Signal processing on simplicial complexes
This work addresses the need for signal processing methods in complex systems with higher-order dependencies, but it is incremental as it surveys and adapts existing ideas to a new domain.
The paper tackles the problem of processing signals on higher-order network structures by extending signal processing techniques from regular domains to simplicial complexes, using the Hodge Laplacian matrix as a key operator for tasks like Fourier analysis and denoising.
Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the context of epidemic spreading processes or opinion formation. In this chapter, we focus on a closely related, but distinct third perspective: how can we use higher-order relationships to process signals and data supported on higher-order network structures. In particular, we survey how ideas from signal processing of data supported on regular domains, such as time series or images, can be extended to graphs and simplicial complexes. We discuss Fourier analysis, signal denoising, signal interpolation, and nonlinear processing through neural networks based on simplicial complexes. Key to our developments is the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.