Document Type

Dissertation

Degree

Doctor of Philosophy

Major

Mathematics

Date of Defense

4-14-2023

Graduate Advisor

Adrian Clingher, Ph.D.

Committee

Haiyan Cai, Ph.D.

David Covert, Ph.D.

Prabhakar Rao, Ph.D.

Abstract

Convolutional Neural Networks (CNNs) have become one of the most commonly used tools for performing image classification. Unfortunately, as with most machine learning algorithms, CNNs suffer from a lack of interpretability. CNNs are trained by using a training data set and a loss function to tune a set of parameters known as the layer weights. This tuning process is based on the classical method of gradient descent, but it relies on a strong stochastic component, which makes the weight behavior during training difficult to understand. However, since CNNs are governed largely by the weights that make up each of the layers, if one can gain an understanding of the space in which these weights lie, then much can be learned about the structure of the CNN and how it calculates its output. Topological Data Analysis (TDA) is a recent addition to the field of data science, which uses ideas from geometry and algebraic topology to create a novel methodology for analyzing high-dimensional datasets. Specifically, TDA offers a mathematically rigorous method for studying the structure of CNN weight spaces. In this thesis, we use TDA to study the weights of a binary classification CNN model trained on a large dataset known as Dogs vs Cats. Our analysis reveals that, during training, the 3x3 convolutional filter weights of the CNN model in question exhibit non-trivial homological properties. Namely, persistent 1-cycles occur within the first homology groups. This structure is similar to the structure that is found in 3x3 high-variance image patches of natural images, demonstrating that a CNN built on this data set learns features of the ambient structure of the image data. This demonstrates the validity of the CNN and, along with work done by Carlsson and Gabrielsson, furthers the hypothesis that convolutional layer weights arising from training a CNN on natural image data lie on a space with non-trivial geometry, in particular a non-empty first homology group.

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