## Date of Award

6-1995

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Department of Mathematics and Statistics

## First Advisor

Mark E. Oxley, PhD

## Abstract

Researchers rely on the mathematics of Vapnik and Chervonenkis to capture quantitatively the capabilities of specific artificial neural network (ANN) architectures. The quantifier is known as the V-C dimension, and is defined on functions or sets. Its value is the largest cardinality 1 of a set of vectors in Rd such that there is at least one set of vectors of cardinality 1 such that all dichotomies of that set into two sets can be implemented by the function or set. Stated another way, the V-C dimension of a set of functions is the largest cardinality of a set, such that there exists one set of that cardinality which can be shattered by the set of functions. A set of functions is said to shatter a set if each dichotomy of that set can be implemented by a function in the set. There is an abundance of research on determining the value of V-C dimensions of ANNs. In this document, research on V-C dimension is refined and extended yielding formulas for evaluating V-C dimension for the set of functions representable by a feed-forward, single hidden-layer perceptron artificial neural network. The fundamental thesis of this research is that the V-C dimension is not an appropriate quantifier of ANN capabilities.

## AFIT Designator

AFIT-DS-ENC-95J-01

## DTIC Accession Number

ADA297408

## Recommended Citation

Carter, Martha A., "The Mathematics of Measuring Capabilities of Artificial Neural Networks" (1995). *Theses and Dissertations*. 6307.

https://scholar.afit.edu/etd/6307