Minimax Wasserstein Testing under Differential Privacy
Abstract
We study Wasserstein-p goodness-of-fit and two-sample testing under differential privacy. For general compact metric spaces, we establish upper and lower bounds on the minimax separation radii in terms of covering numbers, matching up to logarithmic factors. On the hypercube [0,1]d, these bounds match up to constant factors for two-sample testing in all dimensions and for goodness-of-fit testing whenever d≠ 4p, thereby closing logarithmic gaps in existing non-private results. At the critical case (d,p)=(4,1), we further show that a logarithmic factor is unavoidable by establishing matching minimax bounds for non-private Wasserstein-$1$ goodness-of-fit testing under a bounded-density assumption. Finally, for unbounded domains, we establish minimax-optimal separation rates up to logarithmic factors under an exponential tail condition.
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BibTeX
@misc{lee2026minimax,
title = {{Minimax Wasserstein Testing under Differential Privacy}},
author = {Bonwoo Lee and Ilmun Kim and Cheolwoo Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14432},
eprint = {2610.14432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14432},
}
RIS
TY - UNPB
TI - Minimax Wasserstein Testing under Differential Privacy
AU - Bonwoo Lee
AU - Ilmun Kim
AU - Cheolwoo Park
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14432
UR - https://arxiv.org/abs/2610.14432
AB - We study Wasserstein-p goodness-of-fit and two-sample testing under differential privacy. For general compact metric spaces, we establish upper and lower bounds on the minimax separation radii in terms of covering numbers, matching up to logarithmic factors. On the hypercube [0,1]^d, these bounds match up to constant factors for two-sample testing in all dimensions and for goodness-of-fit testing whenever d≠ 4p, thereby closing logarithmic gaps in existing non-private results. At the critical case (d,p)=(4,1), we further show that a logarithmic factor is unavoidable by establishing matching minimax bounds for non-private Wasserstein-$1$ goodness-of-fit testing under a bounded-density assumption. Finally, for unbounded domains, we establish minimax-optimal separation rates up to logarithmic factors under an exponential tail condition.
ER -