# From Optimal Transport to Discrepancy

arXiv: https://arxiv.org/abs/2002.01189 · HTML: https://mlsanity.com/paper/2002.01189/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2020.

## Built on by (most cited first, 7 of 7)

- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [On the Effect of Initialization: The Scaling Path of 2-Layer Neural Networks](https://mlsanity.com/paper/2303.17805/) (2023)
- [Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels](https://mlsanity.com/paper/2402.08425/) (2024)
- [Soft Quantization using Entropic Regularization](https://mlsanity.com/paper/2309.04428/) (2023)
- [Arbitrary Distributions Mapping via SyMOT-Flow: A Flow-based Approach Integrating Maximum Mean Discrepancy and Optimal Transport](https://mlsanity.com/paper/2308.13815/) (2023)
- [Gromov-Wasserstein Quantization and Clustering: Structure, Rates, and Algorithms](https://mlsanity.com/paper/2608.11016/) (2026)

## Abstract (arXiv)

A common way to quantify the ,,distance'' between measures is via their discrepancy, also known as maximum mean discrepancy (MMD). Discrepancies are related to Sinkhorn divergences S_varepsilon with appropriate cost functions as varepsilon → ∞. In the opposite direction, if varepsilon → 0, Sinkhorn divergences approach another important distance between measures, namely the Wasserstein distance or more generally optimal transport ,,distance''. In this chapter, we investigate the limiting process for arbitrary measures on compact sets and Lipschitz continuous cost functions. In particular, we are interested in the behavior of the corresponding optimal potentials hat varphi_varepsilon, hat psi_varepsilon and hat varphiK appearing in the dual formulation of the Sinkhorn divergences and discrepancies, respectively. While part of the results are known, we provide rigorous proofs for some relations which we have not found in this generality in the literature. Finally, we demonstrate the limiting process by numerical examples and show the behavior of the distances when used for the approximation of measures by point measures in a process called dithering.