Advances in nonlinear sufficient dimension reduction: theory and implementation via the nsdr R package

dr package is a widely used tool to implement linear sufficient dimension reduction (SDR), which is a valuable statistical method used to extract core information from a high-dimensional data. However, due to the increasing complexity of big data, especially in cases involving nonlinear structure, some important features of the data may not be fully explained by the linear SDR methods. In such cases, nonlinear SDR methods can be an alternative to address this issue effectively by capturing more flexible and intricate relationships that linear methods may fail to detect. Despite its potential, the theoretical formulation of nonlinear SDR relies on the linear operators in Hilbert space, which hampers many users from applying such nonlinear methods to real data analysis. To address this diffculty, the recently developed nsdr package offers an accessible and userfriendly implementation of nonlinear sufficient dimension reduction techniques in R. In this paper, we compare the theoretical background and numerical results between linear and nonlinear SDR methods by using a widely used dr package and the newly introduced nsdr package. We further demonstrate the advantages of nonlinear SDR methods through a classification task using the wine cultivar dataset, showing that the methods can improve classification performance, offering deeper insight into complex data.