Direction Set Based Methods For Adaptive Least Squares Problems: Improvements And Innovations
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Date
2008-06-01
Authors
Ahmad, Noor Atinah
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
The main objective of this research is to provide a mathematically tractable solutions to
the adaptive filtering problem by formulating the problem as an adaptive least squares
problem. This approach follows the work of Chen (1998) in his study of direction-set
based CDS) adaptive filtering algorithm. Through the said formulation, we relate the DS
algorithm to a class of projection method. In particular, the simplified version of the
algorithm, which is the Euclidean direction search (EDS) algorithm is shown to be related
to a class of iterative methods called relaxation methods. This findings enable us to
improve the EDS algorithm to the accelerated EDS where an acceleration parameter is
introduced to optimize the step size during each line search.
Our formulation also allows us to consider other types of search method, in
particular the conjugate direction based methods. Global convergence of existing
conjugate gradient (CG) based method are proved by viewing the methods as a form of
conjugate gradient method without line search. Another globally convergent descent
algorithm is developed based on pair-wise conjugation of gradient. In stochastic setting,
this new algorithm proved to be comparable to existing CG based method. Furthermore,
due to the absence of explicit computation of the stochastically recursive conjugate
search directions, the algorithm provides a much lower computational complexity, which
makes it more favorable for high speed computing.
We also extend our formulation to the preconditioned adaptive least squares
problem which is shown to be an appropriate formulation for transform-domain adaptive
filtering. Although analyses of alternative pre conditioners are still at an early stage of research, we are able to provide a new preconditioner for the problem derived directly
from the mathematical formulation of the problem.