Functional Principal Subspace Sampling for Large Scale Functional Data Analysis

TL;DR

This paper introduces randomized algorithms for scalable functional principal component analysis and functional linear regression, utilizing a novel importance sampling method to efficiently estimate functional subspaces in large datasets.

Contribution

The paper proposes a functional principal subspace sampling method that improves scalability of FDA techniques by reducing computational costs while maintaining accuracy.

Findings

Algorithms effectively handle large datasets with low intrinsic dimension.

The importance sampling approach accurately estimates functional subspaces.

Experimental results demonstrate improved efficiency and accuracy on synthetic and real data.

Abstract

Functional data analysis (FDA) methods have computational and theoretical appeals for some high dimensional data, but lack the scalability to modern large sample datasets. To tackle the challenge, we develop randomized algorithms for two important FDA methods: functional principal component analysis (FPCA) and functional linear regression (FLR) with scalar response. The two methods are connected as they both rely on the accurate estimation of functional principal subspace. The proposed algorithms draw subsamples from the large dataset at hand and apply FPCA or FLR over the subsamples to reduce the computational cost. To effectively preserve subspace information in the subsamples, we propose a functional principal subspace sampling probability, which removes the eigenvalue scale effect inside the functional principal subspace and properly weights the residual. Based on the operator perturbation analysis, we show the proposed probability has precise control over the first order error of the subspace projection operator and can be interpreted as an importance sampling for functional subspace estimation. Moreover, concentration bounds for the proposed algorithms are established to reflect the low intrinsic dimension nature of functional data in an infinite dimensional space. The effectiveness of the proposed algorithms is demonstrated upon synthetic and real datasets.