Supersmooth functional data analysis and PCA-preprocessing

• Applicant: Professor Dr. Alexander Meister
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 460867398

In this project, we study nonparametric regression and classification for functional data when the underlying regression functions and functional densities, respectively, are supersmooth. Based on a PCA-preprocessing technique, we anticipate polynomial convergence rates of the statistical procedures, which is in strong contrast to the usual asymptotics in nonparametric functional data analysis, where typically only logarithmic rates are available. For a thorough understanding of this phenomena, we aim to uncover the information theoretic complexity of this statistical problem and its interplay with the geometry of the underlying spaces. The second key objective is an early stopping procedure, intrinsically requiring our algorithm to learn an optimal selection of appropriate subspaces based on principal component approximations. In this context, we are particularly interested in computationally effective (sequential) procedures, also being robust with respect to heavy tails or dependencies. Moreover, the corresponding functional regression and classification problems shall be considered when no direct access to the data is available due to privacy constraints.

• DFG Programme: Research Units
• Subproject of: FOR 5381: Mathematical Statistics in the Information Age - Statistical Efficiency and Computational Tractability
• International Connection: Austria
• Partner Organisation: Fonds zur Förderung der wissenschaftlichen Forschung (FWF)
• Cooperation Partner: Professor Dr. Moritz Jirak