# Comparative Analysis of Local Differential Privacy Schemes in Healthcare Datasets
## 2 Preliminaries  
### LDP Overview and the Curse of High Dimensionality  
The Local Differential Privacy (LDP) model comprises two entities: users and a central server, with the objective of enabling users to share information while safeguarding  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Datasets** & **Perturbation** & **Estimation** & **Evaluation** \\ \hline Sung [16] & Synthetic & Normalize & NA & (ML Models) \\  & eICU & Bounded Laplacian & & misclassification rates \\  & & Discretization & & mean squared error \\ \hline Arcolezi [2] & Adults & GRR & NA & (ML Models) \\  & ACS Coverage & BLH & & f1-score \\  & LSAC & OLH & & AUC \\  & & RAPPOR & & DI \\  & & OUE & & SPD \\  & & SS & & EOD \\  & & THE & & OAD \\ \hline LOPUB [6] & Nursery & & LASSO & \\  & Lung Cancer & & with & \\  & Stroke & & EM & \\ \hline LOCOP [10] & NHANES & & LASSO & \\  & Bank & & with & \\  & National & & Gaussian & Joint probability \\  & Massachusetts & Bloom Filter & Copula & distribution \\ \hline  & & & Randomize Response & \\  & & Adults & & \\  & Hypertension & & & \\ \cline{2-5} Ours & Skin Cancer & & BRR [17] & \\  & Diabetes & & & \\  & US CENSUS & & & \\  & Cirrhosis & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Differences between [PERSON] [16], [PERSON] [2], [PERSON] [6], LOCOP [10], and this paper. The datasets pertaining to the healthcare industry are underlined.  
their privacy. The shared information serves various purposes, including computing statistics about the dataset. LDP has proven effective in sharing and recovering the original distribution of a dataset while preserving user privacy. Numerous LDP approaches in the literature employ diverse methods to protect user information [1; 4; 18].  
As a general example, we can think of a dataset where the attributes exhibit the following cardinalities \(\Omega_{gender}=\{\text{male, female}\}\), \(\Omega_{country}=\{USA,CAN,CHN,JAP,MEX\}\), and \(\Omega_{marital}=\{\text{married, widowed, separated, divorced, single}\}\). The objective of LDP is for users to transmit their information to the central server while preserving anonymity. Subsequently, the central server can use the information provided by the users to compute the original distribution of this dataset.  
In this example, if the central server tries to recover the distribution of one attribute, it will face the original cardinalities \(|\Omega_{gender}|=2\), \(|\Omega_{marital}|=5\), and \(|\Omega_{country}|=5\), where the average of these cardinalities is computed as  
\[\overline{\omega_{k}}=\frac{\sum_{k}|\Omega_{(\overset{gender,marital,country} {k})}|}{(\overset{gender,marital,country}{k})}. \tag{1}\]  
Here, \(|\Omega_{(\overset{gender,marital,country}{k})}|\) denotes the cardinality of the unique combinations of the attributes for gender, marital status, and country, while \((\overset{gender,marital,country}{k})\) represents the count of unique combinations for a given value of \(k\). The average of these cardinalities is computed using Equation (1). Substituting (\(k=1\)), the result is \(\overline{\omega_{1}}=\frac{\sum_{1}|\Omega_{(\overset{gender,marital,country} {k})}|}{(\overset{gender,marital,country}{k})}=\frac{2+5+5}{3}=4\), and the cardinality has risen; for two attributes (\(k=2\)), the average is \(\overline{\omega_{2}}=15\). Finally, the cardinality of three attributes (\(k=3\)) is \(\overline{\omega_{3}}=50\). The complexity of the cardinality increases as the number of attributes grows. For instance, consider a real-world dataset with tens of attributes. We can see the challenge when the central server tries to recover \(k\) attributes, as the average cardinality size increases with the data dimensionality.