有向无环图和偏序集上的因果傅里叶分析
Causal Fourier Analysis on Directed Acyclic Graphs and Posets
AI总结：
提出针对边加权有向无环图（DAG）的因果傅里叶分析框架，利用并扩展Moebius反演理论实现移位和卷积算子的特征分解，并成功应用于真实接触追踪数据的感染传播建模与信号学习。
摘要：
我们提出了一种新型傅里叶分析及相关信号处理概念，用于处理以边加权有向无环图（DAG）为索引的信号（或数据）。这意味着我们的傅里叶基对我们所定义的移位和卷积算子的合适概念产生了特征分解。DAG是捕获数据值之间因果关系的常见模型，在这种情况下，我们提出的傅里叶分析在我们定义的线性假设下将数据与其原因相关联。傅里叶变换的定义需要加权DAG的传递闭包，根据边权重的解释，传递闭包可以有多种形式。示例包括影响级别、距离或污染分布。我们的框架不同于先前的GSP：它特定于DAG，并利用并扩展了来自组合学的经典Moebius反演理论。对于一个原型应用，我们考虑了在动态网络中对DAG进行建模，其中边随时间变化。具体来说，我们对从真实世界接触者追踪数据获得的此类DAG上的感染传播进行建模，并假设在傅里叶域中具有稀疏性，从样本中学习感染信号。
英文摘要：
We present a novel form of Fourier analysis, and associated signal processing concepts, for signals (or data) indexed by edge-weighted directed acyclic graphs (DAGs). This means that our Fourier basis yields an eigendecomposition of a suitable notion of shift and convolution operators that we define. DAGs are the common model to capture causal relationships between data values and in this case our proposed Fourier analysis relates data with its causes under a linearity assumption that we define. The definition of the Fourier transform requires the transitive closure of the weighted DAG for which several forms are possible depending on the interpretation of the edge weights. Examples include level of influence, distance, or pollution distribution. Our framework is different from prior GSP: it is specific to DAGs and leverages, and extends, the classical theory of Moebius inversion from combinatorics. For a prototypical application we consider DAGs modeling dynamic networks in which edges change over time. Specifically, we model the spread of an infection on such a DAG obtained from real-world contact tracing data and learn the infection signal from samples assuming sparsity in the Fourier domain.
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