  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.
