Topics in Geometric Machine Learning

Abstract

Recent advances and the widespread adoption of neural networks have revolutionized machine learning and artificial intelligence. These developments demand learning paradigms capable of processing data from diverse applications and sources. In structured domains such as molecules, graphs, sets, and 3D objects, as well as fields such as drug discovery, materials science, and astronomy, models must account for data structures. The emerging field of geometric machine learning has gained attention for enabling neural networks to handle geometric structures, unlocking novel solutions across scientific disciplines. Despite recent advances, theoretical gaps remain. This thesis aims to address these gaps by studying the benefits and limitations of leveraging geometric structures and symmetries in data. We explore sample complexity, generalization bounds, hypothesis testing for the presence of symmetries in data, time complexity of learning under symmetries, and regularization and optimization in symmetric settings. The goal is to build a robust theoretical framework that validates recent successes and sheds light on unexplored aspects, fostering future progress in geometric machine learning.