Garcia, Jason S., authorKrishnaswamy, Nikhil, advisorAdams, Henry, committee memberBeveridge, Ross, committee member2022-05-302022-05-302022https://hdl.handle.net/10217/235253Topological Data Analysis (TDA) uses ideas from topology to study the "shape" of data. It provides a set of tools to extract features, such as holes, voids, and connected components, from complex high-dimensional data. This thesis presents an introductory exposition of the mathematics underlying the two main tools of TDA: Persistent Homology and the MAPPER algorithm. Persistent Homology detects topological features that persist over a range of resolutions, capturing both local and global geometric information. The MAPPER algorithm is a visualization tool that provides a type of dimensional reduction that preserves topological properties of the data by projecting them onto lower dimensional simplicial complexes. Furthermore, this thesis explores recent applications of these tools to natural language processing and computer vision. These applications are divided into two main approaches: In the first approach, TDA is used to extract features from data that is then used as input for a variety of machine learning tasks, like image classification or visualizing the semantic structure of text documents. The second approach, applies the tools of TDA to the machine learning algorithms themselves. For example, using MAPPER to study how structure emerges in the weights of a trained neural network. Finally, the results of several experiments are presented. These include using Persistent Homology for image classification, and using MAPPER to visual the global structure of these data sets. Most notably, the MAPPER algorithm is used to visualize vector representations of contextualized word embeddings as they move through the encoding layers of the BERT-base transformer model.born digitalmasters thesesengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.Text