Functional sliced inverse regression (FSIR) is one of the most popular algorithms for functional sufficient dimension reduction (FSDR). However, the choice of slice scheme in FSIR is critical but challenging. In this paper, we propose a new method called functional slicing-free inverse regression (FSFIR) to estimate the central subspace in FSDR. FSFIR is based on the martingale difference divergence operator, which is a novel metric introduced to characterize the conditional mean independence of a functional predictor on a multivariate response. We also provide a specific convergence rate for the FSFIR estimator. Compared with existing functional sliced inverse regression methods, FSFIR does not require the selection of a slice number. Simulations demonstrate the efficiency and convenience of FSFIR.

翻译：函数型切片逆回归（FSIR）是函数型充分降维（FSDR）中最常用的算法之一。然而，FSIR中切片方案的选择至关重要但困难重重。本文提出一种名为无切片函数型逆回归（FSFIR）的新方法，用于估计FSDR中的中心子空间。FSFIR基于鞅差散度算子，该算子是一种新型度量，用于刻画函数型预测变量对多元响应的条件均值独立性。我们还给出了FSFIR估计量的具体收敛速度。与现有函数型切片逆回归方法相比，FSFIR无需选择切片数量。模拟实验证明了FSFIR的有效性和便捷性。