Electronic Theses and Dissertations

Identifier

252

Date

2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematical Sciences

Concentration

Mathematics

Committee Chair

Maria Fernanda Botelho

Committee Member

James E. Jamison

Committee Member

John R. Haddock

Committee Member

Robert Kozma

Abstract

Learning is a feature of living organisms which is crucial in the process of adaptation. It is understood that a stimulus triggers a chain of neurophysiological reactions in an organism, and as a consequence we say that the organism is learning from that initial exposure. Several researchers have dealt with understanding and modeling learning through mathematical systems. Due to the complexity of the brain, scholars reduced the problem to a simpler mechanism consisting of neurons, which are processing additive units, interconnected with pathways, called synapses. A primary goal was to derive a system of equations that captures the changes synaptic parameters undergo in a learning process, and identify stimuli that generates a flow of changes that will converge over time. This would represent a stable reflection of the learning process. My dissertation explores generalizations of models for unsupervised learning proposed in literature. The first model is due to Oja and Karhunen and reflects the changes of a network connecting weights or synaptic parameters following a Hebbian principle and incorporating a forgetting term to allow convergence. The second model, due to Cox and Adams, generalizes the Oja-Karhunen model by introducing errors in the learning process. These paradigms are presented as systems of differential equations explored in three settings: 1. The finite dimensional Euclidean space over the reals; 2. Then infinite dimensional Hilbert space of square summable sequences equipped with the standard inner product; and 3. The infinite dimensional Banach space of bounded operators on a separable complex Hilbert space. In each setting the existence and uniqueness of local or global solutions is well established, a form for solutions is derived, and the asymptotic behavior is determined. In the third setting we use the polar factorization of operators to decompose the system into two components where an explicit form for solutions is given. In the Cox-Adams model we also explore the impact of the error factor in the long-term behavior of the solutions.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.

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