Topological Data Analysis for Neural Networks

This book offers a comprehensive presentation of methods from topological data analysis applied to the study of neural network structure and dynamics. Using topology-based tools such as persistent homology and the Mapper algorithm, the authors explore the intricate structures and behaviors of fully connected feedforward and convolutional neural networks. 

The authors discuss various strategies for extracting topological information from data and neural networks, synthesizing insights and results from over 40 research articles, including their own contributions to the study of activations in complete neural network graphs. Furthermore, they examine how this topological information can be leveraged to analyze properties of neural networks such as their generalization capacity or expressivity. Practical implications of the use of topological data analysis in deep learning are also discussed, with a focus on areas including adversarial detection and model selection. The authors conclude with a summary of key insights along with a discussion of current challenges and potential future developments in the field.

This monograph is ideally suited for mathematicians with a background in topology who are interested in the applications of topological data analysis in artificial intelligence, as well as for computer scientists seeking to explore the practical use of topological tools in deep learning.

Rubén Ballester is a PhD student in Topological Machine Learning at the Department of Mathematics and Computer Science of the University of Barcelona (UB). He received his bachelor’s degrees in Mathematics and Computer Science from UB in 2021 and completed the Advanced Mathematics and Mathematical Engineering MSc at Universitat Politècnica de Catalunya (UPC) in 2022, achieving the highest master’s degree GPA recognition. He has published articles on the connection between generalizations of neural networks and persistent homology and on the design of neural networks for topological domains. He won the Topological Deep Learning Challenge in the modality of combinatorial complexes, organized within the 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning at ICML 2023. In addition, he has actively contributed to the TopoX software suite for topological neural networks.

Carles Casacuberta is Full Professor of Geometry and Topology at the University of Barcelona (UB) since 2001. He earned his doctoral degree in 1988, specializing in algebraic topology. He has edited ten books and authored 55 research articles in areas such as homotopy theory, category theory, homological algebra, and more recently, topological data analysis. He serves on the editorial board of the Springer Universitext series and two research journals. Currently, he coordinates the Topological Machine Learning Seminar at UB and participates in several Horizon Europe projects focused on applications of artificial intelligence in biomedicine.

Sergio Escalera is Full Professor at the Department of Mathematics and Computer Science of the University of Barcelona. He is action editor of the Journal of Data-centric Machine Learning Research and IEEE Transactions on Pattern Analysis and Machine Intelligence. He is vice-president of ChaLearn Challenges in Machine Learning, leading ChaLearn Looking at People events. He is co-creator of the Codalab open source platform for challenge organization and co-founder of the NeurIPS competition and Datasets and Benchmarks tracks. He has published more than 400 research papers and participated in the organization of scientific events. His research interests include machine learning fundamentals, and inclusive and transparent analysis of humans from visual and multi-modal data by means of deep learning mechanisms.