# Quantifying the Alignment of Graph and Features in Deep Learning

## TL;DR

This paper introduces a subspace alignment measure (SAM) to quantify how the alignment between features, graph structure, and ground truth influences the classification performance of graph convolutional networks, providing new insights into their functioning.

## Contribution

It proposes a novel spectral and geometrical measure (SAM) to analyze the relationship between features, graph, and ground truth in GCNs, enhancing understanding of their performance.

## Key findings

- SAM correlates with GCN classification accuracy.
- Alignment between features, graph, and ground truth impacts performance.
- The measure reveals the relative importance of graph and features.

## Abstract

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.