  This survey is written in summer, 2016. The purpose of this survey is to
briefly introduce nonlinear dimensionality reduction (NLDR) in data reduction.
The first two NLDR were respectively published in Science in 2000 in which they
solve the similar reduction problem of high-dimensional data endowed with the
intrinsic nonlinear structure. The intrinsic nonlinear structure is always
interpreted as a concept in manifolds from geometry and topology in theoretical
mathematics by computer scientists and theoretical physicists. In 2001, the
concept of Manifold Learning first appears as an NLDR method called Laplacian
Eigenmaps. In a typical manifold learning setup, the data set, also called the
observation set, is distributed on or near a low dimensional manifold M
embedded in RD, which yields that each observation has a D-dimensional
representation. The goal of manifold learning is to reduce these observations
as a compact lower-dimensional representation based on the geometric
information. The reduction procedure is called the spectral manifold learning.
In this paper, we derive each spectral manifold learning with the matrix and
operator representation, and we then discuss the convergence behavior of each
method in a geometric uniform language. Hence, the survey is named Geometric
Foundations of Data Reduction.
