Problem
Research questions and friction points this paper is trying to address.
product DAGs
signal processing
causal shifts
linear structural equation model
separability
Signal Processing over Product DAGs: Causal Shifts and Filters
🤖 AI Summary
This study addresses the high computational complexity arising from the non-separability of graph structures in dual directed acyclic graph (DAG) product signal processing. Focusing on dual-domain causal signals described by linear structural equation models (SEMs), this work proposes a novel DAG product operator to construct a separable signal processing framework. By integrating DAG theory with spectral graph signal processing techniques, the proposed method enables the factorized decomposition of transitive closures and spectral analysis. The core contribution lies in overcoming the non-separability bottleneck, thereby achieving efficient cross-factor computation for SEMs, Fourier modes, and causal filters. Ultimately, this approach significantly optimizes the efficiency of Fourier analysis for dual-domain causal signals.
📝 Abstract
We develop a signal processing framework for signals indexed by the product of two directed acyclic graphs (DAGs) and described by a linear structural equation model (SEM). Such a setup arises whenever (linear) causal relations act along two domains, as in component versus manufacturing stage or gene versus experimental condition. Disregarding the factorization of the underlying graph and the native two-axis causal structure requires inverting a weighted transitive closure matrix whose size is the product of the two factor sizes for Fourier analysis. Recognizing that standard graph products fail to yield factorizable transitive closures, we introduce a new DAG product under which separability holds. We motivate the new operator in the vertex domain, and show that it also renders the SEM, the Fourier modes, the causal shifts, and the filters on the product DAG separable across its constituent graph factors.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Product DAGs
Causal Shifts
Signal Processing
Structural Equation Model
Separability
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