Tunable Soft Equivariance with Guarantees

Self-supervised learning (SSL) typically learns representations invariant to semantic-preserving augmentations. While effective for recognition, enforcing strong invariance can suppress transformation-dependent structure that is useful for robustness to geometric perturbations and spatially sensitive transfer. A growing body of work, therefore, augments invariance-based SSL with equivariance objectives, but these objectives are often imposed on the same final representation. We empirically observe a trade-off in this coupled setting: pushing equivariance regularization toward deeper layers improves equivariance scores but degrades ImageNet-1k linear evaluation, motivating a layer-decoupled design. Motivated by this trade-off, we propose Soft Equivariance Regularization (SER), a plug-in regularizer that decouples where invariance and equivariance are enforced: we keep the base SSL objective unchanged on the final embedding, while softly encouraging equivariance on an intermediate spatial token map via analytically specified group actions $ρ_g$ applied directly in feature space. SER learns/predicts no per-sample transformation codes/labels, requires no auxiliary transformation-prediction head, and adds only 1.008x training FLOPs. On ImageNet-1k ViT-S/16 pretraining, SER improves MoCo-v3 by +0.84 Top-1 in linear evaluation under a strictly matched 2-view setting and consistently improves DINO and Barlow Twins; under matched view counts, SER achieves the best ImageNet-1k linear-eval Top-1 among the compared invariance+equivariance add-ons. SER further improves ImageNet-C/P by +1.11/+1.22 Top-1 and frozen-backbone COCO detection by +1.7 mAP. Finally, applying the same layer-decoupling recipe to existing invariance+equivariance baselinesimproves their accuracy, suggesting layer decoupling as a general design principle for combining invariance and equivariance.

Normalization Equivariance (NE) is a structural prior that improves robustness to distribution shift in image-to-image tasks. A function $f$ is normalization equivariant iff $f(a y + b\mathbf{1}) = a f(y) + b\mathbf{1}$ for all $a>0$ and $b\in\mathbb{R}$. Existing NE methods constrain every internal layer to NE-compatible operations. These constraints add runtime cost and exclude standard transformer components such as softmax attention and LayerNorm. We introduce Wrapped Normalization Equivariance (WNE), a parameter-free wrapper that normalizes the input, applies any backbone, and denormalizes the output. We prove every NE function admits this factorization, so the wrapper exactly parameterizes the class of NE functions. On blind denoising, wrapping CNN and transformer architectures improves robustness under noise-level mismatch with no measurable GPU overhead, while architectural NE baselines are up to $1.6\times$ slower.

A latent world model built from an equivariant encoder and predictor inherits a provable symmetry of its training loss: when the dynamics carries a group $G$ acting on latents by an orthogonal representation $ρ(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting a restricted slice of orientations mathematically determines it on the entire orbit. The symmetry survives a real Muon/AdamW+EMA+VICReg run -- composed residual $\sim 10^{-6}$ after training, under any optimiser (intrinsic Vector-Neuron/e3nn parametrisation) -- and one-step error is flat across the group (5-seed medians: equivariant $\times 1.00$ vs a higher-capacity non-equivariant baseline $\times 12.7$ in 2D, $\times 17.2$ in 3D), while that baseline fits the slice but breaks out-of-distribution. The flatness is not a synthetic artefact: on real-robot DROID end-effector trajectories the equivariant model stays flat across the orbit ($\times 1.000$, rotation residual $1.5\times 10^{-16}$) while a $4.5\times$-larger baseline degrades $\times 11$. One caution is load-bearing: flatness is necessary, not sufficient -- the theorem transports the in-distribution error level unchanged but does not lower it (3D relMSE $\approx 0.43$): across-group error is constant, not low. The same isometry lifts to a closed-loop corollary: under a matching equivariant planner the control error is invariant across the group -- float-floor-exact in 2D/SO(2), statistically flat in 3D/SE(3). Stress-tested against Sutton's Bitter Lesson (augmentation, scale, soft-equivariance), each closes at most the across-group task metric, never the float-floor exactness. This is the generalisation-side foundation of a certified-world-models programme (arXiv:2606.13092, 2606.24945, 2606.24946): flatness transports competence, and the trust bounds built on it are downstream products.