Causal Fourier Analysis on Directed Acyclic Graphs and Posets

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 specific to DAGs and leverages, and extends, the classical theory of Moebius inversion from combinatorics. For a prototypical application we consider the reconstruction of signals from samples assuming Fourier-sparsity, i.e., few causes. In particular, we consider DAGs modeling dynamic networks in which edges change over time. 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.