## 4 Experimental Results and Analysis
### Evaluation Metrics  
To effectively evaluate the KBDSP algorithm and other advanced algorithms quantitatively, three public evaluation metrics are adopted, which are clustering accuracy (ACC), normalized mutual information (NMI), and Purity [37].  
Suppose \(n\) is the total number of sample points. We use \(y_{i}\) and \(\hat{y}_{i}\) to denote the cluster label and the ground truth cluster label, respectively. Then accuracy is defined as follows:  
\[\text{ACC}=\frac{\sum_{i=1}^{n}\delta(\hat{y}_{i},map(y_{i}))}{n} \tag{21}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **Type** & **Instances** & **Features** & **Classes** \\ \hline YALE & Face & 165 & 1024 & 15 \\ JAFFE & Face & 213 & 676 & 10 \\ ORL & Face & 400 & 1024 & 40 \\ AR & Face & 840 & 768 & 120 \\ COIL20 & Object & 1440 & 1024 & 20 \\ BA & Handwritten digits & 1404 & 320 & 36 \\ TR11 & Text & 414 & 6429 & 9 \\ TR41 & Text & 878 & 7454 & 10 \\ TR45 & Text & 690 & 8261 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Detailed information of the nine widely used datasets.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Parameter Settings** \\ \hline SC & - \\ KSSC & \(\lambda=\left\{10,\ 30,\ 50\right\}\), \(\rho=\left\{\mathbf{10}^{-4},10^{-3},10^{-2}\right\}\) \\ KLRR & \(\lambda=\left\{1,\ 5,\ 10\right\}\) \\ SLKEs & \(\gamma=\left\{10^{-3},10^{-2},\mathbf{10}^{-1}\right\}\), \(\mu=\left\{10^{-2},\mathbf{10}^{-1},1\right\}\) \\ SLKEr & \(\gamma=\left\{10^{-3},\mathbf{10}^{-2},10^{-1}\right\}\), \(\mu=\left\{10^{-2},\mathbf{10}^{-1},1\right\}\) \\ SLSPs & \(\alpha=\left\{10^{-4},\mathbf{10}^{-3},10^{-2},\ 10^{-1}\right\}\), \(\beta=\left\{10^{-2},\mathbf{10}^{-1},1\right\}\), \(\mu=\left\{10^{-2},\mathbf{10}^{-1},1\right\}\) \\ SLSPr & \(\alpha=\left\{10^{-4},\mathbf{10}^{-3},10^{-2},\ 10^{-1}\right\}\), \(\beta=\left\{10^{-2},\mathbf{10}^{-1},1\right\}\), \(\mu=\left\{10^{-2},\mathbf{10}^{-1},1\right\}\) \\ IBDLR & \(\lambda=\left\{10^{-4},\mathbf{10}^{-3},10^{-2},\ 10^{-1}\right\}\), \(\gamma=\left\{10^{-3},\mathbf{10}^{-2},10^{-1}\right\}\) \\ KBDSP & \(\alpha=\left\{10^{-5},\ 10^{-4},\ 10^{-3},\mathbf{10}^{-2},\ 0.1,\ 1\right\}\), \(\beta=\left\{10^{-5},\ \mathbf{10}^{-3},\ 10^{-1},\ 1\right\}\), \(\gamma=\left\{10^{-2},\ \mathbf{10}^{-1},\ 1,\ 10,\ 30,\ 50\right\}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter settings.  
where \(\delta(x,y)\) denotes the Kronecker delta function, and it takes the value 1 if \(x\) and \(y\) are equal, otherwise it takes the value 0. The \(map(\cdot)\) is a mapping function by which each cluster label can be mapped to ground truth cluster label according to the Kuhn-Munkers algorithm.  
We use \(S\) and \(\hat{S}\) to denote the set of clustering that generated from a clustering algorithm and the ground truth set of clustering, respectively. Then NIM is defined as:  
\[\text{NIM}(S,\hat{S})=\frac{\sum_{s\in S,\;s\in S}p(s,\hat{s})\log\frac{p(s, \hat{s})}{p(\hat{s})p(\hat{s})}}{\max(\text{H}(S),\text{H}(\hat{S}))} \tag{22}\]  
where \(p(s)\) and \(p(\hat{s})\) are the marginal probability distribution functions of \(S\) and \(\hat{S}\), respectively. \(p(s,\hat{s})\) denotes the joint probability density functions of \(S\) and \(\hat{S}\). \(\text{H}(\cdot)\) represents the entropy function. The larger the value of NMI, the higher similar \(S\) and \(\hat{S}\) are.  
Purity is used to measure the overall precision of the clusters, and it is defined as  
\[Purity=\frac{1}{n}{\sum_{i}}\max_{j}\lvert S_{i}\cap\hat{S}_{j}\rvert \tag{23}\]  
The higher these three metrics are, the better the clustering performance.