Breaking Symmetry Bottlenecks in GNN Readouts

TL;DR

This paper identifies a fundamental symmetry-preservation bottleneck in GNN readouts, proves its theoretical basis, and introduces a new projector-based readout method that enhances GNN expressivity and performance.

Contribution

It reveals a previously overlooked symmetry bottleneck at the readout stage of GNNs and proposes a novel invariant readout method to overcome this limitation.

Findings

Reveals the symmetry bottleneck in GNN readouts using representation theory.

Proposes projector-based invariant readouts that preserve symmetry-aware information.

Improves graph separation and benchmark performance by modifying only the readout stage.

Abstract

Graph neural networks (GNNs) are widely used for learning on structured data, yet their ability to distinguish non-isomorphic graphs is fundamentally limited. These limitations are usually attributed to message passing; in this work we show that an independent bottleneck arises at the readout stage. Using finite-dimensional representation theory, we prove that all linear permutation-invariant readouts, including sum and mean pooling, factor through the Reynolds (group-averaging) operator and therefore project node embeddings onto the fixed subspace of the permutation action, erasing all non-trivial symmetry-aware components regardless of encoder expressivity. This yields both a new expressivity barrier and an interpretable characterization of what global pooling preserves or destroys. To overcome this collapse, we introduce projector-based invariant readouts that decompose node representations into symmetry-aware channels and summarize them with nonlinear invariant statistics, preserving permutation invariance while retaining information provably invisible to averaging. Empirically, swapping only the readout enables fixed encoders to separate WL-hard graph pairs and improves performance across multiple benchmarks, demonstrating that readout design is a decisive and under-appreciated factor in GNN expressivity.