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.

翻译：数据日益呈现出复杂的结构，这要求我们发展新的模型与分析工具。图作为一种重要的结构类型，能够编码任意两个实体（节点）之间的关系。然而，图既不允许非二元连接，也不支持节点集合之间的关系。因此，我们转向单纯复形，它能够连接两个以上的节点，并建模单纯形（如边和三角形）之间的关系。于是，我们的数据由位于拓扑空间（以单纯复形表示）上的信号构成。尽管近期许多工作通过确定性框架探索这些拓扑信号，我们提出了一个定义在单纯复形上的随机信号的概率框架。具体而言，我们推广了经典的平稳性概念。借助Hodge理论和Dirac理论的谱对偶性，我们将平稳拓扑信号定义为给定白噪声通过拓扑滤波器的输出。这一定义自然地推广了在时间序列和图信号中均成立的平稳性理想性质。至关重要的是，我们通过清晰的谱表征正确定义了拓扑功率谱密度（PSD）。随后，我们通过PSD的谱特性讨论了拓扑平稳性的优势。此外，我们通过多个合成与真实世界仿真，实证展示了这些优势的实用性。