## 1 Introduction  
Data clustering or segmentation is an active research topic in data mining, signal processing and unsupervised learning [1; 2]. In practice, many data are generated from a space which has intrinsic subspace structure, i.e., the space is composed of union of multiple subspaces. In this case, it is necessary to exploit and reveal the subspace structure, and so different subspace clustering methods were proposed in recent years [3; 4]. In all of the subspace clustering methods, the spectral clustering methods based on the affinity matrix of data are considered to have good prospects. The typical methods are the Sparse Subspace Clustering (SSC) [5] and the Low Rank Representation (LRR) methods [6]. The key problems of the subspace spectral clustering are how to find the proper data representation and how to construct the affinity or similarity matrix of the data for spectral clustering. In SSC and LRR methods, the original data are firstly represented by the self expression model with sparse or low-rank constraints. Then the affinity matrix is constructed from the representation coefficients. Using the affinity matrix as input, the final clustering results can be obtained by common spectral clustering algorithms, such as K-means or NCut methods. It is critical for the spectral subspace clustering methods to construct a proper affinity matrix; one should exploit the data intrinsic properties to form necessary regularization for the data representation model. Based on the assumption that the data sample could only be represented by the samples coming from the same subspace it belongs to, SSC adds sparse constraint onto the representation coefficients matrix by \(l_{1}\) norm [7]. However, the sparse constraint in SSC is an individual constraint for each datum representation. There is no consideration of correlation among the data. For this purpose, LRR adds the holistic constraint of low rank onto the representation coefficients matrix by using nuclear norm \(\|\cdot\|_{*}\), which has been proved to be more helpful to reveal the subspace structure of the data drawn from the true multiple subspaces combined space [6]. Although SSC and LRR clustering methods have shown good clustering performance, the data representation is just formulated by sparse or low rank constraints. The other properties of the data or its representation are not well exploited. For example, in the real world, many high dimensional data are often considered to reside in low-dimensional manifolds space and to have non-linear geometrical structure. The sparse manifold clustering and embedding method [8] showed that introducing the local manifold structure can enhance the clustering performance for the data with non-linear property. However, the basic LRR and SSC methods do not take into account this manifold structure for high dimensional data. As a result, some non-linear metric related properties within data may be corrupted in the procedure of clustering. Observing that, numerous researchers have revised the basic SSC and LRR methods to model the data intrinsic properties by introducing extra constraint or regularization. [PERSON] et al. [9] proposed a graph regularized sparse coding method to obtain data sparse representations with the constraint of local manifold structure. [PERSON] et al. [10] proposed two types of Laplacian regularizer by incorporating a similarity preserving term into the sparse representation model. [PERSON] et al. [11] developed a graph-based approach for non-negative matrix factorization in order to represent the naive geometric structure in data. [PERSON] et al. [12] proposed a Laplacian regularized Gaussian mixture model for data clustering by exploiting the probability distribution of the intrinsic manifold structure of data. Incorporating manifold regularization, [PERSON] et al. [13] proposed a novel low rank matrix factorization model for data representation. [PERSON] et al. [14] proposed a SMooth Representation (SMR) model to enforce grouping effect in the data representation model. [PERSON] et al.