Suppose that Y is a scalar and X is a second-order stochastic process, where Y and X are conditionally independent given the random variables ξ1, …, ξp which belong to the closed span LX2 of X. This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of LX2 with the reproducing kernel Hilbert space of X provides a platform for a seamless extension from the finite- to infinite-dimensional settings. It also facilitates convenient computational algorithms that can be applied to a variety of models.
"An RKHS formulation of the inverse regression dimension-reduction problem." Ann. Statist. 37 (2) 726 - 755, April 2009. https://doi.org/10.1214/07-AOS589