Principal Polynomial Analysis (PPA)
V. Laparra, S. Jimenez, D. Tuia, G. Camps and J. Malo
Paper and Matlab Toolbox
Abstract This
paper (and toolbox) presents a new framework for manifold learning
based on the use of a sequence of principal polynomials that capture
the eventually nonlinear nature of the data. The proposed Principal
Polynomial Analysis (PPA) is shown to generalize PCA by admitting
curves instead of straight lines. As opposed to previous approaches
following the same rationale, PPA reduces to performing canonical,
univariate regressions which make it computationally feasible and easy
to interpret analytically.
We show that the PPA transform is a volumepreserving map, which guarantees the existence of the inverse since the determinant of the Jacobian is bounded. We propose a closedform solution for the inverse map. Invertibility is an important advantage over other nonlinear dimensionality reduction methods because it permits to understand the identified features in the input domain where data have physical meaning. Moreover, invertibility allows to evaluate the dimensionality reduction performance in sensible units. Preserving the volume also allows to compute the reduction in multiinformation achieved by the transform using only marginal operations. Additionally, PPA leads to a clear geometrical interpretation of the manifold: the computation of FrenetSerret frames along the identified curves allow us to obtain generalized curvature and torsion of the manifold. Moreover, the analytical expression of the Jacobian simplifies the computation of the metric induced by the data. Performance in dimensionality reduction and redundancy reduction, as well as the theoretical properties of PPA, are experimentally tested in datasets from the UCI machine learning repository. 

First Principal Curve and generalized curvatures using PPA in 3D Helix 

Discrimination ellipsoids according to the PPA generalized Mahalanobis metric 

Related Papers Principal Polynomial Analysis. V. Laparra, S. Jimenez, D. Tuia, G. CampsValls and J. Malo Accepted in Int. J. Neur. Syst., july 2014 (in press) 

Download Code PPA Toolbox (general purpose code) See README_1st.txt
for an overview of the toolbox
Specific
code to reproduce the experiments in the paper See README_experiments.txt
for an overview. It requires the PPA toolbox 