Some Nonlinear Networks Suggested by Learning Theory

<p>We introduce several systems of nonlinear difference-differential equations and prove oscillation and global ratio limit theorems for some of them. These systems can be interpreted as a learning theory, or alternatively as a nonstationary prediction theory whose goal is to discuss the prediction of individual events, in a fixed order, and at prescribed times. They can also be interpreted as cross-correlated flows on networks, or as deformations of a probabilistic graph. Each system possesses an underlying geometry characterized by a semistochastic matrix, and we study the effect of this geometry on the system's limiting behavior as t→∞. We also investigate the effects which the ratios of solutions of our systems have on the outputs of each system. We show that the average output of each system is not a good index of the mechanism which characterizes its interactions, especially when this average is computed over long time intervals. In particular, the average output is linear whereas the interactions are nonlinear. A system is discussed whose interactions are always locally reversible but whose global interactions are irreversible or not depending on the inputs received by the system. We also find systems whose entropy decreases monotonically in time and connect this phenomenon with the process of learning in these systems.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Thesis

Year

1967

Author dc:creator

• Grossberg, Stephen

Contributors dc:contributor

• Mark Kac

• nonlinear systems
• difference-differential equations
• prediction theory
• semistochastic matrices
• learning dynamics
• entropy
• Physical Sciences and Mathematics

Repository record dc:identifier

https://digitalcommons.rockefeller.edu/student_theses_and_dissertations/561

OAI identifier oai:identifier

oai:digitalcommons.rockefeller.edu:student_theses_and_dissertations-1571