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arxiv: 2606.04683 · v1 · pith:4C7R22NCnew · submitted 2026-06-03 · 🧮 math.ST · stat.TH

Minimax Private Estimation of Smooth Optimal-Transport Maps

Cl\'ement Lalanne , David Rodr\'iguez-V\'itores , Franck Iutzeler , Jean-Michel Loubes This is my paper

Pith reviewed 2026-06-28 04:12 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords differential privacyoptimal transport mapswavelet density estimationminimax ratessmooth mapscentral DPlocal DPnonparametric estimation
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The pith

Differentially private estimators for smooth optimal transport maps achieve near-minimax rates in dimension two and higher, with exact minimax rates in one dimension under central privacy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs estimators for smooth optimal transport maps between probability distributions while enforcing differential privacy constraints. It combines wavelet-based density estimators with stability bounds on the maps to produce private procedures that work in both central and local privacy models. The main estimator attains near-minimax optimal convergence rates when the dimension is at least two. A separate quantile-based estimator reaches exact minimax rates in one dimension under central privacy, and matching lower bounds are proved to show these rates are essentially tight. This matters for any setting where transport maps must be recovered from sensitive data without revealing individual observations.

Core claim

The authors build differentially private estimators for smooth OT maps by privatizing wavelet density estimators and invoking stability bounds; the resulting main procedure attains near-minimax rates in d ≥ 2 under both central and local DP, a quantile-based variant attains minimax rates in d = 1 under central DP, and matching lower bounds confirm that the achieved rates cannot be improved by more than logarithmic factors.

What carries the argument

Privatized wavelet density estimators that are fed into stability-based recovery of the OT map while preserving the original convergence rates under privacy noise.

If this is right

  • Near-minimax rates hold for the main estimator in every dimension d ≥ 2 under both central and local differential privacy.
  • The quantile-based estimator attains exact minimax rates in dimension one under central differential privacy.
  • Matching lower bounds establish that the rates cannot be improved beyond logarithmic factors.
  • The construction supplies the first differentially private OT-map estimator possessing these optimality guarantees.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reliance on wavelets suggests that similar privatized density estimators could be reused for other smooth functionals of distributions.
  • Dimension-dependent rate behavior indicates that one-dimensional private transport problems may admit qualitatively simpler solutions than higher-dimensional ones.
  • The approach leaves open whether the same stability-plus-wavelet route can be adapted to non-smooth or unbounded maps without losing the rate guarantees.

Load-bearing premise

Adding differential privacy noise to the wavelet density estimates does not substantially worsen the accuracy of the recovered optimal transport maps beyond the near-minimax level.

What would settle it

A simulation on a pair of smooth densities in dimension two that measures whether the private estimator's error exceeds the claimed near-minimax rate by more than a small constant factor once the privacy parameter is fixed.

Figures

Figures reproduced from arXiv: 2606.04683 by Cl\'ement Lalanne, David Rodr\'iguez-V\'itores, Franck Iutzeler, Jean-Michel Loubes.

Figure 1
Figure 1. Figure 1: unif-unif, N=2, J=2, for d=2 (a) and d=3 (b). Mean squared error vs. sample size n for varying central privacy budgets ϵ (log-log scale), averaged over 5 independent runs. Solid lines represent the ϵ-DP wavelet estimator; black and red dashed lines denote the OT and EOT baselines, respectively. 6 Conclusion In this work, we studied the private estimation of smooth optimal transport maps and proposed a plug… view at source ↗
Figure 2
Figure 2. Figure 2: Projection onto the subspace of L 2 ([0, 1]2 ) generated by Φ BC N,J,d of the densities of Qa (top) and Qb (bottom), for different values of (N, J). • OT baseline. Compute the OT map Tˆ between Xtest and Y test without regularization (using the standard implementation of POT [43]) and estimate MSE(Tˆ) = 1 nj Xnj i=1 [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: d=2, uniform-uniform, N=2, J=2. Reconstructed target densities for a single repetition across sample sizes n (columns) and central privacy budgets ϵ (rows) [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: d=2, uniform-uniform, N=2, J=2. Plots of the residual Tˆϵ reg(x) − T(x) across sample sizes n (columns) and central privacy budgets ϵ (rows). Vector color is proportional to the norm of the difference The improved performance of the smoothing approach is of great interest and is consistent with the theoretical results of Section 3. However, as suggested by the figure, this improvement is not due to differe… view at source ↗
Figure 5
Figure 5. Figure 5: shows that, in practice, although histograms are not minimax-optimal in theory, they provide a competitive solution in the non-private setting. Among the combinations (N, J) considered, there exist values of N for which the best non-private performance is achieved by scaling wavelet systems with N > 1, while the Haar system remains competitive. However, in the private setting, the Haar system clearly outpe… view at source ↗
Figure 6
Figure 6. Figure 6: d=2, uniform–mixture, N=2, J=2. Mean squared error vs. sample size n for varying central privacy budgets ϵ (log-log scale), averaged over 5 independent runs. Solid lines represent the ϵ-DP wavelet estimator; black and red dashed lines denote the OT and EOT baselines, respectively [PITH_FULL_IMAGE:figures/full_fig_p043_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: d=2, uniform–mixture, N=2, J=2. Reconstructed target densities for a single repetition across sample sizes n (columns) and central privacy budgets ϵ (rows). 43 [PITH_FULL_IMAGE:figures/full_fig_p043_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: d=2, uniform–mixture, N=2, J=2. Plots of the residual Tˆϵ reg(x) − T(x) across sample sizes n (columns) and central privacy budgets ϵ (rows). Vector color is proportional to the norm of the difference. 10 3 10 4 n 10 4 10 3 10 2 10 1 Mean Squared Error = = 10.0 = 5.0 = 1.0 OT baseline EOT baseline [PITH_FULL_IMAGE:figures/full_fig_p044_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: d=2, uniform–mixture, N=2, J=3. Mean squared error vs. sample size n for varying central privacy budgets ϵ (log-log scale), averaged over 5 independent runs. Solid lines represent the ϵ-DP wavelet estimator; black and red dashed lines denote the OT and EOT baselines, respectively. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: d=2, uniform–mixture, N=2, J=3. Reconstructed target densities for a single repetition across sample sizes n (columns) and central privacy budgets ϵ (rows) [PITH_FULL_IMAGE:figures/full_fig_p045_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: d=2, uniform–mixture, N=2, J=3. Plots of the residual Tˆϵ reg(x) − T(x) across sample sizes n (columns) and central privacy budgets ϵ (rows). Vector color is proportional to the norm of the difference 45 [PITH_FULL_IMAGE:figures/full_fig_p045_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: d=2, uniform–mixture, varying N, J. Reconstructed target densities for a single repetition across sample sizes n (columns) and values of (N, J) (rows), for ϵ = ∞. 46 [PITH_FULL_IMAGE:figures/full_fig_p046_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: d=2, uniform–mixture, varying N, J. Reconstructed target densities for a single repetition across sample sizes n (columns) and values of (N, J) (rows), for central ϵ = 3. 47 [PITH_FULL_IMAGE:figures/full_fig_p047_13.png] view at source ↗
read the original abstract

We study the problem of estimating smooth optimal transport (OT) maps between two probability distributions under differential privacy (DP) constraints. Leveraging wavelet-based density estimators and recent stability bounds for smooth OT maps, we propose differentially private estimators that apply to both central and local DP models. Our main estimator achieves near-minimax optimal rates in dimension $d \geq 2$, and we complement it with a quantile-based estimator that attains minimax optimal rates in dimension $d = 1$ under central DP. We further establish matching minimax lower bounds, confirming the near-optimality of our approach. To the best of our knowledge, this constitutes the first differentially private procedure for OT map estimation with minimax optimality guarantees.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops differentially private estimators for smooth optimal transport maps between probability distributions. It combines wavelet-based density estimators with existing stability bounds for smooth OT maps to produce estimators under both central and local DP. The main estimator is claimed to achieve near-minimax optimal rates for dimension d ≥ 2; a separate quantile-based procedure attains exact minimax rates for d = 1 under central DP. Matching minimax lower bounds are established, and the work is presented as the first DP procedure for OT map estimation with minimax optimality guarantees.

Significance. If the composition of privacy noise with the stability bounds preserves the claimed rates, the result would be a notable advance in private nonparametric estimation, providing the first minimax-optimal rates for this problem. The explicit matching lower bounds and the separation of the d=1 case (which avoids the stability step) are strengths. The approach usefully bridges recent OT stability results with wavelet privatization techniques.

major comments (2)
  1. [§4.1–4.2, Theorem 4.1] §4.1–4.2 and the proof of Theorem 4.1: the stability bound (invoked from the cited reference) is applied directly to the output of the privatized wavelet density estimator. The analysis must confirm that the Laplace or Gaussian noise added to the wavelet coefficients preserves the precise Hölder/Besov regularity and norm control required by the stability result; if the noise enters in a norm (e.g., sup-norm) not controlled by the stability theorem, the OT-map error bound inflates and the near-minimax claim for d ≥ 2 fails.
  2. [§4.2 (after Eq. (12))] The rate derivation for the main estimator (displayed after Eq. (12) in §4.2) treats the privacy-induced density error as an additive term whose contribution remains of lower order. An explicit calculation showing that the effective smoothness index after privatization does not drop below the threshold needed for the stability map is required; otherwise the claimed rate for d ≥ 2 is not justified.
minor comments (2)
  1. [§3] Notation for the wavelet basis and the precise DP mechanism (Laplace vs. Gaussian) is introduced in §3 but used inconsistently in the error bounds of §4; a single consistent definition would improve readability.
  2. [§5] The lower-bound construction in §5 is stated for the central DP model; a brief remark on whether the same lower bound extends to local DP would clarify the scope of the optimality result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points about the interaction between privacy noise and the stability bounds that we will clarify in revision. We address each major comment below.

read point-by-point responses

  1. Referee: [§4.1–4.2, Theorem 4.1] the stability bound is applied directly to the output of the privatized wavelet density estimator. The analysis must confirm that the Laplace or Gaussian noise added to the wavelet coefficients preserves the precise Hölder/Besov regularity and norm control required by the stability result; if the noise enters in a norm not controlled by the stability theorem, the OT-map error bound inflates.

    Authors: We agree that an explicit verification is needed. The wavelet coefficients receive independent Laplace (or Gaussian) noise scaled to the privacy budget; because the wavelet basis is unconditional for Besov spaces and the noise is added at each resolution level with amplitude decaying as 2^{-j(s+d/2)} (where s is the smoothness index), the perturbed estimator remains in the same Besov ball up to a logarithmic factor with high probability. The stability theorem (invoked from the cited reference) is stated for perturbations measured in the sup-norm, which is controlled by the wavelet coefficient noise via standard embedding arguments. We will insert a short auxiliary lemma (new Lemma 4.3) that states this preservation and derives the resulting constant factors, thereby justifying direct application of the stability bound. revision: yes

  2. Referee: [§4.2 (after Eq. (12))] The rate derivation treats the privacy-induced density error as an additive term whose contribution remains of lower order. An explicit calculation showing that the effective smoothness index after privatization does not drop below the threshold needed for the stability map is required.

    Authors: The privacy noise level is chosen so that its contribution to the Besov norm is o(2^{-j s}) at the resolution j that balances bias and variance; consequently the effective smoothness index remains strictly above the threshold required by the stability map (s > d/2 + 1). The calculation appears implicitly in the proof of Theorem 4.1 via the triangle inequality separating the wavelet estimation error from the privacy error, but we acknowledge it is not written out as a separate display. In the revision we will add an explicit paragraph immediately after Eq. (12) that computes the post-privatization Besov norm and verifies it stays above the stability threshold, confirming that the privacy term remains lower order for d ≥ 2. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external stability bounds and independent lower bounds

full rationale

The paper's central claims rest on applying wavelet density estimators under DP noise and invoking cited stability results that map density error to OT-map error, followed by separate minimax lower-bound arguments. No equations reduce a claimed rate to a fitted parameter by construction, no self-citation chain is load-bearing for the optimality statement, and the lower bounds are presented as independently derived matching results. The composition of privacy noise with the stability bounds is an assumption whose validity is external to the derivation itself; it does not create a definitional loop or rename a fitted quantity as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; ledger populated from stated technical ingredients.

axioms (2)
  • domain assumption Stability bounds for smooth OT maps exist and survive privatization.
    Leveraged to control error after DP noise addition.
  • domain assumption Wavelet density estimators admit DP versions with controlled rates.
    Core construction for the proposed estimators.

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discussion (0)

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