Simplifying the Theory on Over-Smoothing

TL;DR

This paper simplifies the understanding of over-smoothing in graph convolutions by framing it as a special case of power iteration, introduces a new comprehensive definition and metric, and empirically shows many models suffer from rank collapse.

Contribution

It provides a simplified theoretical framework for over-smoothing, introduces a new definition of rank collapse, and empirically evaluates its prevalence across multiple methods.

Findings

Over-smoothing is a special case of power iteration.

Many models suffer from rank collapse.

The new metric helps identify over-smoothing issues.

Abstract

Graph convolutions have gained popularity due to their ability to efficiently operate on data with an irregular geometric structure. However, graph convolutions cause over-smoothing, which refers to representations becoming more similar with increased depth. However, many different definitions and intuitions currently coexist, leading to research efforts focusing on incompatible directions. This paper attempts to align these directions by showing that over-smoothing is merely a special case of power iteration. This greatly simplifies the existing theory on over-smoothing, making it more accessible. Based on the theory, we provide a novel comprehensive definition of rank collapse as a generalized form of over-smoothing and introduce the rank-one distance as a corresponding metric. Our empirical evaluation of 14 commonly used methods shows that more models than were previously known suffer from this issue.