Symmetry-Aware Feature Learning: A Polynomial Separation for Multi-Index Models
We establish a polynomial sample complexity separation between symmetry-aware and symmetry-agnostic feature learning. We study growing-rank multi-index models with high-dimensional Gaussian covariates in $\mathbb{R}^d$ and $r=Θ(d^δ)$ teacher directions forming a cyclic symmetry orbit, where $0<δ<1/2$. We compare three ways of exploiting this structure: architectural weight sharing, data augmentation over the full symmetry group, and learning without access to the symmetry. In particular, we analyze a symmetry-tied convolutional network, an untied network, and the same untied network trained with full-group data augmentation, using spherical online SGD with correlation loss. For a class of polynomial links with information exponent $p\ge3$, we prove matching sample complexity bounds up to logarithmic factors: the tied and augmented learners achieve weak directional recovery in $\widetildeΘ(d^{p-1})$ samples, whereas the symmetry-agnostic learner requires $\widetildeΘ(rd^{p-1})$. For the pure quadratic Hermite link, the same separation holds for weak recovery of the teacher subspace, with sample complexities $\widetildeΘ(d)$ and $\widetildeΘ(rd)$, respectively. Thus, full-group data augmentation matches the sample efficiency of architectural weight sharing, and both provide a polynomial advantage over training without symmetry. For $p\ge3$, the proof reveals a two-stage mechanism: fluctuations at initialization select one direction in the teacher orbit, after which localized growth amplifies its overlap to the weak recovery scale while competing overlaps remain near their initialization scale.