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.

翻译：我们提出了一种新型傅里叶分析方法及其相关信号处理概念，适用于由边加权有向无环图（DAG）索引的信号（或数据）。这意味着我们定义的傅里叶基能够对恰当定义的移位算子和卷积算子进行特征分解。DAG是刻画数据值之间因果关系的通用模型，在此情况下，我们提出的傅里叶分析在线性假设（由我们定义）下将数据与其成因联系起来。傅里叶变换的定义需要加权DAG的传递闭包，根据边权重的不同解释，存在多种可能的闭包形式。具体示例包括影响程度、距离或污染分布。我们的框架不同于先前的图信号处理（GSP）：它专为DAG设计，并利用并扩展了组合学中经典的莫比乌斯反演理论。作为一个典型应用，我们考虑对边随时间变化的动态网络进行建模的DAG。具体而言，我们基于真实世界接触追踪数据所构建的DAG对感染传播过程进行建模，并利用傅里叶域中的稀疏性假设从样本中学习感染信号。