The Monge Hole: A Regularizer to Be taught All Transport Maps

Optimum transport (OT) idea has been been utilized in machine studying to check and characterize maps that may push-forward effectively a likelihood measure onto one other.
Current works have drawn inspiration from Brenier’s theorem, which states that when the bottom value is the squared-Euclidean distance, the “finest” map to morph a steady measure in into one other should be the gradient of a convex operate.
To use that outcome, , Makkuva et al. (2020); Korotin et al. (2020) contemplate 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 magnificence, becoming OT maps with ICNNs raises many challenges, due notably to the numerous constraints imposed on ; the necessity to approximate the conjugate of ; or the limitation that they solely work for the squared-Euclidean value. 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 strategy 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 observe, we drop all structure necessities for and easily decrease a distance (e.g., the Sinkhorn divergence) between and , regularized by . We research , and present how our easy pipeline outperforms considerably different baselines in observe.