Research map / Learning theory and algorithms / Optimal transport
On the Private Estimation of Smooth Transport Maps
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Proposes an epsilon-differentially private estimator of smooth optimal transport maps via a noisy report-noisy-argmin over a wavelet-based covering of Brenier potentials, with a matching-up-to-a-term minimax lower bound.
Problem
Estimating smooth optimal transport maps (via Brenier potentials) from two i.i.d. samples under differential privacy, with L2(P) error guarantees.
Approach
Combines the empirical semi-dual Brenier potential estimator with wavelet sieve spaces and a report-noisy-argmin mechanism over a finite covering of the admissible potentials, so that privacy is achieved by selecting one candidate potential with Laplace-noised empirical semi-dual scores; the transport map is the gradient of the selected potential.
Key results
- Achieves expected squared L2(P) map error of order n^{-1} ∨ n^{-2α/(2α-2+d)} ∨ (nε)^{-2α/(2α+d)} up to polylog factors (upper bound).
- Minimax lower bound with privacy term (nε)^{-2α/(α-1+d)} that does not match the upper bound's privacy term; the authors conjecture the lower bound is closer to optimal.
- Experiments on a 2D Gaussian attraction/repulsion model show the privately estimated map is close to the optimal map (qualitative only).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| impose ε-differential privacy when estimating smooth transport maps | Smooth transport map estimation · Smooth transport map estimation | squared L2(P) error: (n\epsilon)^{- \frac{2 \alpha}{\alpha - 1 + d}} | Hurt |
Surprising findings
- Privacy provably degrades the estimation rate in some regimes (ε below a threshold in n), even for the best estimator.
Limitations
- Lower and upper bounds on the privacy term do not match; tightness unresolved.
- Estimator is a theoretical report-noisy-argmin over a covering of candidate potentials, which is computationally impractical; practical grid version lacks guarantees beyond privacy.
- Experiments are small-scale and qualitative, only in 2D.
Experimental setup
Datasets: synthetic Gaussian attraction/repulsion potentials with uniform samples on [-1/2,1/2]^2
Benchmarks: synthetic Gaussian attraction/repulsion model (qualitative comparison with true optimal map)
Metrics: L2(P) transport map error (theoretical bound)
Builds on: semi-dual empirical estimator of Brenier potentials (Hütter & Rigollet style min report noisy max mechanism wavelet sieve estimation
Closest papers by meaning
- Minimax Private Estimation of Smooth Optimal-Transport Maps (NeurIPS 2026 · similarity 0.92)
- Estimation of Stochastic Optimal Transport Maps (NeurIPS 2025 · similarity 0.87)
- Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation (NeurIPS 2026 · similarity 0.85)
- A Combinatorial Algorithm for the Semi-Discrete Optimal Transport Problem (NeurIPS 2024 · similarity 0.85)
- Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport (NeurIPS 2026 · similarity 0.85)
- Privately Learning Smooth Distributions on the Hypercube by Projections (ICML 2024 · similarity 0.84)
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