Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions

We describe a simple scheme, based on the Nyström method, for extending empirical functions f defined on a set X to a larger set . The extension process that we describe involves the construction of a specific family of functions that we term geometric harmonics. These functions constitute a generalization of the prolate spheroidal wave functions of Slepian in the sense that they are optimally concentrated on X. We study the case when X is a submanifold of Rn in greater detail. In this situation, any empirical function f on X can be characterized by its decomposition over the intrinsic Fourier modes, i.e., the eigenfunctions of the Laplace–Beltrami operator, and we show that this intrinsic frequency spectrum determines the largest domain of extension of f to the entire space Rn. Our analysis relates the complexity of the function on the training set to the scale of extension off this set. This approach allows us to present a novel multiscale extension scheme for empirical functions.