  Manifold learning is a hot research topic in the field of computer science
and has many applications in the real world. A main drawback of manifold
learning methods is, however, that there is no explicit mappings from the input
data manifold to the output embedding. This prohibits the application of
manifold learning methods in many practical problems such as classification and
target detection. Previously, in order to provide explicit mappings for
manifold learning methods, many methods have been proposed to get an
approximate explicit representation mapping with the assumption that there
exists a linear projection between the high-dimensional data samples and their
low-dimensional embedding. However, this linearity assumption may be too
restrictive. In this paper, an explicit nonlinear mapping is proposed for
manifold learning, based on the assumption that there exists a polynomial
mapping between the high-dimensional data samples and their low-dimensional
representations. As far as we know, this is the first time that an explicit
nonlinear mapping for manifold learning is given. In particular, we apply this
to the method of Locally Linear Embedding (LLE) and derive an explicit
nonlinear manifold learning algorithm, named Neighborhood Preserving Polynomial
Embedding (NPPE). Experimental results on both synthetic and real-world data
show that the proposed mapping is much more effective in preserving the local
neighborhood information and the nonlinear geometry of the high-dimensional
data samples than previous work.
