# Shape And Structure Preserving Differential Privacy

arXiv: https://arxiv.org/abs/2209.12667 · HTML: https://mlsanity.com/paper/2209.12667/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 49 references that are in the map.
- First posted on arXiv 21 Sep 2022.

## Builds on (most cited first, 17 of 17)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Privacy for Free: Posterior Sampling and Stochastic Gradient Monte Carlo](https://mlsanity.com/paper/1502.07645/) (2015)
- [A Continuous-time Perspective for Modeling Acceleration in Riemannian Optimization](https://mlsanity.com/paper/1910.10782/) (2019)
- [Hypothesis Testing For Network Data in Functional Neuroimaging](https://mlsanity.com/paper/1407.5525/) (2014)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA](https://mlsanity.com/paper/1901.10864/) (2019)
- [Wishart Mechanism for Differentially Private Principal Components Analysis](https://mlsanity.com/paper/1511.05680/) (2015)
- [Structure and Sensitivity in Differential Privacy: Comparing K-Norm Mechanisms](https://mlsanity.com/paper/1801.09236/) (2018)
- [Private Approximations of the 2nd-Moment Matrix Using Existing Techniques in Linear Regression](https://mlsanity.com/paper/1507.00056/) (2015)
- [KNG: The K-Norm Gradient Mechanism](https://mlsanity.com/paper/1905.09436/) (2019)
- [Exact Privacy Guarantees for Markov Chain Implementations of the Exponential Mechanism with Artificial Atoms](https://mlsanity.com/paper/2204.01132/) (2022)
- [Elastic depths for detecting shape anomalies in functional data](https://mlsanity.com/paper/1907.06759/) (2019)
- [Mining Frequent Graph Patterns with Differential Privacy](https://mlsanity.com/paper/1301.7015/) (2013)
- [Smooth Sensitivity Based Approach for Differentially Private Principal Component Analysis](https://mlsanity.com/paper/1710.10556/) (2017)

## Built on by (most cited first, 6 of 6)

- [Gaussian Differential Privacy on Riemannian Manifolds](https://mlsanity.com/paper/2311.10101/) (2023)
- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms](https://mlsanity.com/paper/2604.20761/) (2026)
- [Gaussian Differentially Private Human Faces Under a Face Radial Curve Representation](https://mlsanity.com/paper/2409.08301/) (2024)
- [Differentially Private Geodesic Regression](https://mlsanity.com/paper/2504.11304/) (2025)
- [Locally Private Inference for Riemannian Stochastic Optimization](https://mlsanity.com/paper/2609.22642/) (2026)

## Abstract (arXiv)

It is common for data structures such as images and shapes of 2D objects to be represented as points on a manifold. The utility of a mechanism to produce sanitized differentially private estimates from such data is intimately linked to how compatible it is with the underlying structure and geometry of the space. In particular, as recently shown, utility of the Laplace mechanism on a positively curved manifold, such as Kendall's 2D shape space, is significantly influences by the curvature. Focusing on the problem of sanitizing the Fr'echet mean of a sample of points on a manifold, we exploit the characterisation of the mean as the minimizer of an objective function comprised of the sum of squared distances and develop a K-norm gradient mechanism on Riemannian manifolds that favors values that produce gradients close to the the zero of the objective function. For the case of positively curved manifolds, we describe how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism, and demonstrate this numerically on a dataset of shapes of corpus callosa. Further illustrations of the mechanism's utility on a sphere and the manifold of symmetric positive definite matrices are also presented.