Asymptotic Properties of Nonparametric Estimation on Manifold
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Proceedings of the Sixth Workshop on Conformal and Probabilistic Prediction and Applications, PMLR 60:1838, 2017.
Abstract
In many applications, the real highdimensional data occupy only a very small part in the high dimensional ‘observation space’
whose intrinsic dimension is small.
The most popular model of such data is Manifold model which assumes that the data lie on or near an unknown manifold (Data Manifold, DM) of lower dimensionality
embedded in an ambient highdimensional input space (Manifold Assumption about highdimensional data).
Manifold Learning is a Dimensionality Reduction problem under the Manifold assumption about the processed data,
and its goal is to construct a lowdimensional parameterization of the DM (global lowdimensional coordinates on the DM)
from a finite dataset sampled from the DM.
Manifold Assumption means that local neighborhood of each manifold point is equivalent to an area of lowdimensional Euclidean space.
Because of this, most of Manifold Learning algorithms include two parts:
‘local part’ in which certain characteristics reflecting lowdimensional local structure of neighborhoods of all sample points
are constructed via nonparametric estimation,
and ‘global part’ in which global lowdimensional coordinates on the DM are constructed
by solving the certain convex optimization problem for specific cost function depending on the local characteristics.
Both statistical properties of ‘local part’ and its average over manifold are considered in the paper.
The article is an extension of the paper (Yanovich, 2016) for the case of nonparametric estimation.
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