# Differentially Private Geodesic Regression

arXiv: https://arxiv.org/abs/2504.11304 · HTML: https://mlsanity.com/paper/2504.11304/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 34 references that are in the map.
- First posted on arXiv 15 Apr 2025.

## Builds on (most cited first, 18 of 18)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [Privacy for Free: Posterior Sampling and Stochastic Gradient Monte Carlo](https://mlsanity.com/paper/1502.07645/) (2015)
- [Revisiting differentially private linear regression: optimal and adaptive prediction & estimation in unbounded domain](https://mlsanity.com/paper/1803.02596/) (2018)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Differentially Private Ordinary Least Squares](https://mlsanity.com/paper/1507.02482/) (2015)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Differentially Private Simple Linear Regression](https://mlsanity.com/paper/2007.05157/) (2020)
- [Differentially Private Markov Chain Monte Carlo](https://mlsanity.com/paper/1901.10275/) (2019)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [An accelerated first-order method for non-convex optimization on manifolds](https://mlsanity.com/paper/2008.02252/) (2020)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [Regression with Distance Matrices](https://mlsanity.com/paper/1303.3750/) (2013)
- [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)
- [Shape And Structure Preserving Differential Privacy](https://mlsanity.com/paper/2209.12667/) (2022)
- [Gaussian Differential Privacy on Riemannian Manifolds](https://mlsanity.com/paper/2311.10101/) (2023)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)

## Abstract (arXiv)

In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. Classical linear regression, one of the most fundamental methodologies of statistical learning, captures the relationship between an independent variable and a response variable which both are assumed to live in Euclidean space. Thus, geodesic regression emerged as an extension where the response variable lives on a Riemannian manifold. The parameters of geodesic regression, as with linear regression, capture the relationship of sensitive data and hence one should consider the privacy protection practices of said parameters. We consider releasing Differentially Private (DP) parameters of geodesic regression via the K-Norm Gradient (KNG) mechanism for Riemannian manifolds. We derive theoretical bounds for the sensitivity of the parameters showing they are tied to their respective Jacobi fields and hence the curvature of the space. This corroborates, and extends, recent findings of differential privacy for the Fr'echet mean. We demonstrate the efficacy of our methodology on the sphere, S2subsetℝ3, the space of symmetric positive definite matrices, and Kendall's planar shape space. Our methodology is general to any Riemannian manifold, and thus it is suitable for data in domains such as medical imaging and computer vision.