Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes

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.

💡 Research Summary

The paper addresses the problem of modeling and analyzing random signals that reside on simplicial complexes—mathematical structures that capture higher‑order relationships among data points (e.g., nodes, edges, triangles, and higher‑dimensional simplices). While deterministic tools for such “topological signals” have been developed (Hodge decomposition, Dirac‑based filters, etc.), there is a lack of a systematic probabilistic framework that respects the intrinsic geometry of simplicial complexes.

Key Contributions

1. Definition of Topological Weak Stationarity – Extending the classic notion of weak stationarity (time‑invariant correlation) to simplicial complexes, the authors define a random topological signal (s) as weakly stationary with respect to a symmetric shift operator (T) (either a Hodge Laplacian (L_k) for a single order or the Dirac operator (D) for multi‑order signals) if it can be expressed as (s = H w), where (w) is a zero‑mean white simplicial signal ( (\mathbb{E}