An Algorithm for Least Squares Computation Using the Manifold and Hilbert Space Dialectics
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Date
2012-03-12
Authors
OLALEYE, J.B.
ABIODUN, O.E.
OLUSINA, J.O.
Journal Title
Journal ISSN
Volume Title
Publisher
Theoretical Mathematics & Applications, International Scientific Press
Abstract
The method of least squares is widely used in numerical analysis of data in all
applied quantitative fields. Although there is only one least squares criterion,
several schemes have been used for its implementation. The manifold approach
treats the entire least squares process, including the representation of the variables,
the model formation and the computations, in terms of manifolds. A manifold is a
group of variables or functions taken together and treated as an entity in the
computation process. This paper presents the least squares optimization on the
manifolds and shows that the express formation and solution of the usually
formidable normal equations can be avoided by employing the Hilbert space
axioms and methods in the Euclidean space generated by the axial manifolds. The
sequential and systematic approach of the new scheme, the preservation of the
group structure and the analytical insights it provides for understanding the fundamental geometry of the least squares problem, all of which are demonstrated in the sample applications presented, support the conclusions that the manifold
approach is less daunting, requires less core storage space and facilitate better
understanding of the problem and the solution.
Description
Keywords
Manifolds, Inner Product Space, orthogonality, inner product, Basis Manifolds
Citation
J. B. Olaleye, O. E. Abiodun and J. O. Olusina (2012), An Algorithm for Least Squares Computation Using the Manifold and Hilbert Space Dialectics, Journal of Theoretical Mathematics and Applications, Vol. 2. No. 1, pp 143-160.