Learning on manifolds without manifold learning
H. N. Mhaskar Ryan O'Dowd
Function approximation based on data drawn randomly from an unknown
distribution is an important problem in machine learning. In contrast to the
prevalent paradigm of solving this problem by minimizing a loss functional, we
have given a direct one-shot construction together with optimal error bounds
under the manifold assumption; i.e., one assumes that the data is sampled from
an unknown sub-manifold 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 basic quantities
of the data in addition to the errors inherent in function approximation. In
Neural Networks, 132:253268, 2020, we have proposed a one-shot direct method to
achieve function approximation without requiring the extraction of any
information about the manifold other than its dimension. However, one cannot
pin down the class of approximants used in that paper.
  In this paper, we view the unknown manifold as a sub-manifold of an ambient
hypersphere and study the question of constructing a one-shot approximation
using the spherical polynomials based 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.