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| <p> $D=\{d_{1},...,d_{n}\}$ </p> |
| <p> $\hat{ds}$ </p> |
| <p> $\langle T\rangle=\{\langle T\rangle_{1},\langle T\rangle_{2}\}$ </p> |
| <p> $T\neq$ </p> |
| <p> $\mathsf{CentralLap_{\triangle}}$ </p> |
| <p> $\langle g\rangle_{2}=\langle a_{jk}\rangle_{2}-\langle z\rangle_{2}$ </p> |
| <p> $d_{i}^{\prime}\leftarrow d_{i}+\mathsf{Lap}(\frac{1}{\varepsilon_{1}})$ </p> |
| <p> $Lap(\frac{1}{\varepsilon_{1}})$ </p> |
| <p> $\langle d\rangle_{i}=\langle w\rangle_{i}+\langle xy\rangle_{i}g+\langle xz% |
| \rangle_{i}f+\langle yz\rangle_{i}e+\langle x\rangle_{i}fg+\langle y\rangle_{i% |
| }eg+\langle z\rangle_{i}ef+(i-1)efg$ </p> |
| <p> $u_{1}=\langle w\rangle_{1}+\langle xy\rangle_{1}g+\langle xz\rangle_{1}f+% |
| \langle yz\rangle_{1}e$ </p> |
| <p> $re(T,T^{\prime})=\frac{|T-T^{\prime}|}{T}$ </p> |
| <p> $\langle f\rangle_{i}=\langle b\rangle_{i}-\langle y\rangle_{i},$ </p> |
| <p> $Gam_{1}(n,\frac{\triangle}{\varepsilon_{2}})-Gam_{2}(n,\frac{\triangle}{% |
| \varepsilon_{2}})$ </p> |
| <p> $D,D^{\prime}\in\mathcal{X}^{n}$ </p> |
| <p> $A_{ij}==1$ </p> |
| <p> $u=a_{ij}\times a_{ik}\times a_{jk}(i<j<k)$ </p> |
| <p> $A_{i}=\{a_{i1},...,a_{in}\}$ </p> |
| <p> $\varepsilon,n,d_{max},d_{max}^{\prime}$ </p> |
| <p> $y\in range(\mathcal{M})$ </p> |
| <p> $\mathsf{Project}$ </p> |
| <p> $\displaystyle(\mathbb{E}[T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime})]-\hat{% |
| T}(G,d_{max}^{\prime}))^{2}+\mathbb{V}[T^{\prime}(G,\varepsilon_{2},d_{max}^{% |
| \prime})]$ </p> |
| <p> $\mathsf{Gamma}$ </p> |
| <p> $\theta=1000$ </p> |
| <p> $\mathsf{Perturb}$ </p> |
| <p> $Gam_{2}$ </p> |
| <p> $\mathsf{Perturb}()$ </p> |
| <p> $O(nd_{max})$ </p> |
| <p> $\mathsf{GraphProjection}$ </p> |
| <p> $\langle x\rangle_{1},\langle y\rangle_{1},\langle z\rangle_{1},\langle w% |
| \rangle_{1},\langle o\rangle_{1},\langle p\rangle_{1},\langle q\rangle_{1}$ </p> |
| <p> $S\subseteq Range(\mathcal{M}_{i})$ </p> |
| <p> $\mathbb{E}[l_{2}^{2}(T(G),\hat{T}(G,d_{max}^{\prime}))]=0$ </p> |
| <p> $\varepsilon=\varepsilon_{1}+\varepsilon_{2}$ </p> |
| <p> $\langle T^{\prime}\rangle_{1}=\langle T\rangle_{1}+\langle\gamma\rangle_{1}$ </p> |
| <p> $\langle v_{i},v_{j}\rangle\in E$ </p> |
| <p> $\langle y\rangle_{i}$ </p> |
| <p> $ds[j]$ </p> |
| <p> $\langle f\rangle_{1}=\langle a_{ik}\rangle_{1}-\langle y\rangle_{1}$ </p> |
| <p> $Gam_{1}(n,\lambda)$ </p> |
| <p> $\mathsf{project}$ </p> |
| <p> $\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{0\}$ </p> |
| <p> $O(\frac{e^{\varepsilon}}{(e^{\varepsilon}-1)^{2}}(d_{max}^{3}n+\frac{e^{% |
| \varepsilon}}{\varepsilon^{2}}d_{max}^{2}n))$ </p> |
| <p> $u=\langle u\rangle_{1}+\langle u\rangle_{2}=\langle x\rangle_{1}$ </p> |
| <p> $\hat{A_{i}}\leftarrow\hat{A_{i}}\cup\{1\}$ </p> |
| <p> $\langle T\rangle\leftarrow\{\langle T\rangle_{1},\langle T\rangle_{2}\}$ </p> |
| <p> $Laplace$ </p> |
| <p> $Pr[\mathcal{M}(G)=y]\leq e^{\varepsilon}Pr[\mathcal{M}(G^{\prime})=y]$ </p> |
| <p> $\langle x\rangle_{i}$ </p> |
| <p> $\langle T\rangle_{1}$ </p> |
| <p> $1\times\sim 2\times$ </p> |
| <p> $S_{i\in\{1,2\}}$ </p> |
| <p> $A=\{A_{1},...,A_{n}\}$ </p> |
| <p> $\langle T\rangle_{2}\leftarrow\langle T\rangle_{2}+u_{2}$ </p> |
| <p> $G,G^{\prime}\in\mathcal{G}$ </p> |
| <p> $\langle\gamma_{i}\rangle_{2}\rightarrow S_{2}$ </p> |
| <p> $\mathsf{Lap(\frac{1}{\varepsilon_{1}})}$ </p> |
| <p> $O(\frac{d_{max}^{\prime 2}}{\varepsilon^{2}})$ </p> |
| <p> $u_{2}=\langle w\rangle_{2}+\langle xy\rangle_{2}g+\langle xz\rangle_{2}f+% |
| \langle yz\rangle_{2}e$ </p> |
| <p> $\mathbb{E}[l_{2}^{2}(T^{\prime}(G,\varepsilon_{2},d_{max}^{\prime}),\hat{T}(G,% |
| d_{max}^{\prime}))]=O(\frac{d_{max}^{\prime 2}}{\varepsilon_{2}^{2}})$ </p> |
| <p> $\langle v_{i},v_{j}\rangle$ </p> |
| <p> $Gam_{1}$ </p> |
| <p> $l_{2}(T,T^{\prime})=(T-T^{\prime})^{2}$ </p> |
| <p> $=w+xyg+xzf+yze+xfg+yeg+zef+efg$ </p> |
| <p> $\langle u\rangle_{2}$ </p> |
| <p> $D^{\prime}=\{d_{1}^{\prime},...,d_{n}^{\prime}\}$ </p> |
| <p> $\mathsf{Laplace}$ </p> |
| <p> $w=xyz,o=xy,p=xz,q=yz$ </p> |
| <p> $ds[j]\leftarrow\mathsf{DS}(d_{i},d_{j}^{\prime})$ </p> |
| <p> $d_{max}^{\prime}\geq 0$ </p> |
| <p> $\langle f\rangle_{2}=\langle a_{ik}\rangle_{2}-\langle y\rangle_{2}$ </p> |
| <p> $S\subseteq Range(\mathcal{M})$ </p> |
| <p> $\{d_{1}^{\prime},...,d_{n}^{\prime}\}$ </p> |
| <p> $Gam_{2}(n,\lambda)$ </p> |
| <p> $\displaystyle\frac{e^{-\varepsilon_{1}.|d^{\prime}-d_{i}|}}{e^{-\varepsilon_{1% |
| }.|d^{\prime}-d_{i}^{\prime}|}}=e^{\varepsilon_{1}.(|d^{\prime}-d_{i}^{\prime}% |
| |-|d^{\prime}-d_{i}|)}\leq e^{\varepsilon_{1}|d_{i}-d_{i}^{\prime}|}=e^{% |
| \varepsilon_{1}},$ </p> |
| <p> $(r_{1},...,r_{n})$ </p> |
| <p> $\mathsf{view}_{S_{i}}^{\Pi}\approx\mathsf{Sim}_{S_{i}}$ </p> |
| <p> $T(G)=\langle T\rangle_{1}+\langle T\rangle_{2}$ </p> |
| <p> $\mathsf{Project}()$ </p> |
| <p> $Gam_{1}=\mathsf{Gamma}(n,\frac{d_{max}^{\prime}}{\varepsilon_{2}})$ </p> |
| <p> $Gamma(x,n,\lambda)=\frac{(1/\lambda)^{1/n}}{\Gamma(1/n)}x^{\frac{1}{n}-1}e^{-% |
| \frac{x}{\lambda}},$ </p> |
| <p> $g=\langle g\rangle_{1}+\langle g\rangle_{2}$ </p> |
| <p> $d=a\times b\times c$ </p> |
| <p> $Lap(.)$ </p> |
| <p> $ds[k]$ </p> |
| <p> $a_{ij}\times a_{ik}\times a_{jk}=1$ </p> |
| <p> $(D^{\prime},d_{max}^{\prime})\leftarrow\mathsf{Max}(D,\varepsilon_{1})$ </p> |
| <p> $\langle x\rangle=\langle x\rangle_{1}+\langle x\rangle_{2}$ </p> |
| <p> $Lap(\lambda)=\sum_{i=1}^{n}[Gam_{1}(n,\lambda)-Gam_{2}(n,\lambda)],$ </p> |
| <p> $\mathsf{Count}()$ </p> |
| <p> $A=\{A_{1},A_{2},...,A_{n}\}$ </p> |
| <p> $\langle v_{2},v_{5}\rangle$ </p> |
| <p> $DS(d_{1},d_{2})$ </p> |
| <p> $\mathsf{Max}(.)$ </p> |
| <p> $u=a_{ij}\times a_{ik}\times a_{jk}$ </p> |
| <p> $\langle T\rangle\leftarrow\mathsf{Count}(\hat{A})$ </p> |
| <p> $|T(G)-T(G^{\prime})|=\triangle$ </p> |
| <p> $+\langle x\rangle_{1}fg+\langle y\rangle_{1}eg+\langle z\rangle_{1}ef$ </p> |
| <p> $d_{max}^{\prime}\geq d_{max}$ </p> |
| <p> $\gamma_{i}=\langle\gamma_{i}\rangle_{1}+\langle\gamma_{i}\rangle_{2}$ </p> |
| <p> $w=x\times y\times z,o=x\times y,p=x\times z,q=y\times z$ </p> |
| <p> $Pr[\mathcal{M}(G)\in S]\leq e^{\epsilon}Pr[\mathcal{M}(G^{\prime})\in S]$ </p> |
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