Summary
This work studies an extension of a Gaussian privacy mechanism on Riemannian manifolds, so as to extend the notion of mu-GDP for estimators with given global sensitivity. The idea is to sample a random point on the manifold with a Gibbs density (wrt to the Riemannian volume) whose potential is proportional to the squared distance to the given estimator (that would simply be a Gaussian density on a Euclidean space). It is well known that in order for such a distribution to be well defined, it is enough to assume that the sectional (or, equivalenty, Ricci) curvature of the manifold is bounded from below. The main challenges, then, reside in : (1) tuning the temperature parameter of the Gibbs distribution so as to obtain mu-GDP, and (2) sample from such a distribution.
While giving general formulae for the choice of the temperature parameter, the authors study the case of homogeneous manifolds, as well as model manifolds with constant curvature, also providing simulation studies.
Differential privacy on non-linear spaces, in particular Riemannian manifolds, has not been studied a lot yet, and this paper aims at providing new, simple insights.
Strengths
• The question tackled in this paper is important, given the increasing need of dealing with non-linearity in data.
• The paper is written clearly and the main text is easy to follow.
Weaknesses
• While the paper addresses an important question, the execution remains shallow. Laplace and Gaussian mechanisms can be very easily extended to Riemannian manifolds, and the computations that yield to determine the calibration of their parameters are very similar to the Euclidean case. One challenge that is barely addressed here (see the very brief remark just before Definition 4.1) is the understanding of the connexion between these parameters and the geometry (in particular, curvature bounds) of the ambient manifold. Instead, the authors choose to focus on very simple (and perhaps not realistic in most applications) manifolds, where everything becomes much simpler. Hence, I think that a big, if not essential, part of the picture is missing in this work.
• The presence of Definition 3.1 (mu-GDP mechanisms on manifolds) is obscure to me: A definition is already given in Definition 2.2, and I do not see why it should be different on manifolds. Should Definition 3.1 actually be a theorem? I mean that one should check that M is mu-GDP in the sense of Definition 2.2 if and only if it is (eps,delta(eps))-DP for all eps>0, with delta(eps) given by the formula that appears in Definition 3.1 (something similar to Theorem A.1).
• Some proofs seem either wrong or incomplete to me, here are some pointers:
- One line after Theorem A.1, this theorem is interpreted using the O(eps^2) notation, which completely forgets the dependence in mu. Hence, the whole proof of Theorem 3.1 forgets mu and is, therefore, imprecise/incomplete.
- I did not understand the first line of the proof of Theorem 3.1. Also, there, what is x (it does not appear in the formula)?
- In Equation 4, I think that the normalizing constant Z_{eta,sigma} is missing in the upper bound.
Some minor remarks:
• Line 92, there is a typo.
• Line 99: « the » should be replaced with « a ».
• Line 237: Remove « satisfies »
• Every citation should come to a pointer to a specific definition or result, especially when the citation refers to a whole book (e.g., Dudley, 2002)!
• In Theorem A.1, what is the ‘t’ in the subscript of mu_t?
• Also in Theorem A.1, « privacy profile » has not be defined anywhere.
Questions
See paragraph on weaknesses.
Rating
4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.