Document Type
Thesis
Date of Award
2023
Degree Name
Master of Science (MS)
Department
Mathematics
First Advisor
Gabriel Picioroaga
Abstract
In this thesis, the main topic is convolution as a mathematical operation and Convolutional Neural Networks (CNN’s). While convolution is classically defined as a function, it can also be defined as an operator from Lp(R) to itself for 1 ≤ p ≤ 2 where Tw(f ) = f ∗ w given some w ∈ L1(R). CNN’s use convolution in its convolutional layers. Defining a neural network to be the composition of layer maps, we find that the neural network is, by necessity, Lipschitz. While CNN’s can be very powerful for image classification, slight changes to an image can completely fool the network. By augmenting our training data with these modifications, the network’s ability to correctly classify images with these modifications significantly increases.
Subject Categories
Mathematics
Keywords
mathematical operation, Convolutional Neural Networks
Number of Pages
53
Publisher
University of South Dakota
Recommended Citation
McCarter, Daniel, "Mathematical Analysis of Convolutional Neural Networks" (2023). Dissertations and Theses. 114.
https://red.library.usd.edu/diss-thesis/114