Abstract: Functional data often possess nonlinear structures, for example, 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. We generalize a recently developed manifold learning method designed for high-dimensional data into our context, and derive asymptotic convergence results, taking noise into account. The results based on synthetic examples often produce more accurate geodesic distance estimations than those of the traditional functional Isomap method. We further develop a clustering strategy based on the manifold learning outcomes, and demonstrate that our method outperforms others if the data lie on a curved manifold. Two real-data examples are presented for illustration.

Key words and phrases: Geodesic distance, graph clustering, manifold learning, measurement error.