Graph and Simplicial Complex Prediction Gaussian Process via Hodgelet Representations

Predicting labels for graph-structured data is crucial in many scientific applications. Recently, Gaussian processes (GPs) with graph-level inputs have been proposed as flexible, non-parametric models for classification tasks. In this work, we extend this framework to regression tasks and simplicial complexes (SCs), enabling edge-level attributes and attributes supported on higher-order simplices. Drawing on the rich literature on Hodge theory in machine learning, we enhance the resulting SC representations via the Hodge decomposition, capturing homological information such as holes. We introduce Hodgelet representations as rich, learnable topological descriptors of simplicial complexes and show that our framework improves predictions across various applications. This paves the way for broader use of GPs in graph-level and SC-level prediction tasks.