Hodgelets: Localized Spectral Representations of Flows On Simplicial Complexes

We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes. Then, we refine this idea to construct wavelets that respect the Hodge-Helmholtz decomposition. For these Hodgelets, familiar notions of curl-free and divergence-free flows from vector calculus are preserved. We characterize the representational quality of our Hodgelets for edge flows in terms of frame bounds and demonstrate the use of these spectral wavelets for sparse representation of edge flows on real and synthetic data.

1. INTRODUCTION

There has been substantial interest in graph-based techniques to understand data with a complex relational structure [1]–[3], with applications ranging from biology [4] to system robustness [5]. In this context, graph signal processing (GSP) has proven to be a useful way to understand the processing of signals defined on graphs, leveraging ideas from both signal processing and graph theory [6]. The primary focus of GSP has been on signals supported on the nodes of a graph. For such signals, the graph Laplacian and adjacency matrix are natural shift operators, from which we can define notions of filtering and Fourier transformations [6].