Differentially Private Geodesic Regression

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 Fréchet mean. We demonstrate the efficacy of our methodology on the sphere, $S_2\subset\mathbb{R}^3$, 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.

💡 Research Summary

The paper addresses the problem of releasing the parameters of geodesic regression— a regression model where the response variable lies on a Riemannian manifold— under the rigorous privacy guarantees of differential privacy (DP). Classical linear regression assumes both predictor and response live in Euclidean space; however, many modern datasets (directional data, diffusion tensors, shape data) naturally reside on curved manifolds. Geodesic regression, introduced by Fletcher (2011), captures the relationship between a real‑valued predictor and a manifold‑valued response by fitting a geodesic curve parameterized by a footpoint p (the “intercept”) and a shooting vector v (the “slope”).

To privatize (p, v) the authors adopt the K‑Norm Gradient (KNG) mechanism, recently extended to manifolds (Soto et al., 2022). KNG samples from a density proportional to exp{−σ⁻¹‖∇E(z; D)‖_z}, where E is the least‑squares energy of geodesic regression and σ is set to the global sensitivity Δ divided by the privacy budget ε (or 2Δ/ε when the normalizing constant depends on the footpoint). The central technical contribution is a tight bound on Δ for each parameter, expressed in terms of Jacobi fields and the sectional curvature of the underlying manifold.

The authors first derive the Riemannian gradients of the energy with respect to p and v:

∇p E = −(1/n) ∑{i=1}^n d_p Exp(p, x_i v)† ε_i,
∇v E = −(1/n) ∑{i=1}^n x_i d_v Exp(p, x_i v)† ε_i,

where ε_i = Log(Exp(p, x_i v), y_i) and † denotes the adjoint of the differential of the exponential map. The differentials d_p Exp and d_v Exp are shown to be the values at t = 1 of Jacobi fields J(t) along the geodesic γ(t)=Exp(p, t v) with specific initial conditions (J(0)=u₁, J′(0)=0 for d_p Exp; J(0)=0, J′(0)=u₂ for d_v Exp).

Sensitivity analysis proceeds under two mild assumptions: (1) the sectional curvature K of the manifold is bounded between constants κ_l and κ_h, and (2) the data lie inside a geodesic ball of radius r that respects curvature‑dependent limits (r ≤ π/(8√κ_h) for positively curved manifolds, r < τ_m for negatively curved ones) and the fitted geodesic is τ‑close to every observation. Under these conditions, the authors prove that the footpoint sensitivity satisfies

Δ_p ≤ { 2τ/n if κ_l ≥ 0,
   2τ/(n·cosh(2√{-κ_l}(τ_m+τ))) if κ_l < 0 },

and a similar bound for the shooting vector Δ_v that involves the maximal predictor value and a curvature‑dependent constant. The bounds reveal that positive curvature shrinks sensitivity (hence less noise) while negative curvature inflates it, a phenomenon explained by the Rauch comparison theorem: larger curvature forces Jacobi fields to contract, reducing the influence of any single data point on the gradient.

Algorithmically, the procedure is: (i) scale predictors to