First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition

2008 ◽  
Vol 56 (7) ◽  
pp. 3044-3049 ◽  
Author(s):  
Jun Liu ◽  
Xiangqian Liu ◽  
Xiaoli Ma
Keyword(s):  
Singular Value Decomposition ◽  
Perturbation Analysis ◽  
Singular Value ◽  
Order Perturbation ◽  
Singular Vectors ◽  
First Order ◽  
Value Decomposition ◽  
First Order Perturbation

COUPLED SINGULAR VALUE DECOMPOSITION OF A CROSS-COVARIANCE MATRIX

2010 ◽  
Vol 20 (04) ◽  
pp. 293-318 ◽  
Author(s):  
ALEXANDER KAISER ◽  
WOLFRAM SCHENCK ◽  
RALF MÖLLER
Keyword(s):  
Singular Value Decomposition ◽  
Covariance Matrix ◽  
Order Approximation ◽  
Singular Value ◽  
Singular Vectors ◽  
First Order ◽  
Learning Rules ◽  
Cross Covariance ◽  
First Order Approximation ◽  
Value Decomposition

We derive coupled on-line learning rules for the singular value decomposition (SVD) of a cross-covariance matrix. In coupled SVD rules, the singular value is estimated alongside the singular vectors, and the effective learning rates for the singular vector rules are influenced by the singular value estimates. In addition, we use a first-order approximation of Gram-Schmidt orthonormalization as decorrelation method for the estimation of multiple singular vectors and singular values. Experiments on synthetic data show that coupled learning rules converge faster than Hebbian learning rules and that the first-order approximation of Gram-Schmidt orthonormalization produces more precise estimates and better orthonormality than the standard deflation method.

scholarly journals An Elementary Proof of Bevan's Theorem on the Growth of Grid Classes of Permutations

2019 ◽  
Vol 62 (4) ◽  
pp. 975-984
Author(s):  
Michael Albert ◽  
Vincent Vatter
Keyword(s):  
Growth Rate ◽  
Singular Value Decomposition ◽  
Hadamard Product ◽  
Elementary Proof ◽  
Singular Value ◽  
Stirling’S Formula ◽  
Real Numbers ◽  
Singular Vectors ◽  
Value Decomposition ◽  
Method Of Lagrange Multipliers

AbstractBevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's formula, the method of Lagrange multipliers, and the singular value decomposition of matrices. Our proof relies on showing that the maximum over the space of n × n matrices with non-negative entries summing to one of a certain function of those entries, parametrized by the entries of another matrix Γ of non-negative real numbers, is equal to the square of the largest singular value of Γ and that the maximizing point can be expressed as a Hadamard product of Γ with the tensor product of singular vectors for its greatest singular value.

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