  While classical data analysis has addressed observations that are real
numbers or elements of a real vector space, at present many statistical
problems of high interest in the sciences address the analysis of data that
consist of more complex objects, taking values in spaces that are naturally not
(Euclidean) vector spaces but which still feature some geometric structure.
Manifold fitting is a long-standing problem, and has finally been addressed in
recent years by Fefferman et al. (2020, 2021a). We develop a method with a
theory guarantee that fits a $d$-dimensional underlying manifold from noisy
observations sampled in the ambient space $\mathbb{R}^D$. The new approach uses
geometric structures to obtain the manifold estimator in the form of image sets
via a two-step mapping approach. We prove that, under certain mild assumptions
and with a sample size $N=\mathcal{O}(\sigma^{(-d+3)})$, these estimators are
true $d$-dimensional smooth manifolds whose estimation error, as measured by
the Hausdorff distance, is bounded by $\mathcal{O}(\sigma^2\log(1/\sigma))$
with high probability. Compared with the existing approaches proposed in
Fefferman et al. (2018, 2021b); Genovese et al. (2014); Yao and Xia (2019), our
method exhibits superior efficiency while attaining very low error rates with a
significantly reduced sample size, which scales polynomially in $\sigma^{-1}$
and exponentially in $d$. Extensive simulations are performed to validate our
theoretical results. Our findings are relevant to various fields involving
high-dimensional data in machine learning. Furthermore, our method opens up new
avenues for existing non-Euclidean statistical methods in the sense that it has
the potential to unify them to analyze data on manifolds in the ambience space
domain.
