  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.
