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

6 Aug 2020  ·  Alexander Cloninger, Timo Klock ·

We study the approximation of two-layer compositions $f(x) = g(\phi(x))$ via deep networks with ReLU activation, where $\phi$ is a geometrically intuitive, dimensionality reducing feature map. We focus on two intuitive and practically relevant choices for $\phi$: 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 $\phi$ 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 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 $f(x) = g(\phi(x))$ with $\phi$ representing an orthogonal projection onto the same manifold.

PDF Abstract
No code implementations yet. Submit your code now



  Add Datasets introduced or used in this paper

Results from the Paper

  Submit results from this paper to get state-of-the-art GitHub badges and help the community compare results to other papers.