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Vincent DIVOL (Université Paris Dauphine) – Entropic estimation of optimal transport maps in the semi-discrete case

March 25 @ 2:00 pm - 3:15 pm

Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:15 pm
Date: 25th March 2024
Place : 3001


Vincent DIVOL (Université Paris Dauphine) -“Entropic estimation of optimal transport maps in the semi-discrete case”


Abstract: We study the question of estimating the optimal transport map T between two distributions P and Q, based on i.i.d. samples from the two distributions. This problem has gained a lot of traction since J-C. Hütter and P. Rigollet first proposed an estimator in 2021, based on wavelet decompositions. However, all analyses so far crucially rely on the smoothness of the map T, whereas optimal transport maps T are known to be discontinuous in many practical cases (e.g. when P has a connected support and Q has a disconnected support). As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points. We study a computationally efficient estimator of T based on entropic smoothing, and show that it attains optimal, dimension-free rates of convergence. As a byproduct of our analysis, we give a new stability result for the entropic transport map.