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