  In this work, we develop a new theory and method for sufficient dimension
reduction (SDR) in single-index models, where SDR is a sub-field of supervised
dimension reduction based on conditional independence. Our work is primarily
motivated by the recent introduction of the Hellinger correlation as a
dependency measure. Utilizing this measure, we develop a method capable of
effectively detecting the dimension reduction subspace, complete with
theoretical justification. Through extensive numerical experiments, we
demonstrate that our proposed method significantly enhances and outperforms
existing SDR methods. This improvement is largely attributed to our proposed
method's deeper understanding of data dependencies and the refinement of
existing SDR techniques.
