Convolutional Learning on Simplicial Complexes
This work addresses the challenge of incorporating higher-order structures in machine learning for domains like topology and geometry, though it appears incremental as an extension of convolutional methods to simplicial complexes.
The paper tackles the problem of learning data representations on simplicial complexes by proposing a simplicial complex convolutional neural network (SCCNN), which generalizes state-of-the-art methods and shows benefits in tasks like simplex and trajectory prediction.
We propose a simplicial complex convolutional neural network (SCCNN) to learn data representations on simplicial complexes. It performs convolutions based on the multi-hop simplicial adjacencies via common faces and cofaces independently and captures the inter-simplicial couplings, generalizing state-of-the-art. Upon studying symmetries of the simplicial domain and the data space, it is shown to be permutation and orientation equivariant, thus, incorporating such inductive biases. Based on the Hodge theory, we perform a spectral analysis to understand how SCCNNs regulate data in different frequencies, showing that the convolutions via faces and cofaces operate in two orthogonal data spaces. Lastly, we study the stability of SCCNNs to domain deformations and examine the effects of various factors. Empirical results show the benefits of higher-order convolutions and inter-simplicial couplings in simplex prediction and trajectory prediction.