# The Monge Gap: A Regularizer to Learn All Transport Maps

AuthorsThéo Uscidda, Marco Cuturi

content type paperpublished June 2023

AuthorsThéo Uscidda, Marco Cuturi

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another.
Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the "best" map to morph a continuous measure in

Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on

Given a cost

Optimal transport (OT) theory focuses, among all maps that can morph a probability measure onto another, on those that are the "thriftiest", i.e. such that the averaged cost between and its image be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when is the distance, e.g., using entropic maps (Pooladian and Niles-Weed, 2021), or neural networks (Makkuva et al., 2020;
Korotin et al., 2020)…

See paper detailsOptimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. That theory has been mostly used to estimate, given a pair of source and target probability measures , a parameterized map that can efficiently map onto . In many applications, such as predicting cell responses to treatments, the data measures (features of…

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