## 2 Related Works  
Our work is based on the classic SSC and LRR subspace methods, which make impressive progression and have state of the art clustering performance. In the following, we give a review of these two methods and discuss some related subspace clustering works. The SSC and LRR subspace clustering methods are typical spectral clustering methods, which generally have two steps to implement clustering. First, the raw data are represented over a dictionary for obtaining the data representation coefficients and constructing the affinity matrix; Second, based on the affinity matrix, the clustering is implemented by using common spectral clustering methods, e.g., K-mean or NCut. As the dictionary training procedure usually has a high cost of computation for large datasets, an alternative scheme is to use the data itself as the dictionary, known as the self-expression property of data [5]. The first step in SSC and LRR is the core step as the affinity matrix has significant influence for the clustering result. So, many works focus on constructing an optimal affinity matrix by exploiting different constraints or regularizations. In SSC and LRR methods, the sparse and low rank constraints are added for the data representation matrix, respectively. For convenience, we denote the data for clustering by \(\mathbf{X}=[\mathbf{x}_{1},...,\mathbf{x}_{n}]\), which are assumed being generated from \(k\) subspaces, i.e., \(\mathbf{X}=\bigcup_{j=1}^{k}\mathbf{X}_{j}\) s.t. \(\bigcap_{j=1}^{k}\mathbf{X}_{j}=\phi\) and \(\forall\mathbf{x}_{i}\in\mathbf{X}\), \(\exists\mathbf{X}_{j^{*}}\), \(\mathbf{x}_{i}\in\mathbf{X}_{j^{*}}\). The objective of subspace clustering is to find the proper subspace for each datum. For this purpose, SSC implements clustering by representing the data as the sparse coefficients. The original SSC model is usually formulated as follows [5]:  
\[\min_{\mathbf{Z}}\|\mathbf{X}-\mathbf{X}\mathbf{Z}\|_{F}^{2}+\lambda\|\mathbf{ Z}\|_{1},\text{ s.t. }diag(\mathbf{Z})=0, \tag{1}\]  
where \(\mathbf{Z}\) is the sparse representation coefficient matrix and \(\lambda\) is the weight to balance the data self representation error and the sparse representation term. In SSC, the sparse constraint is just applied for the coefficient of each datum. However, the correlation among the whole dataset or its sparse representation coefficients are not described. Based on this observation, to further reveal the holistic sparsity of the data, LRR proposed using low rank constraint for the coefficient matrix. Additionally, to solve the NP-hard optimization problem with low rank constraint, the nuclear norm \(\|\cdot\|_{*}\) is generally used to replace the low rank constraint. Thus the LRR model can be formulated as the following optimization model [6]:  
\[\min_{\mathbf{Z}}\|\mathbf{X}-\mathbf{X}\mathbf{Z}\|_{2,1}+\lambda\|\mathbf{Z} \|_{*}, \tag{2}\]  
where \(\|\cdot\|_{2,1}\) norm is used to obtain robustness for noises and outliers. The above SSC and LRR models only use sparse and low rank constraint to the data coefficients, while the other data intrinsic property, such as the non-linear structure of high dimensional data, is not considered. To reveal the non-linear property of high dimensional data, some researchers proposedto use the Laplacian graph constraint to model the local manifold structure of the data. The SMR method [14] enforces the grouping effect on the data representation matrix by using Laplacian graph constraint as follows:  
\[\min_{\mathbf{Z}}\lVert\mathbf{X}-\mathbf{XZ}\rVert_{F}^{2}+\lambda tr(\mathbf{ ZL_{W}}\mathbf{Z}^{T}), \tag{3}\]  
where \(\mathbf{Z}\) is the data self representation, and \(\mathbf{L_{W}}\) is the Laplacian graph matrix generally constructed from the local neighbours of each data sample. For example, from the local neighbourhood with \(K\) nearest neighbours of each data sample, a weight matrix \(\mathbf{W}\in\mathbb{R}^{n\times n}\) can be constructed with the element \(\mathbf{W}_{ij}\) defined as follows:  
\[\mathbf{W}_{ij}=