Structural adaptation via directional regularity: rate accelerated estimation in multivariate functional data

Abstract: We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view which determines anisotropy as a notion of smoothness along a dimension, directional regularity additionally views anisotropy through the lens of directions. We show that faster rates of convergence can be obtained through a change-of-basis by adapting to the directional regularity of a multivariate process. An algorithm for the estimation and identification of the change-of-basis matrix is constructed, made possible due to the replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. Possible applications of the directional regularity approach are discussed, and we advocate its consideration as a standard pre-processing step in multivariate functional data analysis.