# Sparsity of Quadratically Regularized Optimal Transport: Scalar Case

arXiv: https://arxiv.org/abs/2410.03353 · HTML: https://mlsanity.com/paper/2410.03353/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2024.

## Built on by (most cited first, 4 of 4)

- [Displacement-Sparse Neural Optimal Transport](https://mlsanity.com/paper/2502.01889/) (2025)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)
- [Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis](https://mlsanity.com/paper/2605.24644/) (2026)

## Abstract (arXiv)

The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling πvarepsilon has sparse support for small regularization parameter varepsilon, in contrast to entropic regularization whose solutions have full support for any varepsilon>0. Focusing on continuous and scalar marginals, we provide the first precise description of this sparsity. Namely, we show that the support of πvarepsilon shrinks to the Monge graph at the sharp rate varepsilon1/3. This result is based on a detailed analysis of the dual potential fvarepsilon for small varepsilon. In particular, we prove that fvarepsilon is twice differentiable a.s. and bound the second derivative uniformly in varepsilon, showing that fvarepsilon is uniformly strongly convex. Convergence rates for fvarepsilon and its derivative are also obtained.