Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.

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.

Do q in Line 22 and r in Line 185 denote the same thing? Under what conditions does the equivalence hold? Do these conditions automatically hold for this paper? Is the original dimensionality prohibits the evaluation? Unfortunately, the authors dodged most of my questions that may hurt the paper and my concerns still stand.

Two reviewers support accept, and two reviewers indicate reject. I've looked at the paper, reviews and rebuttal, and recommend an accept decision based on the significance, theoretical grounding and novelty of the contribution. It is refreshing to see a new approach to a problem relevant to the NeurIPS community. However, please revise the paper to provide better explanation of terminology and intuition as to why the method works; the reviewers' additional feedback and post-rebuttal comments should also be carefully considered in the revisions.

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport.