Geometry and Convergence of Quadratically Regularized Optimal Transport II

@misc{gonzalezsanz2026geometry,
  title = {{Geometry and Convergence of Quadratically Regularized Optimal Transport II}},
  author = {Alberto González-Sanz and Marcel Nutz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02077},
  eprint = {2610.02077},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02077},
}

TY  - UNPB
TI  - Geometry and Convergence of Quadratically Regularized Optimal Transport II
AU  - Alberto González-Sanz
AU  - Marcel Nutz
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02077
UR  - https://arxiv.org/abs/2610.02077
AB  - We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization ε, where the support of the optimal coupling is sparse. For smooth marginal densities in ℝ^d, we show that the support contains the graph of the Brenier map. After centering by the Brenier image and rescaling by ℓ=ε^1/(d+2), the sections of the support have explicit ellipsoidal limits for ε→0, except at boundary points, where the limits are half-space profiles. The optimal dual potentials admit expansions at order ℓ^2, uniformly up to the boundary. We identify the leading coefficients, which consist of a common local profile and opposite global corrections determined by a linear Neumann problem. Finally, we analyze two approximations to the Brenier map, namely the gradient of the dual potential and the conditional mean of the coupling. We obtain the sharp L^p rate ℓ^1+1/p with exact leading constants and further identify the leading interior and boundary biases. Taken together, our results illustrate that quadratic regularization induces an accurate sparse approximation of classical optimal transport.
ER  -