# Quadratically regularized optimal transport

arXiv: https://arxiv.org/abs/1903.01112 · HTML: https://mlsanity.com/paper/1903.01112/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2019.

## Built on by (most cited first, 15 of 15)

- [Sparsity-Constrained Optimal Transport](https://mlsanity.com/paper/2209.15466/) (2022)
- [On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error](https://mlsanity.com/paper/2202.03618/) (2022)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)
- [Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport](https://mlsanity.com/paper/2208.00604/) (2022)
- [Bringing regularized optimal transport to lightspeed: a splitting method adapted for GPUs](https://mlsanity.com/paper/2305.18483/) (2023)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)
- [Manifold Learning with Sparse Regularised Optimal Transport](https://mlsanity.com/paper/2307.09816/) (2023)
- [Sliced-Regularized Optimal Transport](https://mlsanity.com/paper/2604.23944/) (2026)
- [Optimal Transport with Adaptive Regularisation](https://mlsanity.com/paper/2310.02925/) (2023)
- [Sparse Partial Optimal Transport via Quadratic Regularization](https://mlsanity.com/paper/2508.08476/) (2025)
- [Quadratically Regularized Optimal Transport: nearly optimal potentials and convergence of discrete Laplace operators](https://mlsanity.com/paper/2212.00103/) (2022)
- [Decentralized and Equitable Optimal Transport](https://mlsanity.com/paper/2403.04259/) (2024)
- [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)

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss-Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).