scholarly journals A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm

2019 ◽  
Vol 124 ◽  
Author(s):  
Jim Lawrence ◽  
Javier Bernal ◽  
Christoph Witzgall
Keyword(s):  
Euclidean Space ◽  
Least Squares ◽  
Rigid Motion ◽  
Singular Value ◽  
Dimensional Euclidean Space ◽  
Least Squares Problem ◽  
Orthogonal Procrustes Problem ◽  
Motion Problem ◽  
Procrustes Problem ◽  
Value Decomposition

The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two matrices of the same order. Over past decades, the algorithm of choice for solving this problem has been the Kabsch-Umeyama algorithm which is essentially no more than the computation of the singular value decomposition of a particular matrix. Its justification as presented separately by Kabsch and Umeyama is not totally algebraic as it is based on solving the minimization problem via Lagrange multipliers. In order to provide a more transparent alternative, it is the main purpose of this paper to present a purely algebraic justification of the algorithm through the exclusive use of simple concepts from linear algebra. For the sake of completeness, a proof is also included of the well-known and widely-used fact that the orientation-preserving rigid motion problem, i.e., the least-squares problem that calls for an orientation-preserving rigid motion that optimally aligns two corresponding sets of points in d-dimensional Euclidean space, reduces to the constrained orthogonal Procrustes problem.

scholarly journals Methods for Updating the Singular Value Decomposition

1974 ◽  
Vol 3 (26) ◽  
Author(s):  
Linda Kaufman
Keyword(s):  
Singular Value Decomposition ◽  
Least Squares ◽  
Singular Values ◽  
Singular Value ◽  
Linear Least Squares ◽  
Least Squares Problem ◽  
Linear Least Squares Problem ◽  
Value Decomposition ◽  
Made In

The linear least squares problem of minimizing ||Ax~ - b~||_(2) where A is an m X n matrix, m >= n, may be solved using the singular value decomposition in approximately 2mn^(3) + 4n^(3) multiplications. In this paper the problem of solving ||A'x~ - b~||_(2) is considered where A' results from deleting or adding a column to A. This might occur when a change is made in the model of a process. Instead of computing the singular value decomposition of A' from scratch, the singular value decomposition of A is updated. Since the updating require about 6n^(3) multiplications the algorithms are useful when m >> n. The problem of recalculation some or all of the singular values of a matrix A', which is obtained by deleting or adding a row or a column from a matrix A, whose singular value decomposition is known, is also studied.

scholarly journals Identification Method for Voltage Sags Based on K-means-Singular Value Decomposition and Least Squares Support Vector Machine

Energies ◽  
2019 ◽  
Vol 12 (6) ◽  
pp. 1137 ◽  
Author(s):  
Haoyuan Sha ◽  
Fei Mei ◽  
Chenyu Zhang ◽  
Yi Pan ◽  
Jianyong Zheng
Keyword(s):  
Support Vector Machine ◽  
Singular Value Decomposition ◽  
Least Squares ◽  
Short Circuit ◽  
Singular Value ◽  
Support Vector ◽  
Voltage Sag ◽  
Svm Classifier ◽  
Identification Algorithm ◽  
Value Decomposition

Voltage sag is one of the most serious problems in power quality. The occurrence of voltage sag will lead to a huge loss in the social economy and have a serious effect on people’s daily life. The identification of sag types is the basis for solving the problem and ensuring the safe grid operation. Therefore, with the measured data uploaded by the sag monitoring system, this paper proposes a sag type identification algorithm based on K-means-Singular Value Decomposition (K-SVD) and Least Squares Support Vector Machine (LS-SVM). Firstly; each phase of the sag sample RMS data is sparsely coded by the K-SVD algorithm and the sparse coding information of each phase data is used as the feature matrix of the sag sample. Then the LS-SVM classifier is used to identify the sag type. This method not only works without any dependence on the sag data feature extraction by artificial ways, but can also judge the short-circuit fault phase, providing more effective information for the repair of grid faults. Finally, based on a comparison with existing methods, the accuracy advantages of the proposed algorithm with be presented.

Gravity interpretation with the aid of quadratic programming

Geophysics ◽  
1980 ◽  
Vol 45 (3) ◽  
pp. 403-419 ◽  
Author(s):  
N. J. Fisher ◽  
L. E. Howard
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Theoretical Data ◽  
Linear Constraints ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Least Squares Problem ◽  
The Matrix ◽  
Gravity Interpretation ◽  
Value Decomposition

The inverse gravity problem is posed as a linear least‐squares problem with the variables being densities of two‐dimensional prisms. Upper and lower bounds on the densities are prescribed so that the problem becomes a linearly constrained least‐squares problem, which is solved using a quadratic programming algorithm designed for upper and lower bound‐type constraints. The solution to any problem is smoothed by damping, using the singular value decomposition of the matrix of gravitational attractions. If the solution is required to be monotonically increasing with depth, then this feature can be incorporated. The method is applied to both field and theoretical data. The results are plotted for (1) undamped, nonmonotonic, (2) damped, nonmonotonic, and (3) damped, monotonic solutions; these conditions illustrate the composite approach of interpretation where both damping techniques and linear constraints are used in refining a solution which at first is unacceptable on geologic grounds while fitting the observed data well.

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