Differentially Private Manifold Denoising

TL;DR

This paper presents a scalable, differentially private manifold denoising framework that corrects noisy data points using reference datasets, ensuring privacy while preserving geometric structure for downstream tasks.

Contribution

It introduces a novel iterative method that privately estimates local geometry and projects query points, providing formal DP guarantees and utility bounds for manifold-based data analysis.

Findings

The method achieves accurate manifold recovery under moderate privacy budgets.

Utility guarantees show convergence of corrected queries toward the manifold.

Simulations demonstrate effective privacy-utility trade-offs in practical scenarios.

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 $(\varepsilon,\delta)$-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.