FUNCTIONAL LINEAR REGRESSION FROM SPARSE TO DENSE DESIGNS: A POOLING-RIDGE METHOD AND MINIMAX OPTIMALITY

Functional Data Analysis (FDA) often deals with data measured at discrete, noisy time points rather than as fully observed continuous functions. While regression models for fully observed functional data are well-understood, optimal estimation for discretely observed data—especially in sparse sampling regimes—has remained an open challenge for over two decades. This paper seeks to bridge the gap between idealized fully observed models and realistic, sparse, noisy observations by developing a unified estimation framework.

The authors propose a pooling-ridge estimator that combines the strengths of Reproducing Kernel Hilbert Space (RKHS) methods with a pooling strategy. Instead of pre-smoothing individual curves—which introduces significant bias in sparse settings—the method pools discrete measurements from all subjects to construct unbiased estimators of the relevant operators. This unified framework is applicable to both scalar-on-function and function-on-function regression models and is designed to handle arbitrary sampling frequencies.

The study establishes the minimax optimal convergence rates for functional linear regression under discrete sampling. The results reveal distinct phase transition phenomena: in scalar-on-function regression, a single phase transition separates the convergence behavior into two regimes based on sampling frequency. For function-on-function regression, the model exhibits up to three phase transitions, determined by the sampling frequencies of both the predictor and response functions. The authors demonstrate that their pooling-ridge estimator achieves these optimal rates, effectively overcoming the "curse of smoothness" where high-frequency information becomes harder to capture as the predictor process becomes smoother.

This work provides the first comprehensive theoretical treatment of functional linear regression with discrete observations, offering a rigorous foundation for practitioners. By accurately localizing phase transition boundaries, the paper clarifies the minimum sampling requirements needed to achieve the same performance as fully observed models. The proposed method is shown to be more stable and accurate than existing pre-smoothing or FPCA-based approaches, particularly in sparse data scenarios common in longitudinal studies.