Online Functional Principal Component Analysis on a Multidimensional Domain

TL;DR

This paper introduces an efficient online method for multidimensional functional principal component analysis using tensor product splines, Riemannian stochastic gradient descent, and adaptive smoothing, suitable for streaming data.

Contribution

It presents a novel online framework with a tensor spline representation, Riemannian optimization, and dynamic smoothing parameter tuning for scalable multidimensional functional data analysis.

Findings

Demonstrates effectiveness through simulations and real data applications.

Achieves scalable and smooth modeling of streaming multidimensional functional data.

Provides a flexible approach adaptable to various scientific fields.

Abstract

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. An efficient Riemannian stochastic gradient descent algorithm is developed, with extensions inspired by adaptive moment estimation and averaging techniques to 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. Extensive simulations and real-world applications demonstrate the flexibility and effectiveness of this framework for analyzing multidimensional functional data.