Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes

TL;DR

This paper introduces a probabilistic framework for analyzing random signals on simplicial complexes, extending stationarity concepts from time-series and graph signals to topological data, with spectral characterization and practical demonstrations.

Contribution

It generalizes the classical notion of stationarity to signals on simplicial complexes using spectral dualities, defining a topological power spectral density and demonstrating its advantages.

Findings

Defined topological stationarity via spectral dualities

Established a spectral characterization of topological PSD

Validated the framework through synthetic and real-world data

Abstract

It is increasingly common for data to possess intricate structure, necessitating new models and analytical tools. Graphs, a prominent type of structure, can encode the relationships between any two entities (nodes). However, graphs neither allow connections that are not dyadic nor permit relationships between sets of nodes. We thus turn to simplicial complexes for connecting more than two nodes as well as modeling relationships between simplices, such as edges and triangles. Our data then consist of signals lying on topological spaces, represented by simplicial complexes. Much recent work explores these topological signals, albeit primarily through deterministic formulations. We propose a probabilistic framework for random signals defined on simplicial complexes. Specifically, we generalize the classical notion of stationarity. By spectral dualities of Hodge and Dirac theory, we define stationary topological signals as the outputs of topological filters given white noise. This definition naturally extends desirable properties of stationarity that hold for both time-series and graph signals. Crucially, we properly define topological power spectral density (PSD) through a clear spectral characterization. We then discuss the advantages of topological stationarity due to spectral properties via the PSD. In addition, we empirically demonstrate the practicality of these benefits through multiple synthetic and real-world simulations.