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

We study the approximation of two-layer compositions f(x) = g(φ(x)) via deep networks with ReLU activation, where φ is a geometrically intuitive, dimensionality reducing feature map. We focus on two intuitive and practically relevant choices for φ: the projection onto a low-dimensional embedded submanifold and a distance to a collection of low-dimensional sets. We achieve near optimal approximation rates, which depend only on the complexity of the dimensionality reducing map φ rather than the ambient dimension. Since φ encapsulates all nonlinear features that are material to the function f, this suggests that deep nets are faithful to an intrinsic dimension governed by f rather than the complexity of the domain of f. In particular, the prevalent assumption of approximating functions on low-dimensional manifolds can be significantly relaxed using functions of type f(x) = g(φ(x)) with φ representing an orthogonal projection onto the same manifold.