Forest-based Approaches for SDR

Abstract Sufficient dimension reduction (SDR) primarily aims to reduce the dimensional-ity of high-dimensional predictor variables while retaining essential informationabout the responses. Traditional SDR methods generally employ kernel weightingfunctions, which unfortunately makes them prone to the curse of dimensional-ity. To address this issue, this paper applies Random Forests as an adaptiveweighted technique for SDR problems, thereby resulting in both improved computational efficiency and predictive accuracy. In the central mean subspace model,by combining the study of Xia et al. (2002) with Mondrian forest weights, wesuggest the forest-based outer product of gradients estimation (mf-OPG) andthe forest-based minimum average variance estimation (mf-MAVE). Moreover, we implement the forest-based techniques (mf-dOPG and mf-dMAVE) to estimate the central subspace based on the study of Xia (2007). Under regularityconditions, we establish the asymptotic properties of the forest-based estimators, as well as the convergence of the affiliated algorithms. The practical efficacy of our proposals is demonstrated via simulation studies and empirical data analysis.Moreover, for a more comprehensive representation, we compare the numericalperformance of forest-based approaches with tree-based methodologies.