Linear Contour Learning: A Method for Supervised Dimension Reduction

TL;DR

This paper introduces General Contour Regression (GCR), a new supervised dimension reduction method that estimates the central space by identifying contour directions, offering robustness and efficiency over existing techniques.

Contribution

The paper presents GCR, a novel contour-based approach for sufficient dimension reduction that guarantees exhaustive estimation under mild assumptions and demonstrates robustness to non-elliptic predictor distributions.

Findings

GCR outperforms traditional methods like OLS and SIR in simulations.

GCR is computationally efficient and robust to departures from ellipticity.

Application to student grades data illustrates practical utility.

Abstract

We propose a novel approach to sufficient dimension reduction in regression, based on estimating contour directions of negligible variation for the response surface. These directions span the orthogonal complement of the minimal space relevant for the regression, and can be extracted according to a measure of the variation in the response, leading to General Contour Regression(GCR). In comparison to exiisting sufficient dimension reduction techniques, this sontour-based mothology guarantees exhaustive estimation of the central space under ellipticity of the predictoor distribution and very mild additional assumptions, while maintaining vn-consisytency and somputational ease. Moreover, it proves to be robust to departures from ellipticity. We also establish some useful population properties for GCR. Simulations to compare performance with that of standard techniques such as ordinary least squares, sliced inverse regression, principal hessian directions, and sliced average variance estimation confirm the advntages anticipated by theoretical analyses. We also demonstrate the use of contour-based methods on a data set concerning grades of students from Massachusetts colleges.