  Traditional functional linear regression usually takes a one-dimensional
functional predictor as input and estimates the continuous coefficient
function. Modern applications often generate two-dimensional covariates, which
become matrices when observed at grid points. To avoid the inefficiency of the
classical method involving estimation of a two-dimensional coefficient
function, we propose a functional bilinear regression model, and introduce an
innovative three-term penalty to impose roughness penalty in the estimation.
The proposed estimator exhibits minimax optimal property for prediction under
the framework of reproducing kernel Hilbert space. An iterative generalized
cross-validation approach is developed to choose tuning parameters, which
significantly improves the computational efficiency over the traditional
cross-validation approach. The statistical and computational advantages of the
proposed method over existing methods are further demonstrated via simulated
experiments, the Canadian weather data, and a biochemical long-range infrared
light detection and ranging data.
