# Quadratically Regularized Optimal Transport: nearly optimal potentials and convergence of discrete Laplace operators

arXiv: https://arxiv.org/abs/2212.00103 · HTML: https://mlsanity.com/paper/2212.00103/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 11 references that are in the map.
- First posted on arXiv 20 Nov 2022.

## Builds on (most cited first, 5 of 5)

- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Dynamical Optimal Transport on Discrete Surfaces](https://mlsanity.com/paper/1809.07083/) (2018)
- [Quadratically regularized optimal transport](https://mlsanity.com/paper/1903.01112/) (2019)
- [Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport](https://mlsanity.com/paper/2208.00604/) (2022)

## Abstract (arXiv)

We consider the conjecture proposed in Matsumoto, Zhang and Schiebinger (2022) suggesting that optimal transport with quadratic regularisation can be used to construct a graph whose discrete Laplace operator converges to the Laplace--Beltrami operator. We derive first order optimal potentials for the problem under consideration and find that the resulting solutions exhibit a surprising resemblance to the well-known Barenblatt--Prattle solution of the porous medium equation. Then, relying on these first order optimal potentials, we derive the pointwise L2-limit of such discrete operators built from an i.i.d. random sample on a smooth compact manifold. Simulation results complementing the limiting distribution results are also presented.