## 4 Experimental Results and Analysis
### Experimental Results and Analysis  
We perform the proposed method on the nine datasets including Yale, JAFFE, ORL, AR, COIL20, BA, TR11, TR41, and TR45. We conduct all the experimental methods on the 12 kernels, and the average experimental results over 12 kernels are computed at each run of the experiment. The experiment is repeated ten times for each experimental method, and the average experimental results are displayed in Table 3. The best results are highlighted with boldface. As can be seen in Table 3, the proposed method presents the best performance in comparison with other methods in most cases, which shows the effectiveness of introducing block diagonal representation term and the similarity-preserving term in our model. Specifically, we obtain the following eight findings from the results listed in Table 3.  
1. SC is a representative clustering approach. Compared to the SC algorithm, our proposed KBDSP obtains better results in terms of three evaluation metrics, namely ACC, NMI, and Purity. From Table 3, we can see the average values of three evaluation metrics of KBDSP are 19%, 18%, and 22%, which are higher than SC, respectively. This is attributed to the advantage that the learned \(Z\) is an input of SC instead of the kernel matrix in SC. 2. KSSC and KLRR methods have made satisfactory results in many fields because they can exploit the nonlinear structure of original data. It is worth mentioning that our method outperforms these two methods in most cases. The superiority of KBDSP derives from our similarity-preserving strategy. 3. Compared to SLKEs and SLKEr, KBDSP obtains a higher performance. The reasons for performance improvement have two aspects: one is that our kernel self-expression frame can preserve some low-order information of input data, which is lost in SLKEs and SLKEr methods; the other is that we can learn a similarity matrix with block diagonal structure by introducing the block diagonal representation term to our model. 4. SLSPs and SLSPr can not only handle nonlinear datasets, but also preserve similar information. The experimental results of the two methods are shown in Table 3, from which we can see they have improved performance compared to SC, KSSC, KLRR, SLKEs and SLKEr. However, KBDSP still outperforms them in most instances. For example, from Table 3, we can see the average values of three evaluation metrics of KBDSP are 14%, 15%, and 2%, which are higher than SLSPs, respectively. These results prove that the introduced block diagonal representation term can help boost performance. 5. IBDLR and KBDSP enable the learning desired affinity matrix with an optimal block diagonal structure by importing block diagonal representation term. From Table 3, it can be seen that KBDSP and IBDLR have better performance than other algorithms except SLSPs on all datasets, which verifies that KBDSP and IBDLR methods with block diagonal representation term are effective. This indicates that the KBDSP and IBDLR are beneficial for datasets that have many classes, and that they can capture the block diagonal structure of data. For COIL20 and BA datasets, which have a greater number of instances, Table 3 shows that our KBDSP achieves nearly 13% higher than other compared methods apart from IBDLR on the COIL20 dataset. In addition, as can be seen in Table 3, we can observe that KBDSP outperforms IBDLR on the COIL20 dataset; this is because our method considers the similarity-preserving strategy, but the IBDLR method does not. 6. For the TR11, TR41, and TR45 datasets, which have high dimensions of features, SLSPr presents better performance than IBDLR because it introduces a similarity-preserving mechanism. Benefiting from the similarity-preserving strategy and block diagonal representation term, the proposed KBDSP consistently outperforms IBDLR and even SLSPr in most cases on the TR11, TR41, and TR45 datasets. This indicates KBDSP is beneficial, performing well with the amount of information provided by datasets that have rich features, such that KBDSP can exploit the intrinsic structure of data.