A nonlinear principal component decomposition

The aim of this paper is to introduce a practical nonlinear generalization of PCA that captures nonlinear forms of dependence and delivers truly independent factors. The output of the method is a low-dimensional curvilinear coordinate system that tracks the important features of the data. The key ingredients of our approach are (i) the reliance on the theory of Brenier maps (Brenier (1991)), which are a natural generalization of monotone functions in multivariate settings, (ii) the use of entropy (Kullback (1959), Csiszar (1991), Golan, Judge, and Miller (1996), Shore and Johnson (1980), Gray (2011), Shannon (1948)) to determine the principal nonlinear components that cap-ture most of the information content of the data and (iii) the introduction of a novel multivariate additive decomposition of the entropy into one-dimensional contributions. The resulting method is computationally attractive, as it reduces to the well-studied problem of computing a Brenier map followed by a suitable matrix diagonalization step. These features distinguish our approach from the numerous other solutions that have been previously proposed in the very active literature seeking nonlinear generalizations of PCA (see, e.g., Lawrence (2012) and Lee and Verleysen (2007) for reviews). An appealing theoretical feature of our approach is the virtual absence of technical regularity conditions for the results to hold - is that is needed is that the data admits a density with respect to the Lebesgue measure.