CW-CNN & CW-AN: Convolutional Networks and Attention Networks for CW-Complexes
Rahul Khorana
TL;DR
This work introduces CW-CNN and CW-AT, the first neural architectures capable of processing CW-complex inputs by combining a Hodge-Laplacian informed convolution with multi-head attention on CW-structures. By defining convolution via the Hodge Laplacian and boundary/coboundary operators, the CW-CNN propagates information along cells while preserving topological structure; the CW-AT extends this with a CW-specific attention mechanism based on incidence relations. The authors validate the approach on a synthetic CW-complex dataset, achieving ultra-low RMSE with a compact CW-CNN and demonstrating feasibility and efficiency for the attention-based variant. The results suggest these networks can capture polyadic relations and higher-order topology, with potential applications in cheminformatics, molecular design, and natural language processing where CW-complex representations are advantageous.
Abstract
We present a novel framework for learning on CW-complex structured data points. Recent advances have discussed CW-complexes as ideal learning representations for problems in cheminformatics. However, there is a lack of available machine learning methods suitable for learning on CW-complexes. In this paper we develop notions of convolution and attention that are well defined for CW-complexes. These notions enable us to create the first Hodge informed neural network that can receive a CW-complex as input. We illustrate and interpret this framework in the context of supervised prediction.