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Exponential-Wrapped Mechanisms: Differential Privacy on Hadamard Manifolds Made Practical

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
Differential PrivacyRiemannian ManifoldHadamard manifoldsFréchet meanSPDM space
Abstract:

We propose a general and computationally efficient framework for achieving differential privacy (DP) on Hadamard manifolds, which are complete and simply connected Riemannian manifolds with non-positive curvature. Leveraging the Cartan-Hadamard theorem, we introduce Exponential-Wrapped Laplace and Gaussian mechanisms that achieve ϵ\epsilon-DP, (ϵ,δ)(\epsilon, \delta)-DP, Gaussian DP (GDP), and Rényi DP (RDP) without relying on computationally intensive MCMC sampling. Our methods operate entirely within the intrinsic geometry of the manifold, ensuring both theoretical soundness and practical scalability. We derive utility bounds for privatized Fréchet means and demonstrate superior utility and runtime performances on both synthetic data and real-world data in the space of symmetric positive definite matrices (SPDM) equipped with three different metrics. To our knowledge, this work constitutes the first unified extension of multiple DP notions to general Hadamard manifolds with practical and scalable implementations.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper proposes a unified framework for differential privacy on Hadamard manifolds, introducing Exponential-Wrapped Laplace and Gaussian mechanisms that achieve multiple DP notions (ε-DP, (ε,δ)-DP, GDP, RDP) without MCMC sampling. It resides in the 'Exponential-Wrapped Mechanisms for Hadamard Manifolds' leaf, which contains only two papers total. This is a notably sparse research direction within the broader taxonomy of 23 papers across differential privacy on non-Euclidean geometries, suggesting the work addresses a relatively underexplored niche in the field.

The taxonomy tree reveals that the paper's immediate sibling category, 'Density-Aware and Conformal Transformation Mechanisms,' contains only one paper focusing on local density calibration. Neighboring branches include local differential privacy with Hadamard-based encoding (5 papers across frequency oracles and itemset mining) and privacy-preserving graph embeddings (3 papers on hierarchical structures). The scope notes clarify that exponential-wrapped approaches differ fundamentally from density-aware methods by operating through distributional wrapping rather than adaptive noise calibration, positioning this work as methodologically distinct from its closest relatives.

Among 30 candidates examined, the contribution-level analysis shows mixed novelty signals. The 'Unified extension of multiple DP notions' examined 10 candidates with 1 appearing to provide overlapping prior work, suggesting some precedent exists for multi-notion DP frameworks on manifolds. However, the 'Exponential-Wrapped mechanisms avoiding MCMC' and 'General framework' contributions each examined 10 candidates with zero refutable matches, indicating these specific technical approaches appear more novel within the limited search scope. The statistics reflect a focused but not exhaustive literature review.

Based on the top-30 semantic matches examined, the work appears to occupy a sparsely populated research direction with limited direct competition. The taxonomy structure confirms that exponential-wrapped mechanisms for Hadamard manifolds constitute a small but distinct methodological branch. While one contribution shows some prior overlap, the core technical mechanisms and general framework appear less anticipated by the examined literature, though the limited search scope precludes definitive claims about absolute novelty across the entire field.

Taxonomy

Core-task Taxonomy Papers
23
3
Claimed Contributions
30
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: Differential privacy on Hadamard manifolds. This field addresses the challenge of protecting sensitive data that naturally resides in non-Euclidean spaces, particularly hyperbolic geometries where hierarchical and graph-structured information is often embedded. The taxonomy reveals several main branches: one focuses on designing differential privacy mechanisms tailored to non-Euclidean geometries, including exponential-wrapped approaches for Hadamard manifolds; another explores local differential privacy with Hadamard-based encoding schemes for decentralized settings; a third examines privacy-preserving embeddings of graphs and hierarchical data into hyperbolic spaces; and additional branches cover federated learning frameworks that leverage hyperbolic geometry, application-specific privacy solutions (such as recommendation systems and health data), and auxiliary topics bridging privacy with hyperbolic methods. Representative works like Hadamard Location Aggregation[2] and Hadamard Itemset Mining[6] illustrate how these geometric tools enable privacy guarantees while preserving the structural properties of complex data. A particularly active line of work centers on exponential-wrapped mechanisms that adapt classical differential privacy noise to the curvature of Hadamard manifolds, balancing utility and privacy in non-flat spaces. Meanwhile, federated learning approaches such as Hyperbolic Federated[3] and application-driven studies like Hyperbolic Cache Privacy[5] explore how hyperbolic embeddings can reduce communication costs and improve model expressiveness under privacy constraints. The original paper, Exponential Wrapped Hadamard[0], sits squarely within the branch of exponential-wrapped mechanisms for Hadamard manifolds, contributing foundational theory for noise injection on curved spaces. Compared to nearby works like Density Aware Manifolds[8], which emphasizes adaptive noise calibration based on local geometry, Exponential Wrapped Hadamard[0] focuses on establishing rigorous privacy guarantees through exponential wrapping techniques. This positions it as a core theoretical contribution that underpins many application-specific extensions across hierarchical data, federated systems, and graph embeddings.

Claimed Contributions

Unified extension of multiple DP notions to general Hadamard manifolds

The authors present the first framework that extends multiple differential privacy notions—including (ε, δ)-DP, Gaussian DP, and Rényi DP—to general Hadamard manifolds. This represents the first RDP mechanism that works beyond Euclidean spaces.

10 retrieved papers
Can Refute
Exponential-Wrapped mechanisms avoiding MCMC sampling

The authors develop Exponential-Wrapped Laplace and Gaussian mechanisms that achieve differential privacy without relying on computationally expensive MCMC sampling. Instead, these mechanisms use efficient sampling from tangent space distributions combined with the exponential map.

10 retrieved papers
General framework for differential privacy on Hadamard manifolds

The authors propose a general and computationally efficient framework that achieves differential privacy on Hadamard manifolds by leveraging the Cartan-Hadamard theorem and operating entirely within the intrinsic geometry of the manifold.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Unified extension of multiple DP notions to general Hadamard manifolds

The authors present the first framework that extends multiple differential privacy notions—including (ε, δ)-DP, Gaussian DP, and Rényi DP—to general Hadamard manifolds. This represents the first RDP mechanism that works beyond Euclidean spaces.

Contribution

Exponential-Wrapped mechanisms avoiding MCMC sampling

The authors develop Exponential-Wrapped Laplace and Gaussian mechanisms that achieve differential privacy without relying on computationally expensive MCMC sampling. Instead, these mechanisms use efficient sampling from tangent space distributions combined with the exponential map.

Contribution

General framework for differential privacy on Hadamard manifolds

The authors propose a general and computationally efficient framework that achieves differential privacy on Hadamard manifolds by leveraging the Cartan-Hadamard theorem and operating entirely within the intrinsic geometry of the manifold.

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