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P.S.Krishnaprasad, University of Maryland. Guest

Posted: Thu Aug 20, 1992 9:02 pm Subject: Papers available




Papers available
Following are abstracts of recent work related to wavelets and frames. Those
interested in obtaining copies of the same may contact the undersigned.

Professor P. S. Krishnaprasad  Tel: (301)4056843
Department of Electrical Engineering &  Fax: (301)4056707
Systems Research Center 
A.V. Williams Building  Rm 2233  Internet:
University of Maryland  krishna@src.umd.edu
College Park, MD 20742.  krishna@poincare.src.umd.edu

SYSTEMS RESEARCH CENTER TECHNICAL REPORT, SRCTR9044r1
University of Maryland, College Park, MD 20742.
(also to appear in IEEE Transactions on Neural Networks)
centerline{f Analysis and Synthesis of Feedforward Neural Networks}
centerline{f Using Discrete Affine Wavelet Transformations}
centerline{by Y.C. Pati and P.S. Krishnaprasad}
In this paper we develop a theoretical description of standard feedfoward
neural networks in terms of discrete affine wavelet transforms. This
description aids in establishing a rigorous understanding of the behavior
of feedforward neural networks based upon the properties of wavelet
transforms. Timefrequency localization properties of wavelet transforms
are shown to be crucial to our formulation. In addition to providing a
solid mathematical foundation for feedforward neural networks, this theory
may prove useful in explaining some of the empirically obtained results in
the field of neural networks. Among the more practical implications of
this work are the following: (1) Simple analysis of training data
provides complete topological definition for a feedforward neural network.
(2) Faster and more efficient learning algorithms are obtained by reducing
the dimension of the parameter space in which interconnection weights are
searched for. This reduction of the weight space is obtained via the same
analysis used to configure the network. Global convergence of the
iterative training procedure discussed here is assured. Moreover, it is
possible to arrive at a noniterative training procedure which involves
solving a system of linear equations. (3) Every feedforward neural
network constructed using our wavelet formulation is equivalent to a
'standard feedforward network.' Hence properties of neural networks, which
have prompted the study of VLSI implementation of such networks are
retained.

SYSTEMS RESEARCH CENTER TECHNICAL REPORT, SRCTR9244,
University of Maryland, College Park, MD 20742.
(shorter version presented at Conference on Information Sciences and Systems,
Princeton, March 1992)
centerline{f Affine Frames of Rational Wavelets in $H2 ( Pi^+)$}
centerline{by Y.C. Pati and P.S. Krishnaprasad}
In this paper we investigate frame decompositions of $H2(Pi^+)$ as a method of
constructing rational approximations to nonrational transfer functions in
$H2(Pi^+ )$. The frames of interest are generated from a single analyzing
wavelet. We consider the case in which the analyzing wavelet is rational
and show that by appropriate grouping of terms in a wavelet expansion,
$H2(Pi^+ )$ can be decomposed as an infinite sum of a rational transfer
functions which are related to one another by dilation and translation.
Criteria for selecting a finite number of terms from such an infinite
expansion are developed using timefrequency localization properties of
wavelets.

SYSTEMS RESEARCH CENTER Ph.D. Dissertation Series, Ph.D. 9213
Title of Dissertation:
Wavelets and TimeFrequency Methods in Linear Systems and Neural Networks
Yagyensh C. Pati, Doctor of Philosophy, 1992
Dissertation directed by:Professor P. S. Krishnaprasad,
Department of Electrical Engineering
In the first part of this dissertation we consider the problem of rational
approximation and identification of stable linear systems.
Affine wavelet decompositions of the Hardy space H$^2(Pi^+)$, are developed as
a means of constructing rational approximations to nonrational transfer functions.
The decompositions considered here are based on frames constructed from
dilations and complex translations of a single rational function.
It is shown that suitable truncations of such decompositions can lead
to low order rational approximants for certain classes of timefrequency
localized systems.
It is also shown that suitably truncated rational wavelet series may be used
as `linearinparameters' black box models for system identification.
In the context of parametric models for system identification, timefrequency
localization afforded by affine wavelets is used to incorporate {em a priori}
knowledge into the formal properties of the model.
Comparisons are made with methods based on the classical Laguerre filters.
The second part of this dissertation is concerned with developing a
theoretical framework for feedforward neural networks which is suitable
for both analysis and synthesis of such networks.
Our approach to this problem is via affine wavelets and the theory of
frames.
Affine frames for L$^2$, are constructed using combinations
of sigmoidal functions and the inherent translations and dilations of
feedforward network architectures.
Timefrequency localization is used in developing methods for the synthesis
of feedforward networks to solve a given problem.
These two seemingly disparate problems both lie within the realm of approximation theory, and our approach to both
is via the theory of frames and affine wavelets. 





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