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Convolutional Learning on Directed Acyclic Graphs

Samuel Rey, Hamed Ajorlou, Gonzalo Mateos

TL;DR

The paper tackles learning from signals on directed acyclic graphs (DAGs), where standard graph convolutions struggle due to nilpotent adjacency. It introduces the DAG Convolutional Network (DCN), built on causal graph filters with a reachability-based operator $W=(I-\mathbf{A})^{-1}$ and node-specific $\mathbf{T}_k=\mathbf{W}\mathbf{D}_k\mathbf{W}^{-1}$, enabling a principled, spectral convolution on DAGs. Each DCN layer uses $\mathbf{X}^{(\ell+1)}=\sigma(\sum_k \mathbf{T}_k \mathbf{X}^{(\ell)} \boldsymbol{\Theta}_k^{(\ell)})$, and complexity can be managed by restricting the GSO set. Empirical results on synthetic network-diffusion tasks show DCN outperforming baselines, highlighting the value of honoring DAG directionality and providing a scalable, causal-aware learning tool for DAG-supported data.

Abstract

We develop a novel convolutional architecture tailored for learning from data defined over directed acyclic graphs (DAGs). DAGs can be used to model causal relationships among variables, but their nilpotent adjacency matrices pose unique challenges towards developing DAG signal processing and machine learning tools. To address this limitation, we harness recent advances offering alternative definitions of causal shifts and convolutions for signals on DAGs. We develop a novel convolutional graph neural network that integrates learnable DAG filters to account for the partial ordering induced by the graph topology, thus providing valuable inductive bias to learn effective representations of DAG-supported data. We discuss the salient advantages and potential limitations of the proposed DAG convolutional network (DCN) and evaluate its performance on two learning tasks using synthetic data: network diffusion estimation and source identification. DCN compares favorably relative to several baselines, showcasing its promising potential.

Convolutional Learning on Directed Acyclic Graphs

TL;DR

The paper tackles learning from signals on directed acyclic graphs (DAGs), where standard graph convolutions struggle due to nilpotent adjacency. It introduces the DAG Convolutional Network (DCN), built on causal graph filters with a reachability-based operator and node-specific , enabling a principled, spectral convolution on DAGs. Each DCN layer uses , and complexity can be managed by restricting the GSO set. Empirical results on synthetic network-diffusion tasks show DCN outperforming baselines, highlighting the value of honoring DAG directionality and providing a scalable, causal-aware learning tool for DAG-supported data.

Abstract

We develop a novel convolutional architecture tailored for learning from data defined over directed acyclic graphs (DAGs). DAGs can be used to model causal relationships among variables, but their nilpotent adjacency matrices pose unique challenges towards developing DAG signal processing and machine learning tools. To address this limitation, we harness recent advances offering alternative definitions of causal shifts and convolutions for signals on DAGs. We develop a novel convolutional graph neural network that integrates learnable DAG filters to account for the partial ordering induced by the graph topology, thus providing valuable inductive bias to learn effective representations of DAG-supported data. We discuss the salient advantages and potential limitations of the proposed DAG convolutional network (DCN) and evaluate its performance on two learning tasks using synthetic data: network diffusion estimation and source identification. DCN compares favorably relative to several baselines, showcasing its promising potential.
Paper Structure (9 sections, 7 equations, 1 figure, 1 table)

This paper contains 9 sections, 7 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and problem statement
  3. Preliminaries: DAGs, graph signals and GNNs
  4. Problem statement
  5. Convolutional learning on DAGs
  6. Graph-shift operators and convolution for DAGs
  7. DAG Convolutional Network (DCN)
  8. Numerical Evaluation
  9. Conclusions

Figures (1)

  • Figure 1: (a) reports the MNSE in the network diffusion task as the noise in the observations increases; For the source identification task, (b) and (c) depict the influence of increasing respectively the proportion of unobserved nodes and the edge probability. We report the median performance across 25 realizations and values between the first and third quartile in the shaded area.