# Signal Processing over Product DAGs: Causal Shifts and Filters

arXiv: https://arxiv.org/abs/2609.40275 · HTML: https://mlsanity.com/paper/2609.40275/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 16 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Graph Signal Processing: History, Development, Impact, and Outlook](https://mlsanity.com/paper/2303.12211/) (2023)
- [Signal Processing on Directed Graphs](https://mlsanity.com/paper/2008.00586/) (2020)
- [A Time-Vertex Signal Processing Framework](https://mlsanity.com/paper/1705.02307/) (2017)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [Product Graph Learning from Multi-domain Data with Sparsity and Rank Constraints](https://mlsanity.com/paper/2012.08090/) (2020)
- [Learning DAGs from Data with Few Root Causes](https://mlsanity.com/paper/2305.15936/) (2023)
- [Learning Product Graphs Underlying Smooth Graph Signals](https://mlsanity.com/paper/2002.11277/) (2020)
- [Sparse Sampling for Inverse Problems with Tensors](https://mlsanity.com/paper/1806.10976/) (2018)
- [Fourier Analysis of Signals on Directed Acyclic Graphs (DAG) Using Graph Zero-Padding](https://mlsanity.com/paper/2311.01073/) (2023)

## Abstract (arXiv)

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.