# Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis

arXiv: https://arxiv.org/abs/2605.24644 · HTML: https://mlsanity.com/paper/2605.24644/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 23 May 2026.

## Builds on (most cited first, 20 of 20)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Quadratically-Regularized Optimal Transport on Graphs](https://mlsanity.com/paper/1704.08200/) (2017)
- [Quadratically regularized optimal transport](https://mlsanity.com/paper/1903.01112/) (2019)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications](https://mlsanity.com/paper/2502.12326/) (2025)
- [ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport](https://mlsanity.com/paper/2403.03777/) (2024)
- [Orlicz space regularization of continuous optimal transport problems](https://mlsanity.com/paper/2004.11574/) (2020)
- [Sparsity of Quadratically Regularized Optimal Transport: Scalar Case](https://mlsanity.com/paper/2410.03353/) (2024)
- [Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport](https://mlsanity.com/paper/2208.00604/) (2022)
- [Quadratically Regularized Optimal Transport: Existence and Multiplicity of Potentials](https://mlsanity.com/paper/2404.06847/) (2024)
- [Sparsity of Quadratically Regularized Optimal Transport: Bounds on concentration and bias](https://mlsanity.com/paper/2410.03425/) (2024)
- [Sample Complexity of Quadratically Regularized Optimal Transport](https://mlsanity.com/paper/2511.09807/) (2025)

## Abstract (arXiv)

Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure. Despite recent progress in one-dimensional settings and general upper bounds, fundamental questions about the localization rate of QOT optimizers around the Monge coupling have remained open. In this work, we establish a general lower bound showing that the support of the QOT optimizer cannot concentrate around the Monge graph faster than order varepsilon^frac1d+2 in the directed Hausdorff distance, matching the conjectured optimal exponent under standard regularity assumptions in citetwiesel2025sparsity. We also show that the QOT value gap controls the mean-squared deviation mathbb Eπ_varepsilon|y-T(x)|2 by the scale of varepsilon^frac2d+2. As a corollary, in the affine Brenier regime, which includes Gaussian-to-Gaussian transport, we derive a sharp pointwise tube bound of order varepsilon^frac1d+2 by reducing the problem to self-transport and applying recent self-transport sparsity results. Finally, we validate our theoretical bound with a synthetic experiment in high-dimensional settings.