D-GRIL: End-to-End Topological Learning with 2-parameter Persistence

In recent years, persistent homology, one of the flagship concepts of Topological Data Analysis (TDA), has found its use in many fields such as neuroscience, material science, sensor networks, shape recognition, gene expression data analysis and many more Giunti et al. (2022). The performance of machine learning models such as Graph Neural Networks (GNNs) can be enhanced by augmenting topological information captured by persistent homology (Hofer et al., 2017; Dehmamy et al., 2019; Carrière et al., 2020; Horn et al., 2022). Classical persistent homology, also known as 1-parameter persistence, captures the evolution of topological structures in a simplicial complex K following a filter function f: K R. The evolution of topological structures, in this case, can be completely characterized and compactly represented as persistence diagrams or equivalently barcodes Zomorodian & Carlsson (2004); Edelsbrunner et al. (2002). These persistence diagrams or barcodes can be vectorized Bubenik (2015); Reininghaus et al. (2015); Adams et al. (2017); Hofer et al. (2019); Carrière et al. (2020); Kim et al. (2020) and used in machine learning pipelines. In most applications, the simplicial complex K is given and the choice of the filter function f depends on the application. Choosing an appropriate filter function can be challenging. To avoid this, in Hofer et al. (2020), the authors proposed an end-to-end learning framework to learn the filter function rather than relying on making the right choice. They showed that learning the filter function performs better than the standard choices of filter functions on many graph datasets.