Functional Slicing-free Inverse Regression via Martingale Difference Divergence Operator

TL;DR

This paper introduces FSFIR, a slicing-free method for functional sufficient dimension reduction that leverages martingale difference divergence, eliminating the need for slice number selection and demonstrating improved efficiency.

Contribution

The paper proposes a novel FSFIR method based on martingale difference divergence, removing the slice scheme requirement in functional inverse regression.

Findings

FSFIR effectively estimates the central subspace without slicing.

The method achieves a specific convergence rate.

Simulations show FSFIR's efficiency and convenience.

Abstract

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.