# Causal Fourier Analysis on Directed Acyclic Graphs and Posets

arXiv: https://arxiv.org/abs/2209.07970 · HTML: https://mlsanity.com/paper/2209.07970/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 66 references that are in the map.
- First posted on arXiv 16 Sep 2022.

## Builds on (most cited first, 20 of 20)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Concave Penalized Estimation of Sparse Gaussian Bayesian Networks](https://mlsanity.com/paper/1401.0852/) (2014)
- [Signal Processing on Directed Graphs](https://mlsanity.com/paper/2008.00586/) (2020)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [Topological Signal Processing over Cell Complexes](https://mlsanity.com/paper/2112.06709/) (2021)
- [Introducing Hypergraph Signal Processing: Theoretical Foundation and Practical Applications](https://mlsanity.com/paper/1907.09203/) (2019)
- [Discrete Signal Processing with Set Functions](https://mlsanity.com/paper/2001.10290/) (2020)
- [Discrete Signal Processing on Meet/Join Lattices](https://mlsanity.com/paper/2012.04358/) (2020)
- [A Directed Graph Fourier Transform with Spread Frequency Components](https://mlsanity.com/paper/1804.03000/) (2018)
- [Graph Fourier Transform: A Stable Approximation](https://mlsanity.com/paper/2001.05042/) (2020)
- [Quiver Signal Processing (QSP)](https://mlsanity.com/paper/2010.11525/) (2020)
- [Digraph Signal Processing with Generalized Boundary Conditions](https://mlsanity.com/paper/2005.09762/) (2020)

## Built on by (most cited first, 12 of 12)

- [Learning DAGs from Data with Few Root Causes](https://mlsanity.com/paper/2305.15936/) (2023)
- [The CausalBench challenge: A machine learning contest for gene network inference from single-cell perturbation data](https://mlsanity.com/paper/2308.15395/) (2023)
- [Non-negative Weighted DAG Structure Learning](https://mlsanity.com/paper/2409.07880/) (2024)
- [Convolutional Learning on Directed Acyclic Graphs](https://mlsanity.com/paper/2405.03056/) (2024)
- [Exploiting Non-Negativity in DAG Structure Learning](https://mlsanity.com/paper/2605.19947/) (2026)
- [Concomitant DAG Learning: On the Roles of Noise Adaptivity, Sparsity, and Non-negativity](https://mlsanity.com/paper/2605.23537/) (2026)
- [Directed Acyclic Graph Convolutional Networks](https://mlsanity.com/paper/2506.12218/) (2025)
- [RKHS Representation of Algebraic Convolutional Filters with Integral Operators](https://mlsanity.com/paper/2602.19094/) (2026)
- [SpinSVAR: Estimating Structural Vector Autoregression Assuming Sparse Input](https://mlsanity.com/paper/2501.03130/) (2025)
- [Non-negative DAG Learning from Time-Series Data](https://mlsanity.com/paper/2512.07267/) (2025)
- [Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions](https://mlsanity.com/paper/2609.36106/) (2026)
- [Signal Processing over Product DAGs: Causal Shifts and Filters](https://mlsanity.com/paper/2609.40275/) (2026)

## Abstract (arXiv)

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.