Learning on manifolds without manifold learning
This addresses the problem of reducing errors in function approximation for data on unknown manifolds, offering a more efficient one-step method that is incremental over existing two-step approaches.
The paper tackles function approximation on data manifolds by projecting the manifold onto a hypersphere and using localized spherical polynomial kernels, avoiding the need for manifold preprocessing. It achieves optimal approximation rates for relatively rough functions.
Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace-Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively ``rough'' functions.