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arxiv: 2604.20761 · v1 · submitted 2026-04-22 · 📊 stat.ML · stat.ME

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Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms

Xiaotian Chang , Yangdi Jiang , Cyrus Mostajeran , Qirui Hu
Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:49 UTC · model grok-4.3

classification 📊 stat.ML stat.ME
keywords Renyi differential privacyRiemannian manifoldsRicci curvatureheat diffusionHarnack inequalitiesFrechet meanLangevin process
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The pith

Renyi differential privacy for manifold-valued data is governed by Ricci curvature via heat diffusion and Harnack inequalities.

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

The paper establishes that Renyi divergence between distributions on Riemannian manifolds can be bounded using dimension-free Harnack inequalities, which in turn yield differential privacy guarantees controlled by the manifold's Ricci curvature. For manifolds with nonnegative Ricci curvature, a heat diffusion mechanism is proposed to achieve these guarantees, while a Langevin process-based mechanism handles general manifolds and supports continuous privacy-utility adjustments without normalization. Detailed utility analyses are provided, along with an application to privacy-preserving estimation of the generalized Frechet mean that includes sensitivity bounds and phase transition results. This matters because it offers a geometry-aware way to protect privacy in data that naturally lives on curved spaces rather than forcing Euclidean embeddings.

Core claim

The central claim is that the Renyi divergence can be characterized via dimension-free Harnack inequalities on Riemannian manifolds, thereby establishing Renyi differential privacy guarantees that are governed by Ricci curvature. For manifolds with nonnegative Ricci curvature, the heat diffusion mechanism provides the privacy guarantee. For general manifolds, the Langevin-process-based approach yields intrinsic mechanisms supporting normalization-free sampling and continuous privacy-utility trade-offs. Utility analyses for both mechanisms are derived, and as a statistical application, privacy-preserving estimation of the generalized Frechet mean is developed with nontrivial sensitivity 1.0px

What carries the argument

Dimension-free Harnack inequalities on Riemannian manifolds that connect Renyi divergence to Ricci curvature for differential privacy.

Load-bearing premise

That heat diffusion and Langevin processes on Riemannian manifolds yield the stated Renyi DP guarantees and utility bounds when governed by Ricci curvature, relying on the applicability of Harnack inequalities without post-hoc adjustments or manifold-specific restrictions.

What would settle it

A concrete counterexample on a manifold with nonnegative Ricci curvature where the heat diffusion mechanism violates the claimed Renyi divergence bound for some pair of distributions.

Figures

Figures reproduced from arXiv: 2604.20761 by Cyrus Mostajeran, Qirui Hu, Xiaotian Chang, Yangdi Jiang.

Figure 1
Figure 1. Figure 1: Left: mechanism-level roadmap of the paper. Right: schematic behavior of the privacy [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The left picture depicts Circle Limit III by Dutch artist M. C. Escher, which was [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Utility comparison of BM and RL mechanisms across different sample size [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Utility comparison of Langevin and RL mechanisms across different sample size [PITH_FULL_IMAGE:figures/full_fig_p034_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Utility comparison of Langevin and EWG mechanisms across different sample size [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
read the original abstract

In this paper, we develop a novel privacy mechanism for Riemannian manifold-valued data. Our key contribution lies in uncovering unexpected connections among geometric analysis, heat diffusion models, and differential privacy (DP). We characterize the Renyi divergence via dimension-free Harnack inequalities on Riemannian manifolds and establish Renyi differential privacy guarantees governed by Ricci curvature. For manifolds with nonnegative Ricci curvature, we propose a mechanism based on heat diffusion. In contrast, for general manifolds we introduce a Langevin-process-based approach that yields intrinsic mechanisms supporting normalization-free sampling and continuous privacy-utility trade-offs. We derive detailed utility analyses for both mechanisms. As a statistical application, we develop privacy-preserving estimation of the generalized Frechet mean, including nontrivial sensitivity analysis and phase transition characterizations. Numerical experiments further demonstrate the advantages of the proposed DP mechanisms over existing approaches.

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 manuscript develops Renyi differential privacy mechanisms for Riemannian manifold-valued data by linking heat diffusion and Langevin processes to geometric analysis. It claims to characterize Renyi divergence between diffused neighboring distributions via dimension-free Harnack inequalities, yielding privacy parameters governed solely by a lower Ricci curvature bound. For nonnegative Ricci curvature manifolds a heat-diffusion mechanism is proposed; for general manifolds a Langevin-based intrinsic mechanism is introduced that supports normalization-free sampling and continuous privacy-utility trade-offs. Detailed utility analyses are derived, followed by an application to privacy-preserving generalized Fréchet mean estimation that includes sensitivity bounds and phase-transition characterizations, together with numerical comparisons to prior DP methods.

Significance. If the claimed characterizations hold without hidden manifold-specific adjustments, the work would provide a geometrically intrinsic approach to Renyi DP whose privacy parameter depends only on Ricci curvature, potentially enabling dimension-free guarantees for data on spheres, hyperbolic spaces, and other manifolds. The Langevin construction and Fréchet-mean application are concrete strengths that could influence private statistical estimation on non-Euclidean domains.

major comments (2)
  1. [Section 3 (characterization of Renyi divergence via Harnack inequalities) and Theorem 1] The central step from pointwise Harnack inequalities to integrated α-Renyi divergence bounds between push-forwards of neighboring initial measures (the step underlying both the heat-diffusion and Langevin guarantees) is not shown to be free of dependence on the support of the initial data or on higher-order curvature terms. Standard Li-Yau-type Harnack estimates control local ratios of the heat kernel but do not automatically deliver dimension-free integrated Renyi bounds for arbitrary neighboring measures; the manuscript must supply the precise passage (including any auxiliary assumptions on the initial measures) that absorbs all such terms into the Ricci lower bound alone.
  2. [Section 5 (privacy-preserving Fréchet mean estimation)] In the utility analysis for the generalized Fréchet mean estimator (Section 5), the claimed phase-transition characterization and sensitivity bound appear to rely on the same Ricci-governed privacy parameter derived in Section 3. If the Renyi-to-privacy translation in Section 3 requires additional manifold-dependent factors, the phase-transition thresholds and utility bounds would need corresponding revision.
minor comments (2)
  1. [Notation and preliminaries] Notation for the order-α Renyi divergence and the diffusion time parameter should be introduced once with explicit dependence on the manifold dimension and curvature bound, then used consistently.
  2. [Numerical experiments] The numerical experiments section would benefit from reporting the exact manifold dimensions, number of Monte-Carlo samples used for utility estimation, and confidence intervals on the reported privacy-utility curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address the two major comments point by point below and will revise the manuscript to supply the requested technical clarifications.

read point-by-point responses

  1. Referee: [Section 3 (characterization of Renyi divergence via Harnack inequalities) and Theorem 1] The central step from pointwise Harnack inequalities to integrated α-Renyi divergence bounds between push-forwards of neighboring initial measures (the step underlying both the heat-diffusion and Langevin guarantees) is not shown to be free of dependence on the support of the initial data or on higher-order curvature terms. Standard Li-Yau-type Harnack estimates control local ratios of the heat kernel but do not automatically deliver dimension-free integrated Renyi bounds for arbitrary neighboring measures; the manuscript must supply the precise passage (including any auxiliary assumptions on the initial measures) that absorbs all such terms into the Ricci lower bound alone.

    Authors: We appreciate the referee pointing out the need for greater explicitness in this derivation. The proof of Theorem 1 performs the integration of the pointwise Harnack ratio to obtain the Renyi divergence between the push-forward measures, but the absorption of support-dependent and higher-order curvature contributions is only sketched. In the revised version we will add a self-contained lemma immediately preceding Theorem 1 that carries out the integration in full detail. Under the standing assumption that the initial neighboring measures are absolutely continuous with respect to the Riemannian volume and have compact support (standard for manifold-valued DP mechanisms), the ratio is uniformly bounded and all additional terms are controlled solely by the lower Ricci curvature bound, yielding the claimed dimension-free estimate. No new assumptions are introduced. revision: yes

  2. Referee: [Section 5 (privacy-preserving Fréchet mean estimation)] In the utility analysis for the generalized Fréchet mean estimator (Section 5), the claimed phase-transition characterization and sensitivity bound appear to rely on the same Ricci-governed privacy parameter derived in Section 3. If the Renyi-to-privacy translation in Section 3 requires additional manifold-dependent factors, the phase-transition thresholds and utility bounds would need corresponding revision.

    Authors: We agree that the utility results in Section 5 are direct consequences of the privacy parameters obtained in Section 3. Once the derivation in Section 3 is made fully explicit via the new lemma, the phase-transition thresholds and sensitivity bounds remain unchanged. In the revision we will insert a short cross-reference paragraph in Section 5 that points to the new lemma, ensuring the dependence is transparent without altering any numerical or asymptotic claims. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation builds from external geometric inequalities to privacy bounds

full rationale

The paper claims to characterize Renyi divergence via dimension-free Harnack inequalities and to obtain DP guarantees governed by Ricci curvature, using heat diffusion for nonnegative Ricci cases and Langevin processes otherwise. No quoted equations or steps reduce a claimed prediction or first-principles result to a fitted parameter, self-defined quantity, or self-citation chain by construction. The abstract and description present the Harnack-to-Renyi passage and curvature-governed mechanisms as derived from standard geometric analysis tools, with utility bounds and Frechet-mean application treated as downstream consequences rather than inputs. This is the normal case of a self-contained derivation resting on independent mathematical facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions from Riemannian geometry (heat kernels, Harnack inequalities) and differential privacy theory, with new mechanisms introduced based on curvature conditions; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Riemannian manifolds admit heat diffusion processes and dimension-free Harnack inequalities that characterize Renyi divergence
    Invoked to establish DP guarantees governed by Ricci curvature for both nonnegative and general curvature cases.

pith-pipeline@v0.9.0 · 5444 in / 1391 out tokens · 32580 ms · 2026-05-09T22:49:16.023719+00:00 · methodology

discussion (0)

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