Optimum transport (OT) idea has been been utilized in machine studying to review and characterize maps that may push-forward effectively a likelihood measure onto one other.
Latest works have drawn inspiration from Brenier’s theorem, which states that when the bottom price is the squared-Euclidean distance, the “finest” map to morph a steady measure in into one other have to be the gradient of a convex perform.
To take advantage of that outcome, , Makkuva et al. (2020); Korotin et al. (2020) think about maps , the place is an enter convex neural community (ICNN), as outlined by Amos et al. 2017, and match with SGD utilizing samples.
Regardless of their mathematical class, becoming OT maps with ICNNs raises many challenges, due notably to the various constraints imposed on ; the necessity to approximate the conjugate of ; or the limitation that they solely work for the squared-Euclidean price. Extra usually, we query the relevance of utilizing Brenier’s outcome, which solely applies to densities, to constrain the structure of candidate maps fitted on samples.
Motivated by these limitations, we suggest a radically totally different method to estimating OT maps:
Given a price and a reference measure , we introduce a regularizer, the Monge hole of a map . That hole quantifies how far a map deviates from the perfect properties we anticipate from a -OT map. In apply, we drop all structure necessities for and easily decrease a distance (e.g., the Sinkhorn divergence) between and , regularized by . We examine , and present how our easy pipeline outperforms considerably different baselines in apply.