Deep Homological Learning

Over the past decade, concepts from the field of algebraic topology have evolved into computationally practical methods to analyze data from a topological perspective. This is now more broadly known as Topological data analysis (TDA). Presumably, the most widely used tool from TDA is persistent homology, which offers a concise summary of homological information at different scales, such as the number of connected components, holes or voids. While summaries of this kind can be highly informative and potentially useful for learning purposes, the connection between TDA and machine learning is still in its infancy. The goal of this project, Deep Homological Learning, is to develop novel and theoretically well-founded approaches to bridge the gap between TDA and recent advances in learning with deep neural networks. This not only includes (1) leveraging information from persistent homology as an additional data source for learning, but also (2) to learn filtrations for persistent homology from data and (3) to use concepts from algebraic topology to study neural network architectures, their capacity and learning progress. We will contribute to the theoretical foundation of learning with persistent homology and to a deeper understanding of neural network capacity and neural network learning behavior from a topological perspective. Advances along the lines proposed in this project (1) have great potential to offer better guidelines for neural network design, for example, informed by topological properties of the data, and (2) will eventually lead to practically useful diagnostic tools to analyze learning progress.

The project's overall goal was to establish a solid, theoretically well-founded bridge between machine learning methods (neural networks in particular) and the relatively new subfield of Topological Data Analysis, focusing primarily on persistent homology. Throughout the project duration, we realized several of those bridges; most notably, we (1) introduced novel construction schemes for so-called "barcode vectorizations", i.e., representations of the prevalent summary representation of topological features in data (barcodes), that can readily be used as novel input layers to neural networks, and (2) we successfully demonstrated that one can use persistent homology during end-to-end training of neural networks, e.g., to promote specific topological properties of a network's internal representation of the data. The latter point, in fact, opened up the path to novel regularizers and established a way forward to study generalization in neural networks from a topological perspective in the future.