TREPH: A Plug-In Topological Layer for Graph Neural Networks

Area of Science:

• Computational Topology
• Machine Learning
• Graph Neural Networks

Background:

• Topological Data Analysis (TDA) uses algebraic topology to analyze data shapes.
• Persistent Homology (PH) is a core TDA technique.
• Combining PH with Graph Neural Networks (GNNs) captures graph topological features but faces limitations.

Purpose of the Study:

• To address the limitations of PH in GNNs, such as incomplete information and irregular outputs.
• To propose a novel plug-in topological layer for GNNs called TREPH.
• To leverage Extended Persistent Homology (EPH) for improved topological feature representation.

Main Methods:

• Developed a plug-in topological layer, TREPH, for GNNs.
• Utilized Extended Persistent Homology (EPH) for uniform topological feature extraction.
• Designed a novel aggregation mechanism to collate topological features of different dimensions.
• Ensured the proposed layer is provably differentiable.

Main Results:

• TREPH offers a more expressive topological representation than PH-based methods.
• The layer demonstrates strictly stronger expressive power than standard message-passing GNNs.
• Experiments show TREPH is competitive with state-of-the-art approaches on graph classification tasks.

Conclusions:

• TREPH effectively integrates EPH into GNNs for superior topological feature learning.
• The proposed method overcomes key limitations of traditional PH in graph analysis.
• TREPH represents a significant advancement in applying topological methods to graph machine learning.