Principal Polynomial Analysis (PPA)
V. Laparra, S. Jimenez, D. Tuia, G. Camps and J. Malo

Paper and Matlab Toolbox



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 volume-preserving map, which guarantees the existence of the inverse since the determinant of the Jacobian is bounded. We propose a closed-form 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 multi-information achieved by the transform using only marginal operations. Additionally, PPA leads to a clear geometrical interpretation of the manifold: the computation of Frenet-Serret 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. Camps-Valls and J. Malo
Accepted in Int. J. Neur. Syst., july 2014 (in press)

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PPA Toolbox (general purpose code) mat   See README_1st.txt for an overview of the toolbox

Specific code to reproduce the experiments in the paper mat   See README_experiments.txt for an overview. It requires the PPA toolbox