Directional Nonlinear Principal and Independent Components: a measure transportation approach

• Department of Mathematics, Université libre de Bruxelles, Belgium and Czech Academy of Sciences Prague, Czech Republic [marc.hallin@ulb.be]

Keywords: - Nonlinear principal components – Nonlinear independent components – Measure transportation – Dimension reduction

Abstract

Traditional Principal and Independent Component Analysis (PCA and ICA) are inherently linear and bidirectional: principal directions, in both cases, are linear combinations defined up to their signs. While this approach is perfectly justified in a linear and symmetric context – essentially, under Gaussian or elliptical symmetry assumptions – a more flexible nonlinear and directional one is more appropriate under more general distributions. Measure transportation is offering the ideal tool for such an extension. Inspired by the measure-transportation-based concepts of Monge-Kantorovich depth and center-outward distribution functions introduced in Chernozhukov et al. [2017] and Hallin et al. [2021], we propose new, nonlinear and directional, notions of principal and independent components (grounded in monotone transports to the uniform over the unit ball ${𝕊}_d$ and to the uniform over the unit cube ${[-1,1]}^d$, respectively). Principal directions, in our approach, are curves originating from a (data-driven) central set (instead of running through some origin) and maximizing the dispersion of appropriate one-sided curvilinear projections; the underlying transports are not necessarily continuous at the center, making one-sidedness a natural feature. Contrary to the classical linear ones, our nonlinear independent components, under absolute continuity assumptions, always exist.