相关网络的邻接光谱嵌入
Adjacency Spectral Embeddings of Correlation Networks
AI总结：
本文提出通过傅里叶基系数学习方法，对具有依赖边噪声的相关网络进行光谱嵌入，从而恢复顶点级别的潜在表示。
摘要：
在许多应用中，基于时间序列数据构建加权网络：每个时间序列与一个顶点相关联，边权重由成对相关性给出。结果是一个边依赖结构违反大多数常见网络模型假设的网络。然而，人们通常使用源自边独立网络模型的嵌入方法来分析这些“相关网络”，基于一种信念，即边近似独立。在本工作中，我们将这种建模选择置于坚实的理论基础上。我们证明当时间序列可以表示为少量傅里叶基元素（或某些其他合适选择的基）时，相关网络对应于具有依赖边噪声的潜在空间网络，在其中顶点级别的潜在变量编码基系数。进一步，我们证明当时间序列受噪声观测时，所得到的噪声相关网络的光谱嵌入在适当假设下仍能恢复这些真实的顶点级别潜在表示。这种将嵌入视为学习傅里叶系数的特征似乎在信号处理社区的主成分分析背景下是常识，但据我们所知，在统计网络分析文献中是新的。
英文摘要：
In many applications, weighted networks are constructed based on time series data: each time series is associated to a vertex and edge weights are given by pairwise correlations. The result is a network whose edge dependency structure violates the assumptions of most common network models. Nonetheless, it is common to analyze these "correlation networks" using embedding methods derived from edge-independent network models, based on a belief that the edges are approximately independent. In this work, we put this modeling choice on firm theoretical ground. We show that when the time series are expressible in terms of a small number of Fourier basis elements (or in some other suitably-chosen basis), correlation networks correspond to latent space networks with dependent edge noise in which the vertex-level latent variables encode the basis coefficients. Further, we show that when time series are observed subject to noise, spectral embedding of the resulting noisy correlation network still recovers these true vertex-level latent representations under suitable assumptions. This characterization of embeddings as learning Fourier coefficients appears to be folklore in the signal processing community in the context of principal component analysis, but is, to the best of our knowledge, new to the statistical network analysis literature.
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