurate and intuitive decompositions of real data, thus demonstrating the power and usefulness of the proposed decompositions.
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The introduction of matrix decomposition into numerical linear algebra revolutionized matrix computations. The article outlines the decompositional approach, comments on its history, and surveys the six most widely used decomposit...
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The introduction of matrix decomposition into numerical linear algebra revolutionized matrix computations. The article outlines the decompositional approach, comments on its history, and surveys the six most widely used decompositions: Cholesky decomposition; pivoted LU decomposition; QR decomposition; spectral decomposition; Schur decomposition; and singular value decomposition.
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This study proposes a data recovery model where substituted values can be further limited by nonlinear and inequality constraints to approximate the ground truth. The objective is to generate substituted values for multifactors wh...
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This study proposes a data recovery model where substituted values can be further limited by nonlinear and inequality constraints to approximate the ground truth. The objective is to generate substituted values for multifactors while considering their lower/upper bounds, data means, and nonlinearity at the same time. This is critical when data need to fall inside a nonlinear range, e.g., a partial hypersphere centered at a given mean. In view of such, this study proposes collaborative filtering with nonlinear inequality constraints to tackle the problem. The proposed method consists of three steps. First, the system finds class-dependent and box-bounded imputation basis factors for an incomplete data set. Class-dependent bases can reflect data domains well. Second, class-dependent imputation coefficients are located by the proposed nonnegative coefficient discovery with nonlinear inequality constraints. This step limits searching space and avoids generating substituted values out of range. Finally, constrained iterative projection pursuit is proposed for measuring the quality of recovered data by examining reconstruction residuals. By using both the nonlinear inequality constraints and the constrained iterative projection pursuit, the system can recover data while satisfying multifactor nonlinear coeffects required by manufacturers. Experimental results showed that the proposed method was capable of generating substituted values with lower root-mean-squared errors. In addition, errors were reduced by at least 10.06% on average, better than those of the baselines. Furthermore, the classification accuracy of the proposed method after data imputation was higher than that of the baselines. Such findings indicated that the proposed method could approximate the characteristics of data when missing values appeared.
Note to Practitioners
—This work was motivated by the problem of missing values in industrial heterogeneous sensor readings. When data recovery is performed, the process should consider multifactor nonlinearity, lower/upper bounds, historical references, and the divergence between substituted values and references at the same time in order to reconstruct original sensor readings as many as possible. Existing approaches generally have solutions to linear or nonlinear equality constraints but not the aforementioned nonlinear inequality ones. This work designs a self-dictionary method—class-dependent and box-bounded imputation basis factors along with constrained iterative projection pursuit—for finding substituted values. Real industrial experiments were conducted based on
$\mathcal {L}_{2}$
-norms. Future research will address the design for other norms.
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This paper sets out a modification of algorithms for calculating the QR decomposition and the singular value decomposition of a polynomial matrix into matrices with elements in the form of rational functions, which is equivalent t...
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This paper sets out a modification of algorithms for calculating the QR decomposition and the singular value decomposition of a polynomial matrix into matrices with elements in the form of rational functions, which is equivalent to the use of IIR rather than FIR filters in processing and generating the signal. The proposed approach can reduce the cost of calculating the decomposition and of further signal processing, primarily by reducing the degree of the polynomials which form the elements of the resultant matrix. This is equivalent to reducing the memory interval of virtual subchannels. This method is based on the polynomial version of the QR algorithm and on Bauer factorization.
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Cholesky decomposition of the atomic two-electron integral matrix has recently been proposed as a procedure for automated generation of auxiliary basis sets for the density fitting approximation F. Aquilante et al., J. Chem. Phys....
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Cholesky decomposition of the atomic two-electron integral matrix has recently been proposed as a procedure for automated generation of auxiliary basis sets for the density fitting approximation F. Aquilante et al., J. Chem. Phys. 127, 114107 (2007). In order to increase computational performance while maintaining accuracy, we propose here to reduce the number of primitive Gaussian functions of the contracted auxiliary basis functions by means of a second Cholesky decomposition. Test calculations show that this procedure is most beneficial in conjunction with highly contracted atomic orbital basis sets such as atomic natural orbitals, and that the error resulting from the second decomposition is negligible. We also demonstrate theoretically as well as computationally that the locality of the fitting coefficients can be controlled by means of the decomposition threshold even with the long-ranged Coulomb metric. Cholesky decomposition-based auxiliary basis sets are thus ideally suited for local density fitting approximations.
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We show that every symplectic matrix is a product of five positive definite symplectic matrices and five is the best in the sense that there are symplectic matrices which are not product of less. (C) 2021 Elsevier Inc. All rights reserved.
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