  Modern sample points in many applications no longer comprise real vectors in
a real vector space but sample points of much more complex structures, which
may be represented as points in a space with a certain underlying geometric
structure, namely a manifold. Manifold learning is an emerging field for
learning the underlying structure. The study of manifold learning can be split
into two main branches: dimension reduction and manifold fitting. With the aim
of combining statistics and geometry, we address the problem of manifold
fitting in the ambient space. Inspired by the relation between the eigenvalues
of the Laplace-Beltrami operator and the geometry of a manifold, we aim to find
a small set of points that preserve the geometry of the underlying manifold.
From this relationship, we extend the idea of subsampling to sample points in
high-dimensional space and employ the Moving Least Squares (MLS) approach to
approximate the underlying manifold. We analyze the two core steps in our
proposed method theoretically and also provide the bounds for the MLS approach.
Our simulation results and theoretical analysis demonstrate the superiority of
our method in estimating the underlying manifold.
