Causal Fourier Analysis on Directed Acyclic Graphs and Posets

TL;DR

The paper addresses learning and processing of signals indexed by directed acyclic graphs (DAGs) by developing a causal Fourier framework built on weighted transitive closures and Moebius inversion. It defines a shift-convolution algebra whose Fourier basis is given by the columns of a weight-derived matrix W, with the spectrum c = W^{-1}s representing causes. The contributions include a complete DAG/poset signal model, a generalized Fourier transform for weighted transitive closures, and demonstrations of Fourier sparsity in synthetic and semi-synthetic dynamic-network applications, notably infection spread. The approach enables causal interpretation of the spectrum, scalable computations, and improved reconstruction/learning of signals when the data exhibit low Fourier rank aligned with causal structure. This has practical impact for causal data analytics in domains such as dynamic networks, epidemiology, and SEM-informed signal processing.

Abstract

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.

Figures (15)

• Figure 1: (a) A DAG ${\cal D}$, (b) its transitive reduction, and (c) its transitive closure $\overline{{\cal D}}$). All three induce the same poset for which (b) is the cover graph, and (c) the reachability graph.
• Figure 2: The weighted transitive closure problem for a simple DAG with three nodes and two edges.
• Figure 3: Generic algorithm to compute various forms of weighted transitive closure of $A$ in $O(n^3)$Abdali.Saunders:1985a. The genericity is in the choice of addition $\oplus$ and multiplication $\otimes$, which need to satisfy a semiring property. Possible choices and the associated results are shown in Table \ref{['tab:ClosedSemirings']}.
• Figure 4: The purpose of product and sum in Table \ref{['tab:ClosedSemirings']}. If these operations satisfy the semiring property, the algorithm in Fig. \ref{['algo:ModifiedFloydWarshall']} works.
• Figure 5: Example DAG with different weights and meanings of weights (left) and their corresponding transitive closures (right). Modified and added weights and edges are colored blue.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2: Fourier basis
• Theorem 3: Fourier transform
• Definition 1
• Theorem 4