Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity

Understand

Topological deep learning (TDL) is a rapidly growing field that seeks to leverage topological structure in data and facilitate learning from data supported on topological objects, ranging from molecules to 3D shapes.

• Most TDL architectures can be unified under the framework of higher-order message-passing (HOMP), which generalizes graph message-passing to higher-order domains.
• In the first part of the paper, we explore HOMP's expressive power from a topological perspective, demonstrating the framework's inability to capture fundamental topological and metric invariants such as diameter, orientability, planarity, and homology.
• In addition, we demonstrate HOMP's limitations in fully leveraging lifting and pooling methods on graphs.