The Singular-Value Decomposition of the First-Order Difference Matrix

1989 ◽  
Vol 5 (1) ◽  
pp. 174-174
Author(s):  
R. W. Farebrother
Keyword(s):  
Singular Value Decomposition ◽  
Singular Value ◽  
Difference Matrix ◽  
Order Difference ◽  
First Order ◽  
Value Decomposition

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 First Order and Second Order Learning Algorithms on the Special Orthogonal Group to Compute the SVD of Data Matrices

Electronics ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 334 ◽  
Author(s):  
Simone Fiori ◽  
Lorenzo Del Rossi ◽  
Michele Gigli ◽  
Alessio Saccuti
Keyword(s):  
Singular Value Decomposition ◽  
Orthogonal Group ◽  
Learning Rule ◽  
Singular Value ◽  
Second Order ◽  
Numerical Schemes ◽  
First Order ◽  
Special Orthogonal Group ◽  
Value Decomposition ◽  
Data Matrices

The present paper deals with neural algorithms to learn the singular value decomposition (SVD) of data matrices. The neural algorithms utilized in the present research endeavor were developed by Helmke and Moore (HM) and appear under the form of two continuous-time differential equations over the special orthogonal group of matrices. The purpose of the present paper is to develop and compare different numerical schemes, under the form of two alternating learning rules, to learn the singular value decomposition of large matrices on the basis of the HM learning paradigm. The numerical schemes developed here are both first-order (Euler-like) and second-order (Runge-like). Moreover, a reduced Euler scheme is presented that consists of a single learning rule for one of the factors involved in the SVD. Numerical experiments performed to estimate the optical-flow (which is a component of modern IoT technologies) in real-world video sequences illustrate the features of the novel learning schemes.

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