We're given a vector $x\in \mathbb{R}^d$ whose coordinates where sampled from a known normal distribution $\mathcal{N}(0, \sigma^2)$.
How should I send this vector while maintaining (local) differential privacy? with some sensitivity over its $\ell_2$ norm (i.e., two close vectors should not be distinguishable). Is there a way to take the fact that we know the source distribution into account?
Thank you!
The fact that $x \sim N(0, \sigma^2 I)$ does not help you much because the sensitivity of two neighboring data points $x$ and $x'$ is unbounded. Differential privacy acts over all possible neighboring datasets, not just likely ones. For a discussion on this (very important) point, see https://differentialprivacy.org/average-case-dp/. In fact, the blog post discusses precisely your setting where the data points are assumed to be sampled from the Gaussian!
You will need to truncate the data points by clipping them to some norm $\Delta$:
$\bar{x} \leftarrow \frac{x}{max\left(1, \frac{\lVert x \rVert_2}{\Delta}\right)}$
Then typically you would add Gaussian noise with scale proportional to $\Delta$ to achieve $(\epsilon, \delta)$-approximate DP. Since Gaussian noise is tailored to zero-concentrated DP you could also use that privacy definition, in which case you will achieve $\rho = \frac{\Delta^2}{2 \sigma^2}$ zero concentrated DP (see Bun & Steinke 2016 proposition 1.6)
