# Deep Subspace Clustering with Block Diagonal Constraint
## 1 Introduction  
Subspace clustering is a significant unsupervised learning method in many applications, such as image representation [1; 2; 3], face clustering [4; 5; 6], motion segmentation [7; 8], bioinformatics [9], medical image analysis [10], etc. Usually, many high-dimensional datasets from the real-world applications are described by a union of low-dimensional subspaces. Thus, the main target of subspace clustering is to segment a series of samples into disjoint clusters such that the samples within each cluster belong to the same subspace [11; 12]. In the past decade, abundant subspace clustering methods have been proposed, and most methods [2; 13; 14; 15] are based on Spectral Clustering, which constructs a representation matrix measuring similarity between data points and segments input data based on this representation matrix. Specifically, the subspace clustering method based on self-expression [2; 15; 16] is one of the most popular clustering methods, and the main idea of this method is that using the self-expression of each data sample obtains the representation coefficient for establishing a representation matrix. Subspace clustering methods based on self-expression include Spares Subspace Clustering (SSC) [17], Low-Rank Representation (LRR) [15], Low Rank Subspace Clustering (LRSC) [18], etc. Among them, SSC and LRR, respectively, give a regular term with sparseness and low-rank to constrain the representation matrix for sparse and low-rank. LRSC decomposes the data matrix into a noise matrix and a clean matrix, and then obtains the low-rank representation matrix forclustering. However, these models are suitable for any representation matrix and are not targeted for clustering application. As is known, the ideal representation matrix should have block diagonal structure for the subspace clustering. Thus, in recent years, many studies [19; 20; 21] utilize the block diagonal prior for the representation matrix. For example, [PERSON] et al. [19] directly pursued the block-diagonal structure by proposing a graph Laplacian constraint based formulation; [PERSON] et al. [20] adopted a block-diagonal regularization directly to construct the representation matrix; and [PERSON] et al. [21] introduced block-diagonal constraints on multi-view representation matrices to obtain accurate heterogeneous information. Those methods demonstrate that the subspace clustering performance is improved by adding the block diagonal prior on the representation matrix. However, it is worth noting that the above algorithms assume that data could be linearly represented with each other in the input space. In addition, most traditional works on subspace clustering [2; 14; 22; 23; 24] are based on the assumption that the data are linearly separable. However, actual data usually do not conform to the linear subspace model. For example, in the task of face clustering, they are affected by the light source, facial geometric features and reflectivity. Under these influences, face data are usually in a nonlinear subspace. Kernel methods [25; 26; 27; 28] are used to implicitly map data to higher-dimensional spaces to better conform to linear models in the resulting spaces. However, the selection of different kernel types is empirical without theoretical guarantee. With the rapid development of deep learning, the deep neural networks have received widespread attention because of their excellent ability to capture complex underlying data structures and learn informative features (for clustering). Inspired by that, subspace clustering based on deep neural networks came into view naturally, and many deep subspace clustering models based on different architectures emerged. These models can better exploit the nonlinear relationships between the sample points [29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40].