# Learning Elastic Costs to Shape Monge Displacements

arXiv: https://arxiv.org/abs/2306.11895 · HTML: https://mlsanity.com/paper/2306.11895/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 59 references that are in the map.
- First posted on arXiv 20 Jun 2023.

## Builds on (most cited first, 26 of 26)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance](https://mlsanity.com/paper/2012.05199/) (2020)
- [Optimal transport map estimation in general function spaces](https://mlsanity.com/paper/2212.03722/) (2022)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Riemannian Convex Potential Maps](https://mlsanity.com/paper/2106.10272/) (2021)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)

## Built on by (most cited first, 6 of 6)

- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (2023)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Displacement-Sparse Neural Optimal Transport](https://mlsanity.com/paper/2502.01889/) (2025)
- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)
- [From Cone Geometry to Monge Structure: Local Defects and Global Regret in High-Dimensional Optimal Transport](https://mlsanity.com/paper/2606.04695/) (2026)

## Abstract (arXiv)

Given a source and a target probability measure supported on ℝd, the Monge problem asks to find the most efficient way to map one distribution to the other. This efficiency is quantified by defining a textitcost function between source and target data. Such a cost is often set by default in the machine learning literature to the squared-Euclidean distance, ℓ22(x,y)=tfrac12|x-y|22. Recently, Cuturi et. al '23 highlighted the benefits of using elastic costs, defined through a regularizer τ as c(x,y)=ℓ22(x,y)+τ(x-y). Such costs shape the textitdisplacements of Monge maps T, i.e., the difference between a source point and its image T(x)-x), by giving them a structure that matches that of the proximal operator of τ. In this work, we make two important contributions to the study of elastic costs: (i) For any elastic cost, we propose a numerical method to compute Monge maps that are provably optimal. This provides a much-needed routine to create synthetic problems where the ground truth OT map is known, by analogy to the Brenier theorem, which states that the gradient of any convex potential is always a valid Monge map for the ℓ22 cost; (ii) We propose a loss to textitlearn the parameter θ of a parameterized regularizer τθ, and apply it in the case where τA(z)=|A^perp z|22. This regularizer promotes displacements that lie on a low dimensional subspace of ℝd, spanned by the p rows of A∈ℝp× d.