The Experts below are selected from a list of 309 Experts worldwide ranked by ideXlab platform
Uwe Prells - One of the best experts on this subject based on the ideXlab platform.
-
Simultaneous tridiagonalization of two symmetric matrices
International Journal for Numerical Methods in Engineering, 2003Co-Authors: Seamus D. Garvey, Françoise Tisseur, Michael I. Friswell, John E. T. Penny, Uwe PrellsAbstract:We show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non-singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric Matrix via Householder reflectors. Two algorithms are proposed, one using non-orthogonal rank-one modifications of the Identity Matrix and the other, more costly but more stable, using a combination of Householder reflectors and non-orthogonal rank-one modifications of the Identity Matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n 3 ) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments.
Ji-hwei Horng - One of the best experts on this subject based on the ideXlab platform.
-
A moment-based approach for deskewing rotationally symmetric shapes
IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 1999Co-Authors: Soo-chang Pei, Ji-hwei HorngAbstract:The covariance Matrix of a pattern is composed by its second order central moments. For a rotationally symmetric shape, its covariance Matrix is a scalar Identity Matrix. In this work, we apply this property to restore the skewed shape of rotational symmetry. The relations between the skew transformation Matrix and the covariance matrices of original and skewed shapes are derived. By computing the covariance Matrix of the skewed shape and letting the covariance Matrix of the original shape be a scalar Identity Matrix, the skew transformation Matrix can be solved. Then, the rotationally symmetric shape can be recovered by multiplying the inverse transformation Matrix with the skewed shape. The method does not rely on continuous contours and is robust to noise, because only the second-order moments of the input shape are required. Experimental results are also presented.
-
ICPR - A moment-based approach for deskewing rotationally symmetric shapes
Proceedings of 13th International Conference on Pattern Recognition, 1996Co-Authors: Soo-chang Pei, Ji-hwei HorngAbstract:The covariance Matrix of a rotationally symmetric shape is a scalar Identity Matrix. We apply this property of covariance Matrix to deskew the skewed shape of rotational symmetry. The parameters of the deskew transformation Matrix are solved by letting the covariance Matrix of the transformed shape be equal to a scalar Identity Matrix. Then, the rotationally symmetric shape can be recovered by the deskew transformation Matrix. The method does not rely on continuous contours, since only second-order moments of the input shape are required to be computed. Experimental results are also presented.
Seamus D. Garvey - One of the best experts on this subject based on the ideXlab platform.
-
Simultaneous tridiagonalization of two symmetric matrices
International Journal for Numerical Methods in Engineering, 2003Co-Authors: Seamus D. Garvey, Françoise Tisseur, Michael I. Friswell, John E. T. Penny, Uwe PrellsAbstract:We show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non-singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric Matrix via Householder reflectors. Two algorithms are proposed, one using non-orthogonal rank-one modifications of the Identity Matrix and the other, more costly but more stable, using a combination of Householder reflectors and non-orthogonal rank-one modifications of the Identity Matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n 3 ) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments.
Wah June Leong - One of the best experts on this subject based on the ideXlab platform.
-
A class of diagonal preconditioners for limited memory BFGS method
Optimization Methods and Software, 2013Co-Authors: Wah June Leong, Chuei Yee ChenAbstract:A major weakness of the limited memory BFGS LBFGS method is that it may converge very slowly on ill-conditioned problems when the Identity Matrix is used for initialization. Very often, the LBFGS method can adopt a preconditioner on the Identity Matrix to speed up the convergence. For this purpose, we propose a class of diagonal preconditioners to boost the performance of the LBFGS method. In this context, we find that it is appropriate to use a diagonal preconditioner, in the form of a diagonal Matrix plus a positive multiple of the Identity Matrix, so as to fit information of local Hessian as well as to induce positive definiteness for the diagonal preconditioner at a whole. The property of hereditary positive definiteness is maintained by a careful choice of the positive scalar on the scaled Identity Matrix while the local curvature information is carried implicitly on the other diagonal Matrix through the variational techniques, commonly employed in the derivation of quasi-Newton updates. Several preconditioning formulae are then derived and tested on a large set of standard test problems to access the impact of different choices of such preconditioners on the minimization performance.
-
A restarting approach for the symmetric rank one update for unconstrained optimization
Computational Optimization and Applications, 2007Co-Authors: Wah June Leong, Malik Abu HassanAbstract:Two basic disadvantages of the symmetric rank one (SR1) update are that the SR1 update may not preserve positive definiteness when starting with a positive definite approximation and the SR1 update can be undefined. A simple remedy to these problems is to restart the update with the initial approximation, mostly the Identity Matrix, whenever these difficulties arise. However, numerical experience shows that restart with the Identity Matrix is not a good choice. Instead of using the Identity Matrix we used a positive multiple of the Identity Matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem--maximize the determinant of the update subject to a bound of one on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of a positive multiple of the Identity Matrix is provided for the SR1 update when it is not positive definite or undefined. Our experiments indicate that with such simple initial scaling the possibility of an undefined update or the loss of positive definiteness for the SR1 method is avoided on all iterations.
-
Scaling symmetric rank one update for unconstrained optimization
2002Co-Authors: Malik Abu Hassan, Mansor Monsi, Wah June LeongAbstract:A basic disadvantage to the symmetric rank one (SR1) update is that the SR1 update may not preserve positive definiteness when starting with a positive definite approximation. A simple remedy to this problem is to restart the update with the initial approximation mostly the Identity Matrix whenever this difficulty arises. However, numerical experience shows that restart with the Identity Matrix is not a good choice. Instead of using the Identity Matrix we used a positive multiple of the Identity Matrix. They Used positive scaling factor is the optimal solution of the measure defined by the problem - maximize the determinant subject to a bound of 1 on the largest eigenn value. This measure is motivated by considering the volume of the symmetric diference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of positive multiple of Identity Matrix is provided for the SRI when it is not positive definite. Our experiments indicate that with such simple scale, the efectiveness of the SRI method is increased dramatically.
Soo-chang Pei - One of the best experts on this subject based on the ideXlab platform.
-
A moment-based approach for deskewing rotationally symmetric shapes
IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 1999Co-Authors: Soo-chang Pei, Ji-hwei HorngAbstract:The covariance Matrix of a pattern is composed by its second order central moments. For a rotationally symmetric shape, its covariance Matrix is a scalar Identity Matrix. In this work, we apply this property to restore the skewed shape of rotational symmetry. The relations between the skew transformation Matrix and the covariance matrices of original and skewed shapes are derived. By computing the covariance Matrix of the skewed shape and letting the covariance Matrix of the original shape be a scalar Identity Matrix, the skew transformation Matrix can be solved. Then, the rotationally symmetric shape can be recovered by multiplying the inverse transformation Matrix with the skewed shape. The method does not rely on continuous contours and is robust to noise, because only the second-order moments of the input shape are required. Experimental results are also presented.
-
ICPR - A moment-based approach for deskewing rotationally symmetric shapes
Proceedings of 13th International Conference on Pattern Recognition, 1996Co-Authors: Soo-chang Pei, Ji-hwei HorngAbstract:The covariance Matrix of a rotationally symmetric shape is a scalar Identity Matrix. We apply this property of covariance Matrix to deskew the skewed shape of rotational symmetry. The parameters of the deskew transformation Matrix are solved by letting the covariance Matrix of the transformed shape be equal to a scalar Identity Matrix. Then, the rotationally symmetric shape can be recovered by the deskew transformation Matrix. The method does not rely on continuous contours, since only second-order moments of the input shape are required to be computed. Experimental results are also presented.
