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

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.