We consider the functional linear regression model with a scalar response and a Hilbert space-valued predictor, a well-known ill-posed inverse problem. We propose a new formulation of the functional partial least-squares (PLS) estimator related to the conjugate gradient method. We shall show that the estimator achieves the (nearly) optimal convergence rate on a class of ellipsoids and we introduce an early stopping rule which adapts to the unknown degree of ill-posedness. Some theoretical and simulation comparison between the estimator and the principal component regression estimator is provided.

翻译：本文考虑标量响应与希尔伯特空间值预测变量的函数线性回归模型，这是一个经典的病态逆问题。我们提出了一种与共轭梯度法相关的函数型偏最小二乘（PLS）估计量的新形式。我们将证明该估计量在一类椭球上达到了（近乎）最优收敛速度，并引入了一种能够适应未知病态程度的早期停止准则。本文还提供了该估计量与主成分回归估计量之间的理论与模拟比较。