Vectorizing Distributed Homology with Deep Set of Set Networks

Abstract

Distributed homology, a topological invariant, holds potential as an instrument for uncov- ering insights into the structural characteristics of complex data. By considering both the density and connectivity of topological spaces, it offers the potential for a more detailed and stable understanding of the underlying structure of data sets. This is particularly beneficial when confronting noisy, real-world data. Despite its potential, the complexity and unstructured nature of distributed homology pose hurdles for practical use. This thesis tackles these issues by proposing a novel pipeline that fuses distributed homology and supervised learning techniques. The goal is to facilitate the effective incorporation of distributed homology into a wide array of supervised learning tasks. Our approach is anchored on the DeepSet network, an architecture adept at managing set inputs. Using this, we devise a comprehensive framework specifically designed to handle inputs composed of a set of sets. Furthermore, we present a dedicated architecture for distributed homology, designed to boost robustness to noise and overall performance. This approach shows marked improvements over full persistent homology methods for both synthetic and real data. While our results may not yet rival state-of-the-art performance on real data, they demonstrate the potential for distributed invariants to enhance the efficiency of topolog- ical approaches. This indicates a promising avenue for future research and development, contributing to the refinement of topological data analysis.