From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport

@article{Carlier2009FromKT,
  title={From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport},
  author={Guillaume Carlier and Alfred Galichon and Filippo Santambrogio},
  journal={SIAM J. Math. Anal.},
  year={2009},
  volume={41},
  pages={2554-2576}
}
A simple procedure to map two probability measures in Rd is the so-called Knothe-Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a… 

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