Nonlinear manifold representations for functional data

TL;DR

This paper introduces nonlinear manifold-based representations for functional data, providing a new way to analyze data on unknown low-dimensional nonlinear spaces, with improved performance over traditional methods.

Contribution

It develops manifold learning methods tailored for functional data, defining new concepts like manifold mean and functional manifold components, with theoretical validation.

Findings

Manifold mean and components outperform traditional methods in simulations.

Proposed estimators are consistent under certain conditions.

Applications demonstrate the effectiveness of nonlinear manifold representations.

Abstract

For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. Our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods, which we modify to address functional data settings. In simulations and applications, we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross-sectional mean and functional principal components. We also include consistency proofs for our estimators under certain assumptions.