\section{Preliminary}
A sequence $\{ X_n \}$ of random variables in $\mathbb{R}^d$ is defined to \textit{ converge in probability} towards the random variable $X$ if for all $\beta > 0$,
$$
\lim_{n\to\infty}\bbP\left[ \lVert X_n - X\rVert > \beta \right] = 0.
$$
The norm notation $\lVert\cdot\rVert$ denotes $\ell^2$ norm in our paper. We denote this convergence as $\plim_{n\to \infty}X_n = X$.

\textit{Differential privacy} (DP; \citep{dwork2006calibrating, dwork2014algorithmic}) is a quantifiable and rigorous privacy framework, which is formally defined as follows.
\begin{definition}[$(\varepsilon,\delta)$-differential privacy]
	A randomized mechanism $\calM:\calD \to \cal R$ with domain $\calD$ and range $\calR$ satisfies $(\varepsilon,\delta)$-differential privacy if for any two adjacent datasets $D, D'\in\calD$, which differ at exactly one data point, and for any subset of outputs $S\subseteq \calR$, it holds that
	$$
	\bbP[\calM(D)\in S] \leq e^{\varepsilon}\cdot\bbP[\calM(D')\in S] + \delta.
	$$
\end{definition}

\textit{Gaussian mechanism}~\citep{dwork2014algorithmic} is a post-hoc mechanism to convert a deterministic real-valued function $f:\calD \to \bbR^m$ to a randomized algorithm with differential privacy guarantee. 
It relies on \emph{sensitivity} of $f$, denoted by $S_f$, which is defined as the maximum difference of output $\lVert f(D) - f(D') \rVert$.
We define Gaussian mechanism for differential privacy as below.

\begin{lemma}[Gaussian mechanism]
\label{lem:gm}
For any deterministic real-valued function $f:\calD \to \bbR^m$ with sensitivity $S_f$, we can define a randomized function by adding Gaussian noise to $f$:
$$
f^{dp}(D):=f(D) + R,
$$
where R is sampled from a multivariate normal distribution $\calN\left(\mathbf{0}, S_f^2 \sigma^2\cdot I\right)$. 
When $\sigma \geq \frac{\sqrt{2\log\left(1.25 / \delta\right)} }{\varepsilon}$, $f^{dp}$ is $(\varepsilon,\delta)$-differentially private for $0 < \varepsilon \leq 1$ and $\delta>0$.
\end{lemma}
To simplify notations, we define $\sigma_{\varepsilon, \delta}:=\frac{\sqrt{2\log\left(1.25 / \delta\right)} }{\varepsilon}$.

\textit{Johnson-Lindenstrauss lemma}~(JL; \citep{johnson1984extensions, achlioptas2003database}) is a technique to compress a set of vectors $S=\{v_1, \cdots, v_l \}$ with dimension $d$ to a lower dimension space $k < d$. 
With a proper selection $k$, it is able to approximately preserve the inner product between any two vectors in the set $S$ with high probability. 
We specifically introduce the Bernoulli version of JL Lemma, which is extended from Theorem 1.1 in \cite{achlioptas2003database}.

\begin{lemma}[JL Lemma for inner-product preserving (Bernoulli)]
\label{lem:jl}
Suppose $S$ is an arbitrary set of $l$ points in $\bbR^d$ and suppose $s$ is an upper bound for the maximum $\ell^2$-norm for vectors in $S$. Let $B$ be a $k\times d$ random matrix, where $B_{ij}$ are independent random variables taking value from $1$ or $-1$ with probability $1/2$ respectively.
With the probability at least $ 1 - (l+1)^2\exp\left(-k\left(\frac{\beta^2}{4} - \frac{\beta^3}{6}\right)\right)$, $\forall \mathbf{u},\mathbf{v}\in S$, we have
$$
\frac{\mathbf{u}^\top\mathbf{v}}{s^2} - 4\beta \leq \frac{\left( B\mathbf{u}/\sqrt{k}\right)^\top\left( B\mathbf{v} / \sqrt{k}\right)}{s^2} \leq \frac{\mathbf{u}^\top\mathbf{v}}{s^2} + 4\beta.
$$
\end{lemma}

