Fréchet kernel sliced inverse regression

Projection expectile regression for sufficient dimension reduction

“…This motivated the development of Fréchet dimension reduction which extends existing traditional sufficient dimension reduction (SDR) methods to random response objects in non-Euclidean metric spaces such as probability distributions, symmetric positive definite matrices, and spheres. See Zhang et al (2021) and Dong & Wu (2022) for examples. The goal of the Fréchet SDR is to find the smallest set of linear combinations of X, which captures the relevant information in X needed to predict Y without loss of information.…”

“…where " ⊥ ⊥ " means statistical independence and P S denotes the projection onto S with respect to the inner product in R p . Cook (1996) and Yin et al (2008) can be extended to show that under some mild conditions, a unique S with the smallest column space exists, Both Zhang et al (2021) and Dong & Wu (2022) proposed to use universal kernels…”