Abstract: 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. We theoretically show
the proposed dimension reduction method can consistently estimate the most
``informative'' projection direction in each iteration. Furthermore, the PPMM
algorithm weakly convergences to the target large-scale OTM in a reasonable
number of steps. Empirically, PPMM is computationally easy and converges fast.
We assess its finite sample performance through the applications of Wasserstein
distance estimation and generative models.