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We study the approximation of two-layer compositions $f(x) = g(\phi(x))$ viadeep networks with ReLU activation, where $\phi$ is a geometrically intuitive,dimensionality reducing feature map. We focus on two intuitive and practicallyrelevant choices for $\phi$: the projection onto a low-dimensional embeddedsubmanifold and a distance to a collection of low-dimensional sets. We achievenear optimal approximation rates, which depend only on the complexity of thedimensionality reducing map $\phi$ rather than the ambient dimension. Since$\phi$ encapsulates all nonlinear features that are material to the function$f$, this suggests that deep nets are faithful to an intrinsic dimensiongoverned by $f$ rather than the complexity of the domain of $f$. In particular,the prevalent assumption of approximating functions on low-dimensionalmanifolds can be significantly relaxed using functions of type $f(x) =g(\phi(x))$ with $\phi$ representing an orthogonal projection onto the samemanifold.

A deep network construction that adapts to intrinsic dimensionality beyond the domain