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In much of the literature on function approximation by deep networks, thefunction is assumed to be defined on some known domain, such as a cube or asphere. In practice, the data might not be dense on these domains, andtherefore, the approximation theory results are observed to be tooconservative. In manifold learning, one assumes instead that the data issampled from an unknown manifold; i.e., the manifold is defined by the dataitself. Function approximation on this unknown manifold is then a two stageprocedure: first, one approximates the Laplace-Beltrami operator (and itseigen-decomposition) on this manifold using a graph Laplacian, and next,approximates the target function using the eigen-functions. Alternatively, oneestimates first some atlas on the manifold and then uses local approximationtechniques based on the local coordinate charts.  In this paper, we propose a more direct approach to function approximation on\emph{unknown}, data defined manifolds without computing theeigen-decomposition of some operator or an atlas for the manifold, and withoutany kind of training in the classical sense. Our constructions are universal;i.e., do not require the knowledge of any prior on the target function otherthan continuity on the manifold. We estimate the degree of approximation. Forsmooth functions, the estimates do not suffer from the so-called saturationphenomenon. We demonstrate via a property called good propagation of errors howthe results can be lifted for function approximation using deep networks whereeach channel evaluates a Gaussian network on a possibly unknown manifold.

A direct approach for function approximation on data defined manifolds