Int J Neural Syst - Principal polynomial analysis.

Tópicos

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{ result(1111) use(1088) new(759) }
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Resumo

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.

Resumo Limpo

paper present new framework manifold learn base sequenc princip polynomi captur possibl nonlinear natur data propos princip polynomi analysi ppa general pca model direct maxim varianc mean curv instead straight line contrarili previous approach ppa reduc perform simpl univari regress make comput feasibl robust moreov ppa show number interest analyt properti first ppa volumepreserv map turn guarante exist invers second invers can obtain close form invert import advantag learn method permit understand identifi featur input domain data physic mean moreov allow evalu perform dimension reduct sensibl inputdomain unit volum preserv also allow easi comput inform theoret quantiti reduct multiinform transform third analyt natur ppa lead clear geometr interpret manifold allow comput frenetserret frame local featur general curvatur point space fourth analyt jacobian allow comput metric induc data thus general mahalanobi distanc properti demonstr theoret illustr experiment perform ppa evalu dimension redund reduct synthet real dataset uci repositori

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