Functional Inverse Regression in an Enlarged Dimension Reduction Space
This work offers a theoretical extension for dimension reduction in functional data analysis, but it appears incremental as it builds on existing concepts without clear empirical validation.
The authors tackled the problem of functional inverse regression by proposing an enlarged dimension reduction space using operator and functional analysis, which provides a unified framework for classical methods like SIR and linear discriminant analysis.
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.