6533b7d8fe1ef96bd126b730

RESEARCH PRODUCT

PRINCIPAL POLYNOMIAL ANALYSIS

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© 2014 World Scientific Publishing Company. 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.