# Quantifying the Alignment of Graph and Features in Deep Learning

arXiv: https://arxiv.org/abs/1905.12921 · HTML: https://mlsanity.com/paper/1905.12921/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2019.

## Built on by (most cited first, 5 of 5)

- [No Metric to Rule Them All: Toward Principled Evaluations of Graph-Learning Datasets](https://mlsanity.com/paper/2502.02379/) (2025)
- [Understanding Deep Learning via Decision Boundary](https://mlsanity.com/paper/2206.01515/) (2022)
- [Structure-Aware DropEdge Towards Deep Graph Convolutional Networks](https://mlsanity.com/paper/2306.12091/) (2023)
- [Exact Generalisation Error Exposes Benchmarks Skew Graph Neural Networks Success (or Failure)](https://mlsanity.com/paper/2509.10337/) (2025)
- [Mini-Batch Class Composition Bias in Link Prediction](https://mlsanity.com/paper/2604.25978/) (2026)

## Abstract (arXiv)

We show that the classification performance of graph convolutional networks (GCNs) is related to the alignment between features, graph, and ground truth, which we quantify using a subspace alignment measure (SAM) corresponding to the Frobenius norm of the matrix of pairwise chordal distances between three subspaces associated with features, graph, and ground truth. The proposed measure is based on the principal angles between subspaces and has both spectral and geometrical interpretations. We showcase the relationship between the SAM and the classification performance through the study of limiting cases of GCNs and systematic randomizations of both features and graph structure applied to a constructive example and several examples of citation networks of different origins. The analysis also reveals the relative importance of the graph and features for classification purposes.