A Note on Sliced Inverse Regression with Regularizations

TL;DR

This paper examines the theoretical properties of the ridge SIR estimator, revealing that the associated minimization problem is degenerate, leading to either non-existence or a zero estimator, contrasting with its practical effectiveness.

Contribution

It provides a theoretical analysis showing the minimization problem for ridge SIR is degenerate, a novel insight into its mathematical properties.

Findings

The minimization problem is degenerate, with only two possible outcomes.

The ridge SIR estimator either does not exist or is zero.

Contrasts practical performance with theoretical limitations.

Abstract

In "Li, L. and Yin, X. (2008). Sliced Inverse Regression with Regularizations. Biometrics, 64(1):124--131" a ridge SIR estimator is introduced as the solution of a minimization problem and computed thanks to an alternating least-squares algorithm. This methodology reveals good performance in practice. In this note, we focus on the theoretical properties of the estimator. Is it shown that the minimization problem is degenerated in the sense that only two situations can occur: Either the ridge SIR estimator does not exist or it is zero.