## 3 Peak Clustering Based Data Restoration  
Based on the proliferation result of data samples by introducing generator discussed above, a so-called peak clustering based incomplete data restoration method is proposed in this section. In order to overcome the restoration failures of traditional algorithms with linear inseparable data, the proposed method constructs as few as possible open intervals with a fixed neighborhood radius for all of the data samples. Then, the set of open intervals form a finite open coverage, avoiding the interference of linear inseparable data samples on clustering results, as shown in Figure 3.  
Inspired by [PERSON]'s work in Reference [23], peak clustering algorithm is proposed for incomplete data restoration to improve the calculation efficiency, while sustaining the restoration precision. Supposing the finite open coverage \(Coverage_{i}(d)\) contains \(n_{i}\) data samples \(\left\{d_{j}\right\}_{j=1}^{n_{i}}\), calculate the peak distance of density peaks (distance between data sample and density peak point) \(Dist(d_{j},\;Temp\_Peaks_{i}(d))\). Then, \(n_{i}\) clusters are constructed according to the phase angle with the density peak point as the center, and each cluster contains only one sample [24]. If the absolute value of the peak distance difference of the cluster with similar phase angle is smaller than or equal to the threshold value, the two classes are merged, and the distance between the peak point of the new class and the peak to peak value of the density is calculated.  
Figure 2: Diagram of generative adversarial networks (GANs) calculation processing.  
If the absolute value of the peak distance difference of the clusters with the similar phase angle is no larger than the threshold value \(k\), combine the two clusters and calculate new density peak points and peak distances. Repeat the operations above, until the absolute value of the peak distance difference of the clusters with the similar phase angle becomes larger than the threshold value \(k\), or the total cluster number becomes 1. Then, end the iteration and output the clustering result of the last iteration.  
Finally, after the peak clustering of targeted data samples, the weighted averages of corresponding values in complete samples could be used as the predictive values of missing data. The concept of entropy in information theory is introduced, and the weighting coefficient is determined by the similarity between data samples. Generally speaking, the process of peak clustering based data restoration could be shown in the following Table 1.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Inputs** & Establish the combined dataset \(D_{GANs}=\{d_{i}\}_{i=1}^{N+M}\); initialize threshold values \(R_{1}\), \(R_{2}\) and clustering threshold value \(k\); \\ \hline \multirow{5}{*}{**Step 1.1**: & Establish the finite open coverage of the targeted combined dataset: \\  & **Step 1.1**: if \(\textit{Centre}\
eq\varphi\), randomly select \(d\in\textit{Centre}\) from the central point set; otherwise, go to Step 1.4. \\  & **Step 1.2**: calculate \(i\)-th open interval: \\  & \(\textit{Corelarge}_{i}(d)=\{d^{\prime}\in D_{GANs}|Dist(d,d^{\prime})\leq R_{1}\}\); \(\textit{Coverage}=\{\textit{Coverage}_{i}(d)\}\) \\  & and renew the temporary peak point set \(\textit{Temp\_P\_P\_R\_R\_}\leftarrow\{