The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
Tong Zhang - One of the best experts on this subject based on the ideXlab platform.
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Improved Analyses of the Randomized Power Method and Block Lanczos Method
arXiv: Numerical Analysis, 2015Co-Authors: Shusen Wang, Zhihua Zhang, Tong ZhangAbstract:The Power Method and block Lanczos Method are popular numerical algorithms for computing the truncated singular value decomposition (SVD) and eigenvalue decomposition problems. Especially in the literature of randomized numerical linear algebra, the Power Method is widely applied to improve the quality of randomized sketching, and relative-error bounds have been well established. Recently, Musco & Musco (2015) proposed a block Krylov subspace Method that fully exploits the intermediate results of the Power iteration to accelerate convergence. They showed spectral gap-independent bounds which are stronger than the Power Method by order-of-magnitude. This paper offers novel error analysis techniques and significantly improves the bounds of both the randomized Power Method and the block Lanczos Method. This paper also establishes the first gap-independent bound for the warm-start block Lanczos Method.
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Truncated Power Method for sparse eigenvalue problems
Journal of Machine Learning Research, 2013Co-Authors: Xiao-tong Yuan, Tong ZhangAbstract:This paper considers the sparse eigenvalue problem, which is to extract dominant (largest) sparse eigenvectors with at most k non-zero components. We propose a simple yet effective solution called truncated Power Method that can approximately solve the underlying nonconvex optimization problem. A strong sparse recovery result is proved for the truncated Power Method, and this theory is our key motivation for developing the new algorithm. The proposed Method is tested on applications such as sparse principal component analysis and the densest k-subgraph problem. Extensive experiments on several synthetic and real-world data sets demonstrate the competitive empirical performance of our Method.
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Truncated Power Method for Sparse Eigenvalue Problems
arXiv: Machine Learning, 2011Co-Authors: Xiao-tong Yuan, Tong ZhangAbstract:This paper considers the sparse eigenvalue problem, which is to extract dominant (largest) sparse eigenvectors with at most $k$ non-zero components. We propose a simple yet effective solution called truncated Power Method that can approximately solve the underlying nonconvex optimization problem. A strong sparse recovery result is proved for the truncated Power Method, and this theory is our key motivation for developing the new algorithm. The proposed Method is tested on applications such as sparse principal component analysis and the densest $k$-subgraph problem. Extensive experiments on several synthetic and real-world large scale datasets demonstrate the competitive empirical performance of our Method.
Xiao-tong Yuan - One of the best experts on this subject based on the ideXlab platform.
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Truncated Power Method for sparse eigenvalue problems
Journal of Machine Learning Research, 2013Co-Authors: Xiao-tong Yuan, Tong ZhangAbstract:This paper considers the sparse eigenvalue problem, which is to extract dominant (largest) sparse eigenvectors with at most k non-zero components. We propose a simple yet effective solution called truncated Power Method that can approximately solve the underlying nonconvex optimization problem. A strong sparse recovery result is proved for the truncated Power Method, and this theory is our key motivation for developing the new algorithm. The proposed Method is tested on applications such as sparse principal component analysis and the densest k-subgraph problem. Extensive experiments on several synthetic and real-world data sets demonstrate the competitive empirical performance of our Method.
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Truncated Power Method for Sparse Eigenvalue Problems
arXiv: Machine Learning, 2011Co-Authors: Xiao-tong Yuan, Tong ZhangAbstract:This paper considers the sparse eigenvalue problem, which is to extract dominant (largest) sparse eigenvectors with at most $k$ non-zero components. We propose a simple yet effective solution called truncated Power Method that can approximately solve the underlying nonconvex optimization problem. A strong sparse recovery result is proved for the truncated Power Method, and this theory is our key motivation for developing the new algorithm. The proposed Method is tested on applications such as sparse principal component analysis and the densest $k$-subgraph problem. Extensive experiments on several synthetic and real-world large scale datasets demonstrate the competitive empirical performance of our Method.
Todd C. Headrick - One of the best experts on this subject based on the ideXlab platform.
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A Characterization of Power Method Transformations through The Method of Percentiles
Communications in Statistics - Simulation and Computation, 2017Co-Authors: Tzu-chun Kuo, Todd C. HeadrickAbstract:ABSTRACTThis article derives closed-form solutions for fifth-ordered Power Method polynomial transformations based on the Method of Percentiles (MOP). A proposed MOP univariate procedure is compared with the Method of Moments (MOM) in the context of distribution fitting and estimating the shape functions. The MOP is also extended from univariate to multivariate data generation. The MOP procedure has an advantage because it does not require numerical integration to compute intermediate correlations and can be applied to distributions, where conventional moments do not exist. Simulation results demonstrate that the proposed MOP procedure is superior in terms of estimation, bias, and error.
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A Doubling Technique for the Power Method Transformations
Applied mathematical sciences, 2012Co-Authors: Mohan D. Pant, Todd C. HeadrickAbstract:Power Method polynomials are used for simulating non-normal distributions with specied product moments or L-moments. The Power Method is capable of producing distributions with extreme values of skew (L-skew) and kurtosis (L-kurtosis). However, these distributions can be extremely peaked and thus not representative of real-world data. To obviate this problem, two families of distributions are introduced based on a doubling technique with symmetric standard normal and logistic Power Method distributions. The primary focus of the Methodology is in the context of L-moment theory. As such, L-moment based systems of equations are derived for simulating univariate and multivariate non-normal distributions with specied values of L-skew, L-kurtosis, and L-correlation. Evaluation of the proposed doubling technique indicates that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional product-moments in terms of relative bias and relative eciency when extreme non-normal distributions are of concern.
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A Characterization of Power Method Transformations through L-Moments
Journal of Probability and Statistics, 2011Co-Authors: Todd C. HeadrickAbstract:Power Method polynomial transformations are commonly used for simulating continuous nonnormal distributions with specified moments. However, conventional moment-based estimators can (a) be substantially biased, (b) have high variance, or (c) be influenced by outliers. In view of these concerns, a characterization of Power Method transformations by L-moments is introduced. Specifically, systems of equations are derived for determining coefficients for specified L-moment ratios, which are associated with standard normal and standard logistic-based polynomials of order five and three. Boundaries for L-moment ratios are also derived, and closed-formed formulae are provided for determining if a Power Method distribution has a valid probability density function. It is demonstrated that L-moment estimators are nearly unbiased and have relatively small variance in the context of the Power Method. Examples of fitting Power Method distributions to theoretical and empirical distributions based on the Method of L-moments are also provided.
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Statistical Simulation: Power Method Polynomials and Other Transformations
2009Co-Authors: Todd C. HeadrickAbstract:Introduction The Power Method Transformation Univariate Theory Third-Order Systems Fifth-Order Systems Mathematica(R) Functions Limitations Multivariate Theory Using the Power Method Transformation Introduction Examples of Third- and Fifth-Order Polynomials Remediation Techniques Monte Carlo Simulation Some Further Considerations Simulating More Elaborate Correlation Structures Introduction Simulating Systems of Linear Statistical Models Methodology Numerical Example and Monte Carlo Simulation Some Additional Comments Simulating Intraclass Correlation Coefficients Methodology Numerical Example and Monte Carlo Simulation Simulating Correlated Continuous Variates and Ranks Methodology Numerical Example and Monte Carlo Simulation Some Additional Comments Other Transformations: The g-and-h and GLD Families of Distributions Introduction The g-and-h Family The Generalized Lambda Distributions (GLDs) Numerical Examples Multivariate Data Generation References Index
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Simulating Controlled Variate and Rank Correlations Based on the Power Method Transformation
Communications in Statistics - Simulation and Computation, 2008Co-Authors: Todd C. Headrick, Simon Y. Aman, T. Mark BeasleyAbstract:The Power Method transformation is a popular algorithm used for simulating correlated non normal continuous variates because of its simplicity and ease of execution. Statistical models may consist of continuous and (or) ranked variates. In view of this, the Methodology is derived for simulating controlled correlation structures between non normal (a) variates, (b) ranks, and (c) variates with ranks in the context of the Power Method. The correlation structure between variate-values and their associated rank-order is also derived for the Power Method. As such, a measure of the potential loss of information is provided when ranks are used in place of variate-values. The results of a Monte Carlo simulation are provided to confirm and demonstrate the Methodology.
Ammar Daskin - One of the best experts on this subject based on the ideXlab platform.
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Combinatorial optimization through variational quantum Power Method
arXiv: Quantum Physics, 2020Co-Authors: Ammar DaskinAbstract:The Power Method (or iteration) is a well-known classical technique that can be used to find the dominant eigenpair of a matrix. Here, we present a variational quantum circuit for the quantum Power Method that can be used to find eigenpairs of unitary matrices. We apply the circuit to the combinatorial optimization and discuss its complexity. We show that the circuit can generate a solution to the optimization problem with only a few number of iterations. The accuracy of the generated solution is determined by the accuracy of the measurement of the single qubit probabilities at the end of the circuit.
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On the quantum version of the shifted Power Method
2018Co-Authors: Ammar DaskinAbstract:In this paper, we present a direct quantum adaptation of the classical shifted Power Method. The Method is very similar to the iterative phase estimation algorithm; however, it does not require any initial estimate of an eigenvector and as in the classical case its convergence (the success probability) is directly related to the eigengap. If the amount of the gap is polynomial in the number of qubits $n$, then the algorithm can converge to the dominant eigenvalue in $O(poly(n))$ time. The Method can be potentially used in any eigenvalue related problems and to find minimum/maximum of a quantum state in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with $n$ parameters can be encoded as the eigenspace of an $2^n$ dimensional unitary operator in $O(poly(n))$ time, then the solution for such a problem can be found in $O(poly(n))$ if the eigengap is polynomial. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as eigenvectors of a diagonal unitary matrix and find the solution for the problem.
Michael J Panza - One of the best experts on this subject based on the ideXlab platform.
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application of Power Method and dominant eigenvector eigenvalue concept for approximate eigenspace solutions to mechanical engineering algebraic systems
American journal of mechanical engineering, 2018Co-Authors: Michael J PanzaAbstract:This paper shows how the concept of dominant eigenvector/eigenvalue and the Power Method can be used for understanding the solution to practical problems in a broad class of mechanical engineering systems described by linear algebraic equations. An analytical mathematical procedure is developed to obtain a reasonably accurate approximate eigenspace solution to the system Ax=b by transforming the non-symmetric system matrix into a form where the Power Method is used several times to compute the contribution of only one or two eigenvector/eigenvalue pairs that dominate the solution. The complete set of eigenvalues and eigenvectors is not required. The intent is to provide a novel application of and show the importance of the significance of dominant eigenvector/eigenvalue pairs and to use the Power Method in the analysis and understanding of mechanical engineering systems for both education and practice. Typically, the concepts of both eigenvector expansion and dominant contribution are not included in mechanical engineering education. The scope of application is a broad area of six practical mechanical engineering problems including translational and rotational dynamics, statics of structures, thermal energy balance, fluid continuity, and feedback control. These general mechanical engineering systems naturally contain numerical A and b matrices that fit the scope suitable for providing feasible accuracy. The systems range from fourth order to two hundredth order. Results indicate that accuracy greatly improves as the A matrix contains elements of the same order of numerical magnitude, either naturally from the physics of the problem, or through a transformation to dimensionless parameters. Motivation for using the Power Method for dominant eigenvector/eigenvalue calculation comes from Methods used in web page rank algorithms and a desire to expand mechanical engineering students’ education in understanding the role of the eigenstate expansion Method for the solution of algebraic equations without computing all of its eigenvalues and eigenvectors. An in depth quantitative assessment of the approximate dominant eigenspace Method accuracy is obtained by testing the mechanical engineering examples against a near exact solution via a software solver. This error assessment is based on parameters useful in the design and understanding of mechanical systems. The accuracy achieved is feasible for education and for other uses of the Method by practicing mechanical engineers such as when a simple analytical based approximate model may be more suited for inclusion into larger system based models.
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Application of Power Method and Dominant Eigenvector/Eigenvalue Concept for Approximate Eigenspace Solutions to Mechanical Engineering Algebraic Systems
American journal of mechanical engineering, 2018Co-Authors: Michael J PanzaAbstract:This paper shows how the concept of dominant eigenvector/eigenvalue and the Power Method can be used for understanding the solution to practical problems in a broad class of mechanical engineering systems described by linear algebraic equations. An analytical mathematical procedure is developed to obtain a reasonably accurate approximate eigenspace solution to the system Ax=b by transforming the non-symmetric system matrix into a form where the Power Method is used several times to compute the contribution of only one or two eigenvector/eigenvalue pairs that dominate the solution. The complete set of eigenvalues and eigenvectors is not required. The intent is to provide a novel application of and show the importance of the significance of dominant eigenvector/eigenvalue pairs and to use the Power Method in the analysis and understanding of mechanical engineering systems for both education and practice. Typically, the concepts of both eigenvector expansion and dominant contribution are not included in mechanical engineering education. The scope of application is a broad area of six practical mechanical engineering problems including translational and rotational dynamics, statics of structures, thermal energy balance, fluid continuity, and feedback control. These general mechanical engineering systems naturally contain numerical A and b matrices that fit the scope suitable for providing feasible accuracy. The systems range from fourth order to two hundredth order. Results indicate that accuracy greatly improves as the A matrix contains elements of the same order of numerical magnitude, either naturally from the physics of the problem, or through a transformation to dimensionless parameters. Motivation for using the Power Method for dominant eigenvector/eigenvalue calculation comes from Methods used in web page rank algorithms and a desire to expand mechanical engineering students’ education in understanding the role of the eigenstate expansion Method for the solution of algebraic equations without computing all of its eigenvalues and eigenvectors. An in depth quantitative assessment of the approximate dominant eigenspace Method accuracy is obtained by testing the mechanical engineering examples against a near exact solution via a software solver. This error assessment is based on parameters useful in the design and understanding of mechanical systems. The accuracy achieved is feasible for education and for other uses of the Method by practicing mechanical engineers such as when a simple analytical based approximate model may be more suited for inclusion into larger system based models.