New forest-based approaches for sufficient dimension reduction

Sufficient dimension reduction (SDR) primarily aims to reduce the dimensionality of high-dimensional predictor variables while retaining essential information about the responses. Traditional SDR methods typically employ kernel weighting functions, which unfortunately makes them susceptible to the curse of dimensionality. To address this issue, we in this paper propose novel forest-based approaches for SDR that utilize a locally adaptive kernel generated by Mondrian forests. Overall, our work takes the perspective of Mondrian forest as an adaptive weighted kernel technique for SDR problems. In the central mean subspace model, by integrating the methods from Xia et al. (J R Stat Soc Ser B (Stat Methodol) 64(3):363–410, 2002. https://doi.org/10.1111/1467-9868.03411) with Mondrian forest weights, we suggest the forest-based outer product of gradients estimation (mf-OPG) and the forest-based minimum average variance estimation (mf-MAVE). Moreover, we substitute the kernels used in nonparametric density function estimations (Xia in Ann Stat 35(6):2654–2690, 2007. https://doi.org/10.1214/009053607000000352), targeting the central subspace, with Mondrian forest weights. These techniques are referred to as mf-dOPG and mf-dMAVE, respectively. Under regularity conditions, we establish the asymptotic properties of our forest-based estimators, as well as the convergence of the affiliated algorithms. Through simulation studies and analysis of fully observable data, we demonstrate substantial improvements in computational efficiency and predictive accuracy of our proposals compared with the traditional counterparts.