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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$\xi_1,...,\xi_p$ which belong to the closed span $L_X^2$ of $X$. This paperinvestigates a unified framework for the inverse regression dimension-reductionproblem. It is found that the identification of $L_X^2$ with the reproducingkernel Hilbert space of $X$ provides a platform for a seamless extension fromthe finite- to infinite-dimensional settings. It also facilitates convenientcomputational algorithms that can be applied to a variety of models.

An RKHS formulation of the inverse regression dimension-reduction problem