Helmhotlz-Hodge Laplacians: edge flows and simplicial learning

The Laplacian operator is a ubiquitous tool in signal processing. Generalizations of this operator to manifolds (i.e., the Laplace-Beltrami operator) and graphs (i.e., the graph Laplacian) have proved to be very useful for learning on non-euclidean domains. Helmholtz-Hodge theory can further generalize these operators to higher-order differential forms.  For example, the continuous 1-Laplacian can be used to analyze vector-fields on manifolds [1] and edge flows on directed or undirected graphs can be analyzed via wavelet-like bases generated by the Hodge-Laplacian [3]. Similarly, the dth-Laplacian can be used to define convolutional operations on d-degree complexes [2]. This project will use these operators to develop methods for solving signal processing problems such as denoising and in-painting on simplicial domains focussing on edge-based and face-based  signals. We do so using both traditional variational methods and neural network-based approaches.

References

[1] Y.-C. Chen, M. Meila and I. G. Kevrekidis, Helmholtzian eigenmap: Topological feature discovery and edge flow learning from point cloud data, arXiv preprint arXiv:2103.07626, (2021).

[2] S. Ebli, M. Defferrard, and G. Spreemann, Simplicial neural networks, arXiv preprint arXiv:2010.03633, (2020).

[3] T. M. Roddenberry, F. Frantzen, M. T. Schaub, and S. Segarra, Hodgelets: Localized spectral representations of flows on simplicial complexes, arXiv preprint arXiv:2109.08728, (2021).