Unsupervised Functional Data Analysis via Nonlinear Dimension Reduction
This work addresses the challenge of applying and evaluating nonlinear dimension reduction techniques for functional data, which is relevant for researchers and practitioners working with continuous observational data.
This paper investigates the application of nonlinear dimension reduction methods, originally designed for tabular and image data, to functional data. It establishes a theoretical framework for this transfer and demonstrates the successful use of manifold methods in this new context. The authors also thoroughly evaluate performance assessment and tuning strategies for functional data, identifying weaknesses and proposing a nuanced approach for objective comparison of nonconforming embeddings.
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.