Quantitative stability of optimal transport via regularization
Abstract
We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an Lp density, p geqslant 1, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the W2 Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the Lp norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For Lp changes of Gaussian measure, the factor in the bounded-target estimate is d(p−1)/(4p−2). For each p, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.
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BibTeX
@misc{wang2026quantitative,
title = {{Quantitative stability of optimal transport via regularization}},
author = {Songbo Wang and Xiaozhen Wang},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09640},
eprint = {2610.09640},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05782900v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantitative stability of optimal transport via regularization
AU - Songbo Wang
AU - Xiaozhen Wang
PY - 2026
DA - 2026/10/07
PB - HAL
DO - 10.48550/arxiv.2610.09640
UR - https://hal.science/hal-05782900v1
AB - We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an L^p density, p geqslant 1, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the W_2 Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the L^p norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For L^p changes of Gaussian measure, the factor in the bounded-target estimate is d^(p−1)/(4p−2). For each p, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.
ER -