  In recent years, manifold methods have moved into focus as tools for
dimension reduction. Assuming that the high-dimensional data actually lie on or
close to a low-dimensional nonlinear manifold, these methods have shown
convincing results in several settings. This manifold assumption is often
reasonable for functional data, i.e., data representing continuously observed
functions, as well. However, the performance of manifold methods recently
proposed for tabular or image data has not been systematically assessed in the
case of functional data yet. Moreover, it is unclear how to evaluate the
quality of learned embeddings that do not yield invertible mappings, since the
reconstruction error cannot be used as a performance measure for such
representations. In this work, we describe and investigate the specific
challenges for nonlinear dimension reduction posed by the functional data
setting. The contributions of the paper are three-fold: First of all, we define
a theoretical framework which allows to systematically assess specific
challenges that arise in the functional data context, transfer several
nonlinear dimension reduction methods for tabular and image data to functional
data, and show that manifold methods can be used successfully in this setting.
Secondly, we subject performance assessment and tuning strategies to a thorough
and systematic evaluation based on several different functional data settings
and point out some previously undescribed weaknesses and pitfalls which can
jeopardize reliable judgment of embedding quality. Thirdly, we propose a
nuanced approach to make trustworthy decisions for or against competing
nonconforming embeddings more objectively.
