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This paper studies the estimation of large-scale optimal transport maps(OTM), which is a well-known challenging problem owing to the curse ofdimensionality. Existing literature approximates the large-scale OTM by aseries of one-dimensional OTM problems through iterative random projection.Such methods, however, suffer from slow or none convergence in practice due tothe nature of randomly selected projection directions. Instead, we propose anestimation method of large-scale OTM by combining the idea of projectionpursuit regression and sufficient dimension reduction. The proposed method,named projection pursuit Monge map (PPMM), adaptively selects the most``informative'' projection direction in each iteration. We theoretically showthe proposed dimension reduction method can consistently estimate the most``informative'' projection direction in each iteration. Furthermore, the PPMMalgorithm weakly convergences to the target large-scale OTM in a reasonablenumber of steps. Empirically, PPMM is computationally easy and converges fast.We assess its finite sample performance through the applications of Wassersteindistance estimation and generative models.

Large-scale optimal transport map estimation using projection pursuit