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GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras

ICML 2025 · Architectures and efficiency › CNN design · approach: Group symmetry equivariant networks · 0 citations (OpenAlex)

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Proposes GLGENN, a parameter-light Clifford-algebra equivariant network built on generalized Lipschitz groups and weight sharing across four grade-involution/reversion subspaces, matching or beating CGENN with far fewer parameters.

Problem

Geometric-algebra equivariant networks (e.g., CGENN) are overparameterized and overfit on small datasets typical of natural-science applications, and train slowly.

Approach

Introduces generalized Lipschitz groups preserving the four quaternion-type subspaces (defined by grade involution and reversion) under a twisted adjoint action, and builds layers (conjugation, subspace-linear, geometric product, normalization) that project onto these four subspaces with weight sharing instead of fixed grades, yielding equivariance to pseudo-orthogonal groups with fewer parameters

Key results

• On O(5,0) regression with 30 training samples, GLGENN reaches MSE 0.1055 vs CGENN 0.0791 (CGENN better), while at 300 samples GLGENN gets 0.0020 vs CGENN 0.0089, using about 0.6K GA-layer parameters vs CGENN's 1.8K.
• On 16-point convex hull with large training sets, GLGENN matches or beats CGENN (e.g., 2.61 vs 2.08 MSE at 2^17 samples, so CGENN better there; GLGENN better at 2^8 samples: 16.94 vs 18.71) with 24.1K vs 58.8K parameters.
• On 256- and 512-point convex hull tasks, GLGENN outperforms CGENN (e.g., 2908 vs 5177 MSE at 2^10 samples for K=256) with about half the parameters.

Measured effects reported in the paper

Surprising findings

• CGENN outperforms GLGENN on the O(5,0) regression at the smallest sample size (30 samples) and on the 16-point convex hull at the largest sample counts.
• Scaling CGENN down to GLGENN's parameter budget degrades its performance, suggesting the gain is not purely from fewer parameters.

Limitations

• Experiments are restricted to the non-degenerate case Cl_{p,q}; degenerate algebra experiments are deferred to future work.
• Nonlinearities applied only to scalars isolate grades, and grade mixing relies on geometric product layers.
• Evaluated only on synthetic benchmarks; no real-world data experiments.

Experimental setup

Datasets: synthetic O(5,0) regression data synthetic convex hull volume data in R^{5,0} and R^{7,0} N-body simulation data in R^{5,0}

Benchmarks: O(5,0)-regression O(5,0) convex hull volume estimation O(7,0) convex hull volume estimation O(5,0) N-body

Metrics: MSE

Baselines: MLP MLP+Aug EMLP-O(5) EMLP-SO(5) CGENN Geometric Vector Perceptrons Vector Neurons

Builds on: CGENN Clifford Group Equivariant Neural Networks Lipschitz groups EMLP

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