# Differentially Private Fr'echet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric

arXiv: https://arxiv.org/abs/2208.04245 · HTML: https://mlsanity.com/paper/2208.04245/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 73 references that are in the map.
- First posted on arXiv 8 Aug 2022.

## Builds on (most cited first, 12 of 12)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Lognormal Distributions and Geometric Averages of Positive Definite Matrices](https://mlsanity.com/paper/1407.6383/) (2014)
- [Second-order Convolutional Neural Networks](https://mlsanity.com/paper/1703.06817/) (2017)
- [tensorflow-riemopt: A Library for Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2105.13921/) (2021)
- [ICLR 2021 Challenge for Computational Geometry & Topology: Design and Results](https://mlsanity.com/paper/2108.09810/) (2021)

## Built on by (most cited first, 8 of 8)

- [DPZero: Private Fine-Tuning of Language Models without Backpropagation](https://mlsanity.com/paper/2310.09639/) (2023)
- [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)
- [Low-dimensional approximations of the conditional law of Volterra processes: a non-positive curvature approach](https://mlsanity.com/paper/2405.20094/) (2024)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)
- [Random matrix theory improved Fr'echet mean of symmetric positive definite matrices](https://mlsanity.com/paper/2405.06558/) (2024)
- [Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms](https://mlsanity.com/paper/2604.20761/) (2026)
- [Locally Private Inference for Riemannian Stochastic Optimization](https://mlsanity.com/paper/2609.22642/) (2026)

## Abstract (arXiv)

Differential privacy has become crucial in the real-world deployment of statistical and machine learning algorithms with rigorous privacy guarantees. The earliest statistical queries, for which differential privacy mechanisms have been developed, were for the release of the sample mean. In Geometric Statistics, the sample Fr'echet mean represents one of the most fundamental statistical summaries, as it generalizes the sample mean for data belonging to nonlinear manifolds. In that spirit, the only geometric statistical query for which a differential privacy mechanism has been developed, so far, is for the release of the sample Fr'echet mean: the Riemannian Laplace mechanism was recently proposed to privatize the Fr'echet mean on complete Riemannian manifolds. In many fields, the manifold of Symmetric Positive Definite (SPD) matrices is used to model data spaces, including in medical imaging where privacy requirements are key. We propose a novel, simple and fast mechanism - the tangent Gaussian mechanism - to compute a differentially private Fr'echet mean on the SPD manifold endowed with the log-Euclidean Riemannian metric. We show that our new mechanism has significantly better utility and is computationally efficient -- as confirmed by extensive experiments.