A data-adaptive dimension reduction for functional data via penalized low-rank approximation

We introduce a data-adaptive nonparametric dimension reduction tool to obtain a low-dimensional approximation of functional data contami- nated by erratic measurement errors following symmetric or asymmetric distributions. We propose to apply robust submatrix completion tech- niques to matrices consisting of coefficients of basis functions calcu- lated by projecting the observed trajectories onto a given orthogonal basis set. In this process, we use a composite asymmetric Huber loss function to accommodate domain-specific erratic behaviors in a data- adaptive manner. We further incorporate the L1 penalty to regularize the smoothness of latent factor curves. The proposed method can also be applied to partially observed functional data, where each tra- jectory contains individual-specific missing segments. Moreover, since our method does not require estimating the covariance operator, the extension to any dimensional functional data observed over a contin- uum is straightforward. We demonstrate the empirical performance in estimating lower-dimensional space and reconstruction of trajectories of the proposed method through simulation studies. We then apply the proposed method to two real datasets, one-dimensional Advanced Metering Infrastructure (AMI) data and two-dimensional max precip- itation spatial data collected in North America and South America.