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.

翻译：传统函数线性回归通常以一维函数预测变量作为输入，并估计连续的系数函数。现代应用常产生二维协变量，在网格点观测时表现为矩阵形式。为避免传统方法估计二维系数函数的低效性，我们提出函数双线性回归模型，并创新性地引入三项惩罚项以在估计过程中施加粗糙度惩罚。所提出的估计量在再生核希尔伯特空间框架下具有预测的极小化最优性质。我们开发了迭代广义交叉验证方法选择调优参数，该方法相比传统交叉验证方法显著提升了计算效率。通过模拟实验、加拿大气象数据以及生化远红外激光雷达数据的对比分析，进一步验证了所提方法相较于现有方法的统计与计算优势。