Geometry and Convergence of Quadratically Regularized Optimal Transport I

@misc{gonzalezsanz2026geometry,
  title = {{Geometry and Convergence of Quadratically Regularized Optimal Transport I}},
  author = {Alberto González-Sanz and Marcel Nutz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.20400},
  eprint = {2609.20400},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20400},
}

TY  - UNPB
TI  - Geometry and Convergence of Quadratically Regularized Optimal Transport I
AU  - Alberto González-Sanz
AU  - Marcel Nutz
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.20400
UR  - https://arxiv.org/abs/2609.20400
AB  - We establish sharp convergence rates for quadratically regularized optimal transport with quadratic cost in the regime of small regularization ε. In particular, we quantify the sparsity of the support of the regularized optimizer. For smooth marginal densities in ℝ^d, this support lies within a distance of order ε^1/(d+2) from the Brenier graph, and every section of the support is sandwiched between balls with radii of that order. The geometry of the support is closely linked to the dual potentials. We show that the potentials satisfy uniform two-sided Hessian bounds and converge uniformly at rate ε^2/(d+2), while their gradients converge at rate ε^1/(d+2). The rates are sharp, and we further identify the regularity threshold where convergence breaks down.
ER  -