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

Included in

Mathematics Commons

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