The Cost of Discretization in Functional Linear Regression: Minimax Rates and Adaptation

Abstract

We study scalar-on-function linear regression when each covariate curve is observed only through finitely many noisy point evaluations. Our goal is to characterize the minimax estimation and prediction risks as joint functions of the number of trajectories $n$ and the within-trajectory resolution $m$. Working in a fixed trigonometric eigenbasis, with covariance eigenvalues decaying at rate $α$ and slope function of Sobolev smoothness $s$, we derive matching minimax upper and lower bounds under two canonical sampling schemes. Under an independent random design, the minimax prediction rate is $n^{-\frac{2α+2s}{2α+2s+1}} + (nm)^{-\frac{2α+2s}{4α+2s+1}}$. The first term is the fully observed functional linear regression benchmark, while the second term captures the cost of noisy point evaluations after amplification by the inverse covariance operator. Under a common design on an equally spaced grid, the shared sampling geometry introduces additional obstructions, and the minimax prediction rate becomes $n^{-\frac{2α+2s}{2α+2s+1}} + (nm)^{-\frac{2α+2s}{4α+2s+1}} + m^{-(2α+2s)} + m^{-4α}$. Here the third term represents discretization error induced by the fixed grid, whereas the fourth reflects the cost of identifying unknown eigenvalues from observations on a common grid. We further construct data-driven adaptive estimators that screen the covariance scale and threshold blockwise prediction energy, attaining these rates without prior knowledge of the eigenvalue sequence or the smoothness indices. The results reveal a sharp phase transition that depends on the sampling resolution under independent design and a richer phase diagram under common design. Numerical simulations and a real data example illustrate the theoretical findings.