Signal Processing On Cell Complexes

The processing of signals supported on non-Euclidean domains has attracted large interest recently. Thus far, such non-Euclidean domains have been abstracted primarily as graphs with signals supported on the nodes, though the processing of signals on more general structures such as simplicial complexes has also been considered. In this paper, we give an introduction to signal processing on (abstract) regular cell complexes, which provide a unifying framework encompassing graphs, simplicial complexes, cubical complexes and various meshes as special cases. We discuss how appropriate Hodge Laplacians for these cell complexes can be derived. These Hodge Laplacians enable the construction of convolutional filters, which can be employed in linear filtering and non-linear filtering via neural networks defined on cell complexes.

1. INTRODUCTION

Graph signal processing (GSP) has become important to study signals defined over non-Euclidean spaces. GSP provides tools for processing signals supported on discrete spaces abstracted as graphs, leveraging ideas from both signal processing and graph theory [1]. Recently, there has been increased interest in extending these ideas to more general topological spaces described, e.g., via simplicial complexes (SCs) [2]–[9]. Underpinning this development is the notion that encoding a richer set of relationships can a) help to capture relevant invariants in data supported on complex spaces more faithfully; b) enable the extraction of a richer set of structural features, ultimately leading to more powerful signal processing methods.