## 1 Introduction
In the research works of the literature [8; 14; 15; 16; 17; 18; 19; 20; 21; 22], all kinds of norm regularization terms in self-expressive models have been used to learn a block diagonal coefficient matrix, such as \(l_{1}\) norm, \(l_{2}\) norm, and nuclear norm. However, those regularization methods have two drawbacks: one is that the number of blocks in the coefficient matrix cannot be controlled, and another is that the learned coefficient matrix may not be an optimal block diagonal because of the noise in the data. To remedy these drawbacks, the block diagonal representation (BDR) subspace clustering algorithms [13; 27; 37; 38; 39; 40; 41] are proposed to pursue a \"good\" block diagonal structure of the coefficient matrix, such as implicit block diagonal low-rank representation (IBDLR) [40]. IBDLR constructs a novel model by integrating the block diagonal prior and implicit feature representation into the low-rank representation model. These algorithms based on the block diagonal representation gradually improve clustering performance. However, these subspace clustering methods based on BDR are not introduced into the similarity-preserving mechanism. To resolve the above problems, this paper first constructs a representation, which can better capture the rich structure information in original data. Second, to promote the affinity matrix from the self-expression learning framework to obey a \"good\" block diagonal structure for spectral clustering, we introduce the block diagonal representation (BDR) term to our model. Third, the kernel self-expressing framework is introduced into our model, which can efficiently process the nonlinear structure data. By fully considering these issues, we propose a novel robust subspace clustering method termed kernel block diagonal representation subspace clustering with similarity preservation (KBDSP) by embedding the block diagonal representation term and similarity-preserving regularizer into the kernel self-expressing framework. Experiments on nine datasets prove the effectiveness and robustness of our proposed KBDSP method. To sum up, the main contributions of this work can be summarized as follows:  
1. To capture the nonlinear structure of original data, a subspace clustering frame based on kernel self-expressing is introduced into our model. 2. To better preserve the similarity information between original data, a similarity-preserving regularizer is constructed and introduced to our model by minimizing the difference between inner products of original data and inner products of reconstructed data in the kernel space. 3. To obtain the similarity matrix with optimal block diagonal structure, a block diagonal representation is introduced to our model. Then, the optimal block diagonal matrix is directly captured by optimizing the objective function, overcoming the drawback that sparse or low-rank-based subspace clustering methods cannot obtain the optimal block diagonal matrix. 4. Our KBDSP model is proposed by integrating the block diagonal representation term and similarity-preserving regularizer into the kernel self-expressing frame. The proposed optimization model is solved by applying the alternating direction method of multipliers (ADMM). This paper includes the following five sections. Self-expression-based subspace clustering, kernel-based subspace clustering, and block diagonal representation term are briefly introduced in Section 2. Section 3 describes the proposed KBDSP method in detail. Section 4 presents the experimental results on nine datasets. Section 5 gives the conclusions.