Functional Inverse Regression in an Enlarged Dimension Reduction Space

TL;DR

This paper introduces an enlarged dimension reduction space for functional inverse regression, providing a unified, operator-based framework that extends classical methods to a broader functional setting.

Contribution

It develops a rigorous operator and functional analysis framework for functional inverse regression in an enlarged space, unifying classical techniques within this new setting.

Findings

Framework facilitates classical methods like SIR in the enlarged space

Links to linear discriminant analysis of Gaussian measures

Enables more flexible dimension reduction functions

Abstract

We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions belong. Our formulation provides a unified framework so that the classical notions, such as covariance standardization, Mahalanobis distance, SIR and linear discriminant analysis, can be naturally and smoothly carried out in our enlarged space. This enlarged dimension reduction space also links to the linear discriminant space of Gaussian measures on a separable Hilbert space.