## 1 Introduction  
In real applications, high-dimensional datasets need to be processed [1, 2]. In recent years, subspace clustering developed as a hot research topic and was introduced to process high-dimensional data efficiently [3, 4, 5]. Subspace clustering has been widely applied in many application areas, such as computer vision [6], image representation and compression [7], motion segmentation [8], and face clustering [9]. The existing subspace clustering methods can be divided into five categories: iterative models [10]; statistical models [11]; algebraic models [12]; spectral clustering-based models [1, 13, 14, 15]; and deep learning-based models [15, 16]. Notably, the spectral clustering-based subspace clustering approaches have achieved outstanding performance. These methods have two main parts: (1) affinity matrix construction; (2) spectral clustering [17]. Recently, many spectral clustering methods have emerged [13, 18, 19, 20, 21], and the two most representative models are sparse subspace clustering (SSC) [14] and low-rank representation (LRR) [22]. SSC and LRR have obtained great success in many research areas. These methods based on sparse or low-rank representation techniques can obtain a coefficient matrix by using a self-expression model. The coefficient matrix is named the similarity matrix [23]. The similarity matrix is adaptively calculated and can effectively avoid the drawback that the number of neighbors is difficult to set for a graph in subspace clustering methods. However, all these methods can only process linear subspaces and they cannot handle the nonlinear data; but in fact, most data are nonlinear structures in the real world [24; 25]. In addition, although a lot of recent clustering studies based on deep learning are proposed [15; 16], the topics of these research studies are not the emphasis of this paper. To overcome the drawback that the existing linear subspace clustering methods cannot deal with nonlinear data, some kernel self-expression methods [26; 27; 28; 29; 30; 31] extended linear subspace clustering to nonlinear subspace clustering by employing the \"kernel strategy\", where the linear subspace clustering can be carried out. The two typical methods are kernelized SSC (KSSC) [30] and kernelized LRR (KLRR) [14]. KSSC and KLRR have captured nonlinear structure information in the input space and have made the most development in many applications. Although these kernel subspace clustering methods based on kernel self-expression can efficiently process the nonlinear structure data, some similarity information between samples may be lost when reconstructing the original data in the kernel space. In practice, real data with manifold structures present complex structures aside from sparse or low rank [32]. Hence, it is essential to construct a representation that can sufficiently implant the rich structure information of original data. Many methods discover the underlying structure by exploring data relations [33; 34; 35]. Some subspace clustering methods based on structure learning have been proposed, such as similarity learning via kernel-preserving embedding (SLKE) [35] and structure learning with similarity preserving (SLSP) [36]. SLKE constructs a model by retaining the similarity information between data, which conveys better performance. SLSP constructs a structure learning framework that contains the similarity information of original data, which overcomes the drawback that the SLKE algorithm might lose some low-order information of original data. In summary, although these methods can obtain good performance, they are subject to the learned similarity matrix that does not have an optimal block diagonal structure. Hence, the construction of the block diagonal similarity matrix is a hot topic that recently emerged in the field of subspace clustering methods.