Tunable Equivariance via Learnable Order-Dependent Scaling

Equivariance is a powerful inductive bias in geometric deep learning, but real-world data often only approximately respects known symmetries. Strictly equivariant models can therefore be overly restrictive, while unconstrained models fail to exploit useful geometric structure. Existing relaxed-equivariance approaches mainly interpolate between equivariant and non-equivariant components, without modulating the internal structure of the equivariant features. We introduce a **tunable-equivariance** framework in which a single learnable parameter $\delta \in [0,1]$ controls both the unconstrained branch and the relative strength of irrep components, weighting order $\ell$ by ${\delta }^{\ell }$. This yields a structured, continuous modulation of equivariance, motivated by a torus–hub geometric interpretation within an **associative operator algebra**. We learn $\delta$ jointly with the task objective under an empirical equivariance constraint. We instantiate the framework on $\mathrm{SO}(3)$-equivariant architectures for N-body dynamics, molecular property prediction, 3D shape classification, and LiDAR pose estimation. Across these benchmarks, the learned degree of equivariance is task-dependent. It generally improves performance over strict-equivariant baselines and outperforms learned relaxed-equivariance baselines where directly compared. Higher-order irrep components strengthen with $\delta$, and the **single-layer equivariance error** is consistent with the predicted $\mathcal{O}(1-\delta )$ bound. Code will be released upon acceptance.