  This paper presents a new framework for manifold learning based on a sequence
of principal polynomials that capture the possibly nonlinear nature of the
data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by
modeling the directions of maximal variance by means of curves, instead of
straight lines. Contrarily to previous approaches, PPA reduces to performing
simple univariate regressions, which makes it computationally feasible and
robust. Moreover, PPA shows a number of interesting analytical properties.
First, PPA is a volume-preserving map, which in turn guarantees the existence
of the inverse. Second, such an inverse can be obtained in closed form.
Invertibility is an important advantage over other learning methods, because it
permits to understand the identified features in the input domain where the
data has physical meaning. Moreover, it allows to evaluate the performance of
dimensionality reduction in sensible (input-domain) units. Volume preservation
also allows an easy computation of information theoretic quantities, such as
the reduction in multi-information after the transform. Third, the analytical
nature of PPA leads to a clear geometrical interpretation of the manifold: it
allows the computation of Frenet-Serret frames (local features) and of
generalized curvatures at any point of the space. And fourth, the analytical
Jacobian allows the computation of the metric induced by the data, thus
generalizing the Mahalanobis distance. These properties are demonstrated
theoretically and illustrated experimentally. The performance of PPA is
evaluated in dimensionality and redundancy reduction, in both synthetic and
real datasets from the UCI repository.
