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且颇具挑战。本文提出一种名为函数型无切片逆回归(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——一种为刻画函数型预测变量对多元响应的条件均值独立性而引入的新型度量。我们同时给出了FSFIR估计量的具体收敛速率。与现有的函数型切片逆回归方法相比，FSFIR无需选择切片数。仿真实验验证了FSFIR的高效性与便捷性。