Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes

TL;DR

The paper extends classical stationarity to random signals on simplicial complexes by defining stationary topological processes as outputs of topological filters driven by white noise, yielding a PSD that is diagonal in the topological Fourier basis. Using Hodge and Dirac spectral geometry, it derives both nonparametric and parametric covariance estimation methods and generalizes Wiener filtering, reconstruction, and Gaussianity results to the topological domain. Key contributions include MA/AR topological process models, non-polynomial and kernel parameterizations, and extensive numerical validation on synthetic and real-world data demonstrating practical benefits in uncertainty-aware high-order signal processing. This framework enables efficient, spectrally informed analysis of higher-order interactions and supports data-scarce scenarios through low-dimensional parametric models and spectral-domain processing.

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.

Figures (5)

• Figure 1: Examples of simplicial complexes of different orders. (a) A simplicial complex of order $2$ with one triangle $\{1,2,3\}$. The edge $\{1,2\}$ is aligned with $\{1,2,3\}$ but $\{1,3\}$ is anti-aligned. (b) A simplicial complex of order $2$ with one tetrahedron $\{1,2,3,4\}$. The triangle $\{1,2,4\}$ is aligned with $\{1,2,3,4\}$ but $\{1,2,3\}$ is anti-aligned.
• Figure 2: Covariance estimation error as the number of observed signals $M$ increases for (a) MA topological signals and (b) AR topological signals. (c) Runtime comparison in seconds as the number of observed signals $M$ increases.
• Figure 3: Covariance estimation error from noisy or filtered signals as SNR varies for (a) MA topological signals and (b) AR topological signals. (c) Denoising error as the SNR varies for both MA and AR topological signals.
• Figure 4: Reconstruction error as the percentage of observed simplices increases for (a) MA topological signals, (b) AR topological signals, and (c) low-pass topological signals.
• Figure 5: (a) Denoising error as the SNR varies for keyword signals. (b) Reconstruction error as the percentage of observed simplices increases for keyword signals. (c) Reconstruction error as the percentage of observed $1$-simplices increases for keyword signals.

Theorems & Definitions (3)

• Corollary 1
• proof
• Remark 1: Single vs. Multiorder Models