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Permutation Equivariant Neural Networks for Symmetric Tensors

ICML 2025 · Learning theory and algorithms › Matrix and tensor methods · approach: Tensor network decompositions · 0 citations (OpenAlex)

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Characterises all linear S_n-equivariant maps between symmetric powers of R^n via orbit and diagram bases (bipartition diagrams), introduces map label notation for memory-efficient computation, and validates on two toy tasks.

Problem

Permutation equivariant linear layers acting on symmetric tensors (inputs in symmetric powers of R^n) had no exact characterisation, and prior equivariant work on symmetric tensors focused on Euclidean groups.

Approach

Gives two bases (orbit basis and diagram basis) of all S_n-equivariant linear maps between symmetric power spaces, indexed by (k,l)-bipartitions with at most n blocks, and map label notation that expresses the layer action as index equations so weight matrices need not be stored and the same equations work for any n.

Key results

• On diagonal extraction (S_8-equivariant, 8x8x8 tensors), SymmPermEquiv reaches test MSE 0.0035 versus 0.0447 for PermEquiv and 0.6486 for MLP.
• The SymmPermEquiv model trained at n=8 attains test MSE 0.0048 at n=16 and 0.0088 at n=32 without retraining, indicating generalisation across tensor sizes.
• The paper reports the S_12-invariant task as showing high data efficiency relative to a standard MLP, with 95% confidence intervals over 3 runs (no exact numbers given in the text).

Measured effects reported in the paper

Limitations

• Evaluation limited to two toy synthetic tasks
• Only linear equivariant layers are characterised; no full deep network experiments on real data
• Generalisation to larger n degrades (MSE rises from 0.0035 to 0.0088 at n=32)

Experimental setup

Datasets: synthetic S_12-invariant task data synthetic diagonal extraction data for 8x8x8 symmetric tensors

Benchmarks: synthetic S_12-invariant task synthetic S_8-equivariant diagonal extraction task

Metrics: test mean squared error data efficiency

Baselines: MLP PermEquiv

Builds on: Deep Sets spherical partition algebra

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