Going beyond persistent homology using persistent homology
This work addresses a foundational limitation in graph learning by theoretically characterizing and improving the integration of topological features, though it is incremental as it builds on existing PH methods.
The paper tackles the problem of identifying which attributed graphs can be recognized by persistent homology (PH) in graph neural networks, resolving it by introducing color-separating sets and establishing necessary and sufficient conditions for graph distinction based on connected components persistence. It proposes RePHINE, which combines vertex- and edge-level PH to enhance expressive power, achieving gains over standard PH on graph classification benchmarks.
Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent homology (PH) has gained prominence, but identifying the class of attributed graphs that PH can recognize remains open. We introduce a novel concept of color-separating sets to provide a complete resolution to this important problem. Specifically, we establish the necessary and sufficient conditions for distinguishing graphs based on the persistence of their connected components, obtained from filter functions on vertex and edge colors. Our constructions expose the limits of vertex- and edge-level PH, proving that neither category subsumes the other. Leveraging these theoretical insights, we propose RePHINE for learning topological features on graphs. RePHINE efficiently combines vertex- and edge-level PH, achieving a scheme that is provably more powerful than both. Integrating RePHINE into MP-GNNs boosts their expressive power, resulting in gains over standard PH on several benchmarks for graph classification.