Statistical Properties of the rectified flow
Speaker: Gonzalo Mena
Abstract: Transport maps are important statistical objects, chief among them the optimal transport map. A nice nonparametric theory on estimation of optimal transport maps have recently been developed, showing that estimation can be performed stlightly faster than the usual non-parametric rate, as a consequence of the smoothness property of the optimal transport. However, computational challenges prevent us from computing the optimal transport efficiently, and people have suggested the rectified flow as a cheaper surrogate. This estimator can be estimated via non-parametric regression. In this talk I will introduce the rectified transport framework and discuss its smoothness properties in bounded and unbounded domains. Based on that I will comment on a bias and variance analysis of the associated non-parametric estimator. This also enjoys benefits over the usual kernel regression estimator