Supplementary Material to " Sufficient dimension reduction for classification using principal optimal transport direction "

X. Without loss of generality, we assume S(B) = S Hence, to prove Theorem 1, it is sufficient to show that S(B) = S(Σ) holds. To verify S(B) = S(Σ), we only need to show the following two results hold: (I). We now begin with the statement (I). This completes the proof for Statement I. We then turn to Statement II. This leads to a contradiction with (H.2) where the structure dimension is r.