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Differentially private inference framework of Riemannian manifold data

This paper proposes a novel differentially private inference framework for Riemannian manifold data by designing geometry-aware mechanisms for Fréchet mean and variance, establishing their consistency and central limit theorems, and validating the approach through extensive simulations and real-world medical and sociological datasets.

Original authors: Yangdi Jiang, Xiaotian Chang, Qirui Hu

Published 2026-05-15

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CC BY 4.0

Original authors: Yangdi Jiang, Xiaotian Chang, Qirui Hu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the "average" location of a group of hikers, but they aren't walking on a flat, straight road (like a city street). Instead, they are walking on the surface of a giant, curved mountain or a sphere. In math, this curved surface is called a Riemannian manifold.

If you tried to calculate their average position using standard "flat" math (Euclidean geometry), you would get the wrong answer because the math doesn't account for the curve. You might end up calculating a point that is actually floating in the air inside the mountain, rather than on the surface where the hikers actually are.

Furthermore, imagine these hikers are carrying sensitive personal data (like medical records or private opinions). You want to share the "average" location with the public, but you must protect their privacy so no one can figure out exactly where any single hiker was.

This paper proposes a new toolkit to solve both problems at once: how to calculate the true "average" on a curved surface while keeping the data private.

Here is a breakdown of their solution using simple analogies:

1. The Problem: Curved Roads and Privacy

The Curve: In the real world, data often lives on curves. For example, medical imaging (like MRI scans) uses data that looks like a curved surface, and social science data (like directions people face) lives on a sphere. Standard statistics fail here because they assume everything is flat.
The Privacy: To protect privacy, we usually add "noise" (random static) to the data. On a flat road, you just add random noise in a straight line. But on a curved mountain, if you add noise in a straight line, you might push the data off the mountain entirely, which breaks the math.

2. The Solution: Two Special "Noise" Machines

The authors designed two special ways to add noise that respects the shape of the mountain:

The "Riemannian Gaussian" (For Curved Mountains): Imagine you are on a sphere. Instead of pushing the data in a straight line, this method pushes it along the curve of the sphere itself. It's like rolling a marble along the surface rather than shooting a bullet through the air. This works well for data with "positive curvature" (like a sphere).
The "Exponential-Wrapped Gaussian" (For Saddle Shapes): Imagine a saddle shape (like a Pringles chip or a hyperbolic surface). This method takes a flat piece of paper (a tangent plane), adds noise there, and then "wraps" it back onto the curved surface, like wrapping a gift. This works for "Hadamard manifolds" (surfaces that curve away from each other).

3. The "Privacy Budget" (The Cost of Secrecy)

In privacy math, there is a "budget" (called $\mu$ ).

High Budget: You can be very accurate, but you reveal more about the individuals.
Low Budget: You are very private, but your "average" will be blurry and less accurate.

The authors figured out a precise formula to calculate exactly how much noise to add based on the shape of the mountain and the privacy budget. Before this paper, figuring this out required complex, slow computer simulations. The authors found a direct, fast mathematical shortcut.

4. The Result: Trustworthy Averages

Once they added the noise, the authors proved that their new "Private Averages" still work mathematically:

Consistency: If you have enough data, the private average gets closer and closer to the true average.
Confidence Regions: They can draw a "circle of uncertainty" around the average. Even with the privacy noise added, they proved that this circle is the right size to tell you how confident you can be in the result.

5. Real-World Tests

The team tested their method on two types of real data:

Medical Images: They looked at retinal scans (OCT images) from a database. They treated the images as points on a curved surface and calculated the "average eye" for different disease categories while protecting patient privacy.
Sociology Data: They analyzed how people judge different jobs (like "Earnings" vs. "Social Status"). This data lives on a sphere (directions). They showed that even with privacy noise, they could still see that "Earnings" and "Social Status" are judged differently, even if the privacy budget was tight.

Summary

Think of this paper as inventing a new GPS for curved worlds that has a "blur" feature for privacy.

Before this, if you wanted to find the average of curved data privately, you had to guess or use very slow computer methods. This paper gives you a fast, reliable map and a rulebook for how much to blur the map so that you can still find the destination without revealing who the travelers were. They proved that even with the blur, the map remains accurate enough to make scientific decisions.

Technical Summary: Differentially Private Inference Framework of Riemannian Manifold Data

Problem Statement
As data collection advances, complex datasets increasingly reside in non-Euclidean spaces, specifically Riemannian manifolds. Prominent examples include the space of Symmetric Positive Definite Matrices (SPDMs) used in medical imaging (e.g., Diffusion Tensor Imaging) and the hypersphere used in directional statistics (e.g., sociological judgment data). While Differential Privacy (DP) has been established as a formal framework for protecting Euclidean data, extending it to manifold-valued data remains largely unexplored. The primary challenges include the absence of a vector space structure (preventing direct addition of Euclidean noise), the analytical complexity introduced by manifold curvature, and the lack of statistical inference procedures (such as consistency and Central Limit Theorems) that guarantee privacy on these geometries. This paper addresses the inference problem for the Fréchet mean and variance of confidential data defined on a $d$ -dimensional Riemannian manifold under a DP framework.

Methodology
The authors propose a systematic framework for differentially private (DP) estimation and inference of the population Fréchet mean ( $\eta$ ) and Fréchet variance ( $V$ ). The methodology is built upon two distinct DP mechanisms tailored to specific geometric structures:

Riemannian Gaussian (RG) Mechanism: Designed for homogeneous manifolds with positive curvature (e.g., spheres). This mechanism injects noise directly on the manifold using a Riemannian Gaussian distribution, $N^{RG}(\eta, \sigma)$ , defined by a density proportional to $\exp(-\rho^2(y, \eta)/2\sigma^2)$ .
Exponential-Wrapped Gaussian (EWG) Mechanism: Designed for Hadamard manifolds (complete, simply connected manifolds with non-positive sectional curvature, e.g., SPDM spaces with affine-invariant metrics). This mechanism leverages the tangent space structure by pushing forward a tangent-space Gaussian distribution through the exponential map ( $\exp_p$ ).

Key Technical Contributions:

Analytic Privacy Calibration: Unlike prior work that relied on computationally expensive Markov Chain Monte Carlo (MCMC) procedures to calibrate noise levels for constant curvature manifolds, this paper derives an explicit, analytic relationship between the sensitivity of the Fréchet mean, the Riemannian geometry, and the privacy budget ( $\mu$ ). This allows for direct computation of the noise scale parameter $\sigma$ without inverting the privacy budget numerically.
Asymptotic Theory: The authors establish the $\sqrt{n}$ -consistency of the proposed DP estimators. Furthermore, they prove Central Limit Theorems (CLTs) for both the DP Fréchet mean and variance. These results account for the intrinsic geometry of the manifold, specifically the expected Hessian of the squared distance function, and address the phenomenon of "smeariness" (slower convergence rates) when standard assumptions fail.
Inference Procedures: The framework provides a complete suite for statistical inference, including the construction of asymptotic confidence regions for the Fréchet mean and confidence intervals for the Fréchet variance.
Private Variance Estimation: To enable valid inference, the paper proposes consistent DP estimators for the asymptotic variance components (matrices $\Lambda$ and $C$ ) and the limiting variance of the variance estimator, utilizing Gaussian mechanisms on the vectorized forms of these statistics.

Results
The proposed methodologies were validated through extensive numerical experiments and real-world applications:

Simulations: Experiments on the unit sphere ( $S^2$ ) and the SPDM space ( $S^+_2$ ) demonstrated that the gap between DP and non-DP estimators shrinks as the privacy budget increases. The coverage rates of the constructed DP confidence regions remained close to the nominal level (e.g., 95%) across various privacy budgets.
Real-World Applications:
- Medical Imaging: Applied to the OCTMNIST dataset (retinal OCT scans mapped to SPDMs). The framework successfully constructed DP confidence regions for different diagnosis classes, showing that moderate privacy budgets ( $\mu=1, 2$ ) yield regions close to non-DP counterparts, while strong privacy ( $\mu=0.1$ ) appropriately inflates uncertainty.
- Sociological Data: Applied to occupational judgment data on a sphere. The DP confidence regions for different criteria (Earnings, Social Status, etc.) maintained the ability to distinguish between latent patterns even under privacy constraints, with regions contracting toward non-DP estimates as $\mu$ increased.

Significance and Claims
The paper claims to be the first to address the DP inference problem for Fréchet mean and variance on Riemannian manifolds using novel privacy mechanisms and a general, implementable framework. Its significance lies in:

Bridging Geometry and Privacy: It successfully adapts the spirit of Euclidean DP to curved spaces by tying noise calibration to intrinsic geometric sensitivity, respecting curvature and the lack of linear structure.
Practicality: By providing analytic calibration formulas, the framework eliminates the need for bespoke MCMC procedures or opaque budget inversions, making DP inference on manifolds computationally feasible for real-world applications.
Statistical Rigor: It delivers root- $n$ consistency, CLTs, and valid uncertainty quantification (confidence regions) under privacy constraints, including handling geometric complexities like smeariness.

The authors note limitations, specifically that the framework operates under a central DP model with a trusted aggregator and does not currently provide local privacy guarantees against the curator, which presents challenges for future work due to the nonlinear geometry of manifolds.