Nonlinear dimension reduction for functional data with application to clustering

This is the authors' abstract. We don't add key points for this paper.

Functional data often possess nonlinear structures, e.g. phase variation, for which linear dimension reduction techniques can be ineffective. We study nonlinear dimension reduction for functional data based on the assumption that the data lie on an unknown manifold contaminated with noise. A recently developed manifold learning method designed for high dimensional data is generalized into our context. Asymptotic convergence results taking noise into account are derived. We find in synthetic examples that it often produces more accurate geodesic distance estimation than the traditional functional Isomap. We further develop a clustering strategy based on the manifold learning outcomes, and demonstrate that our method can perform much better than others if the data indeed lie on a curved manifold. Two real data examples are presented for illustration.