Training Dynamics of Learning 3D-Rotational Equivariance

• URL: http://arxiv.org/abs/2512.02303v1
• Date: Tue, 02 Dec 2025 00:48:09 GMT
• Title: Training Dynamics of Learning 3D-Rotational Equivariance
• Authors: Max W. Shen, Ewa Nowara, Michael Maser, Kyunghyun Cho,
• Abstract summary: We investigate how quickly symmetry-agnostic models learn to respect symmetries.<n>For 3D rotations, the loss penalty for non-equivariant models is small throughout training.
• Score: 36.537155535159435
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: While data augmentation is widely used to train symmetry-agnostic models, it remains unclear how quickly and effectively they learn to respect symmetries. We investigate this by deriving a principled measure of equivariance error that, for convex losses, calculates the percent of total loss attributable to imperfections in learned symmetry. We focus our empirical investigation to 3D-rotation equivariance on high-dimensional molecular tasks (flow matching, force field prediction, denoising voxels) and find that models reduce equivariance error quickly to $\leq$2\% held-out loss within 1k-10k training steps, a result robust to model and dataset size. This happens because learning 3D-rotational equivariance is an easier learning task, with a smoother and better-conditioned loss landscape, than the main prediction task. For 3D rotations, the loss penalty for non-equivariant models is small throughout training, so they may achieve lower test loss than equivariant models per GPU-hour unless the equivariant ``efficiency gap'' is narrowed. We also experimentally and theoretically investigate the relationships between relative equivariance error, learning gradients, and model parameters.

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