After Rebuttal Thank you for the response and the clarifications. We still have some concerns about the complexity of the algorithm and its dependence on the iteration count -- in particular in cases where this may scale with the dimension d based on the specified termination criteria. This being said, it would be great to comment on this in the numerical results and include more timing studies to confirm this dependence. We also strongly suggest that the authors include the extensions listed in the improvement in the final revision. Before Rebuttal The authors present a novel algorithm with both theoretical analysis and empirical results. We have a few comments and suggestions for the work: In the introduction, we recommend that the authors also note alternative versions of finding OTM that are not based on solving a linear program (including non-discrete versions of OT based on solving ODEs or finding parametrized maps).

This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most informative'' projection direction in each iteration.

This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most informative'' projection direction in each iteration.