Geometric Stability and Sample Complexity for Compositional Function Priors

For compositional models f(x)=g(Ax+Δh(x)), with A∈R k×d linear, Δh a nonlinear correction, and g an outer function, we prove covering-number bounds showing that statistical complexity interpolates between an intrinsic regime---parametric in A, nonparametric only in the intrinsic dimension k---and an ambient d-dimensional rate, controlled by the correction size ϵ=∥Δh∥. Geometry explains when ϵ is small: for data in a tube around a low-curvature manifold, the target factorizes exactly through a nearly linear feature map with ϵ linear in curvature at the localization scale, yielding a finite-sample excess-risk bound over an atlas of local models whose ambient term vanishes with curvature. For a one-dimensional model problem we prove a converse: on curved data every affine compositional predictor faces an error floor unless its correction budget scales with curvature, while the matching curved class attains the intrinsic rate. Finally, ϵ is exactly an invariance defect, with factorization through a quotient at ϵ=0.