Functional Diffusion Maps
This work addresses the limitation of linear assumptions in functional data analysis for researchers and practitioners, though it is incremental as it adapts an existing method to a new data type.
The authors tackled the problem of dimensionality reduction for functional data, which often lies on nonlinear manifolds, by extending Diffusion Maps to functional data and comparing it to Functional PCA, showing improved performance in simulated and real examples.
Nowadays many real-world datasets can be considered as functional, in the sense that the processes which generate them are continuous. A fundamental property of this type of data is that in theory they belong to an infinite-dimensional space. Although in practice we usually receive finite observations, they are still high-dimensional and hence dimensionality reduction methods are crucial. In this vein, the main state-of-the-art method for functional data analysis is Functional PCA. Nevertheless, this classic technique assumes that the data lie in a linear manifold, and hence it could have problems when this hypothesis is not fulfilled. In this research, attention has been placed on a non-linear manifold learning method: Diffusion Maps. The article explains how to extend this multivariate method to functional data and compares its behavior against Functional PCA over different simulated and real examples.