An RKHS formulation of the inverse regression dimension-reduction problem

TL;DR

This paper introduces a unified RKHS-based framework for inverse regression dimension reduction, extending finite-dimensional methods to infinite-dimensional stochastic processes with practical computational algorithms.

Contribution

It develops an RKHS formulation that unifies finite and infinite-dimensional inverse regression, enabling new computational approaches.

Findings

RKHS provides a seamless extension from finite to infinite dimensions.

The framework facilitates practical computational algorithms.

It offers a unified approach for diverse models.

Abstract

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 paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of $L_X^2$ 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.