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Differentially Private Manifold Denoising

Jiaqi Wu, Yiqing Sun, Zhigang Yao

Abstract

We introduce a differentially private manifold denoising framework that allows users to exploit sensitive reference datasets to correct noisy, non-private query points without compromising privacy. The method follows an iterative procedure that (i) privately estimates local means and tangent geometry using the reference data under calibrated sensitivity, (ii) projects query points along the privately estimated subspace toward the local mean via corrective steps at each iteration, and (iii) performs rigorous privacy accounting across iterations and queries using $(\varepsilon,δ)$-differential privacy (DP). Conceptually, this framework brings differential privacy to manifold methods, retaining sufficient geometric signal for downstream tasks such as embedding, clustering, and visualization, while providing formal DP guarantees for the reference data. Practically, the procedure is modular and scalable, separating DP-protected local geometry (means and tangents) from budgeted query-point updates, with a simple scheduler allocating privacy budget across iterations and queries. Under standard assumptions on manifold regularity, sampling density, and measurement noise, we establish high-probability utility guarantees showing that corrected queries converge toward the manifold at a non-asymptotic rate governed by sample size, noise level, bandwidth, and the privacy budget. Simulations and case studies demonstrate accurate signal recovery under moderate privacy budgets, illustrating clear utility-privacy trade-offs and providing a deployable DP component for manifold-based workflows in regulated environments without reengineering privacy systems.

Differentially Private Manifold Denoising

Abstract

We introduce a differentially private manifold denoising framework that allows users to exploit sensitive reference datasets to correct noisy, non-private query points without compromising privacy. The method follows an iterative procedure that (i) privately estimates local means and tangent geometry using the reference data under calibrated sensitivity, (ii) projects query points along the privately estimated subspace toward the local mean via corrective steps at each iteration, and (iii) performs rigorous privacy accounting across iterations and queries using -differential privacy (DP). Conceptually, this framework brings differential privacy to manifold methods, retaining sufficient geometric signal for downstream tasks such as embedding, clustering, and visualization, while providing formal DP guarantees for the reference data. Practically, the procedure is modular and scalable, separating DP-protected local geometry (means and tangents) from budgeted query-point updates, with a simple scheduler allocating privacy budget across iterations and queries. Under standard assumptions on manifold regularity, sampling density, and measurement noise, we establish high-probability utility guarantees showing that corrected queries converge toward the manifold at a non-asymptotic rate governed by sample size, noise level, bandwidth, and the privacy budget. Simulations and case studies demonstrate accurate signal recovery under moderate privacy budgets, illustrating clear utility-privacy trade-offs and providing a deployable DP component for manifold-based workflows in regulated environments without reengineering privacy systems.

Paper Structure

This paper contains 60 sections, 8 theorems, 142 equations, 6 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. From noisy observations to geometry: non-private methods.
  3. From sensitive data to releases: differentially private methods.
  4. Model and setup.
  5. Our contributions.
  6. Differential privacy
  7. Sensitivity and Gaussian mechanisms.
  8. Zero-concentrated DP (zCDP).
  9. Public--private split.
  10. DP tangent space estimation
  11. Sensitivity of the local projector.
  12. DP manifold denoising
  13. Weighted residual.
  14. Sensitivity of the weighted projector and mean.
  15. Algorithm.
  16. ...and 45 more sections

Key Result

Lemma 1

Assume the model in eq:model. Suppose $\mathbf{z}\in\mathcal{M}_{{\sigma}}$ is sampled independently from the same distribution as $\{\mathbf{y}_i\}_{i=1}^n$ and bandwidth $h$ satisfies Let $\widehat{\mathbf{P}}_{\mathbf{z}}=\mathbf{V}_{\mathbf{z},d}\mathbf{V}_{\mathbf{z},d}^{\top}$ be the rank-$d$ projector onto the top-$d$ eigenvectors of $\widehat{\boldsymbol{\Sigma}}_{\mathbf{z}}$. For each $

Figures (6)

  • Figure 1: Synthetic manifold denoising results. (A--C) Circle, torus, and Swiss roll: geometric visualization and error trends. (D--E) Sphere error vs ambient dimension $D$ and privacy-utility tradeoff.
  • Figure 2: Manifold denoising preserves local geometric stability while remaining compatible with downstream biomedical risk modeling. (A) PCA embedding of the raw biomarker space, with subject-level displacement vectors induced by denoising. (B) Local geometry diagnostics showing bounded distortion and stable neighborhood structure relative to raw references. (C) Changes in out-of-sample risk discrimination across a pre-specified panel of clinically interpretable cardio-metabolic endpoints. Detailed results are provided in Fig. \ref{['fig:S4_icd_fullpanel']} and Table \ref{['tab:si_endpointset_cindex_ci']} of the Appendix. (D) Subject-level illustration linking coordinated biomarker shifts to coherent movement along a clinically meaningful risk axis.
  • Figure S.1: Robustness under Gaussian ambient noise across different manifolds. Rows correspond to circle, Swiss roll, and torus manifolds. Left panels show reconstruction error as a function of sample size $n$ under fixed noise scale, and right panels show reconstruction error as a function of noise scale $\sigma$ under fixed sample size. Results are averaged over repeated trials; shaded regions indicate 95% confidence intervals.
  • Figure S.2: Scalability with ambient dimension on high-dimensional spheres. (Top) Reconstruction error as a function of ambient dimension $D$ under Gaussian noise. (Bottom left) Average neighborhood size used for local geometry estimation as $D$ increases. (Bottom right) Computational runtime as a function of $D$ under bounded noise. Sample size is fixed across experiments.
  • Figure S.3: Privacy--utility tradeoff on nontrivial manifolds. Reconstruction error as a function of the privacy budget for (top) Swiss roll and (bottom) torus manifolds. Curves compare the non-private manifold denoising method with its differentially private counterpart calibrated under the $\rho$-zCDP mechanism.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 1: High-probability sensitivity
  • Remark 2
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Corollary 1
  • Lemma 3
  • Lemma 4
  • Lemma 5