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Interpreting Equivariant Representations

ICML 2024 · Label-efficient and robust learning › Recognition and long-tail · approach: Contrastive representation theory · 0 citations (OpenAlex)

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Shows that analyzing equivariant latent representations naively is misleading and proposes invariant projections (sorting-based isometric cross-section for permutation-equivariant graph VAEs, random invariant linear projections for rotation-equivariant classifiers) to make latent analysis meaningful.

Problem

Latent representations of equivariant models contain multiple equivalent codes per input (one per group orbit), so standard latent analysis (visualization, interpolation, kNN, clustering) gives ambiguous or misleading results and degraded downstream performance.

Approach

Treat equivariant latents through their quotient space and apply a post hoc invariant map: a sorting function that gives an isometric cross-section for permutation groups, and random invariant linear projections as a general continuous invariant map; also shows the same ambiguity appears for augmentation-trained invariant networks.

Key results

• On QM9, kNN regression from the sorted invariant representation achieves lower MAE than from the equivariant representation and common invariant pooling operations across molecular properties.
• Interpolation in the sorted invariant space yields more stable decoded molecules, with smaller pairwise Hamming distances between consecutive molecules than equivariant interpolation.
• For the SO(2) rotation-invariant MNIST classifier, kNN F1 score is higher on random invariant projections than on the equivariant representation.

Surprising findings

• Equivariant latent spaces can appear to show no structure in PCA plots even when the model has learned meaningful structure, because orbit ambiguity hides it.
• Augmentation-trained invariant CNNs show a similar ambiguity pattern in t-SNE of intermediate latents.

Limitations

• Invariant random projections are not generally invertible, so they do not support interpolation-based analysis.
• Choosing a suitable invariant map depends on the architecture and is not always clear.
• Isometric cross-section is shown only for permutation groups; general groups may lack isometric embeddings.

Experimental setup

Datasets: QM9 MNIST

Benchmarks: QM9 MNIST

Metrics: mean absolute error F1 score pairwise Hamming distance

Baselines: equivariant latent representation invariant pooling operations (sum, mean, max) augmentation-trained CNN

Builds on: variational autoencoder E(n)-equivariant graph neural networks ESCNN steerable CNNs

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