Online Learning of Functional Principal Component Analysis for Multidimensional Functional Data

Multidimensional functional data streams arise in diverse scientific fields, yet their analysis poses significant challenges. We propose a novel online framework for functional principal component analysis that enables efficient and scalable modeling of such data. Our method represents functional principal components using tensor product splines, enforcing smoothness and orthonormality through a penalized framework on a Stiefel manifold. We develop an efficient Riemannian stochastic gradient descent algorithm and a Riemannian adaptive gradient (AdaGrad) variant, both utilizing iterative averaging techniques to stabilize the estimation and accelerate convergence. Additionally, a dynamic tuning strategy for smoothing parameter selection is developed based on a rolling averaged block validation score that adapts to the streaming nature of the data. Furthermore, we derive the asymptotic normality of the estimators and construct pointwise confidence intervals to quantify the uncertainty of the estimated functional principal components. Extensive simulations and real-world applications demonstrate the flexibility and effectiveness of this framework for analyzing multidimensional functional data.