simplicial complexes上的随机信号的平稳性与频谱特性
Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes
AI总结：
本文提出了一种基于simplicial complexes的随机信号平稳性定义及其频谱特性分析方法，通过拓扑滤波器和白噪声构建平稳信号，并通过实验验证其有效性。
摘要：
随着数据复杂结构的日益普遍，需要新的模型和分析工具。图作为一种重要的结构类型，可以编码任意两个实体（节点）之间的关系。然而，图不允许非二元连接，也不允许节点集合之间的关系。因此，我们转向simplicial complexes来连接超过两个节点以及建模简单形之间的关系，如边和三角形。我们的数据则由位于拓扑空间上的信号组成，这些拓扑空间由simplicial complexes表示。最近许多工作探讨了这些拓扑信号，但主要通过确定性方法。我们提出了一种针对simplicial complexes上随机信号的概率框架。具体来说，我们推广了经典的平稳性概念。通过Hodge和Dirac理论的谱对偶性，我们将平稳拓扑信号定义为由拓扑滤波器给白噪声所产生的输出。这种定义自然地扩展了平稳性在时间序列和图信号中所具有的良好性质。关键的是，我们通过清晰的谱特性正确地定义了拓扑功率谱密度（PSD）。然后，我们通过PSD讨论了拓扑平稳性的优势。此外，我们通过多个合成和实际模拟验证了这些优势的实用性。
英文摘要：
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.
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