# Deep Subspace Clustering with Block Diagonal Constraint
## 1 Introduction
The commonly used deep architectures for these models are: Auto-encoders (AEs), Variational Autoencoders (VAEs) [39; 41] and Generative Adversarial Networks (GANs) [37; 38; 40; 42], which are used to produce feature representations suitable for clustering [43]. As good generative models, VAEs and GANs have an intrinsic difference. VAEs aim to find a probability density through an explicit method and obtain the optimal solution by minimizing the lower limit of the log likelihood function. GANs are an adversarial way to find a balance, and they do not need to be given an explicit probability density function. [PERSON] et al. [37] proposed an approach for the problem of clustering in the Generative Adversarial Network (GAN)-defined latent space of GANs to unveil the patterns in multimodal distributions, via a latent space inversion network. For the subspace clustering problem, the main tasks are feature extraction and affinity matrix construction. Therefore, as a simple generative and feature extraction model, AEs are often used as the deep architecture combined with subspace clustering. [PERSON] et al. [29] presented the first deep learning-based subspace clustering method that progressively transforms input data into nonlinear latent space. Deep Subspace Clustering Network (DSC-Net) [30] firstly designs the self-expression and the corresponding loss function for the self-expressiveness matrix of data with deep auto-encoder. [PERSON] et al. [31] learned a set of explicit transformations to progressively map input data points into nonlinear latent spaces while preserving the local and global subspace structure. [PERSON] et al. [32] adopted a dual self-supervision structure, which uses the division results of spectral clustering to simultaneously supervise the representation process and the self-expression process. [PERSON] et al. [33] re-formulated subspace clustering as a classification problem, which in turn removes the spectral clustering step from the computations. [PERSON] et al. [34] utilized GANs to guide the learning of DSC-Net, so that the network can be trained in the direction of better results every time. [PERSON] et al. [36] presented the first work revealing the sample-assignment invariance prior based on the idea of treating labels as ideal representations. These deep learning-based methods outperform other state-of-the-art subspace clustering methods significantly. Because of the good performance demonstrated by the deep neural network and block-diagonal constraint, this paper employs the deep neural network to set up the subspace clustering model with Block Diagonal Constraint (DSC-BDC), in which the block diagonal regularization is used to restrain the self-expression matrix. At the same time, to improve the flexibility and ability of self-expression, we conduct a separation strategy to indirectly restrain the block diagonal structure of the representation matrix. Specifically, the main contributions of this paper can be summarized as follows. * A deep subspace clustering method based on an auto-encode is proposed, and block-diagonal constraints on representation matrix are used for better cluster performance. * A separation strategy on the block diagonal constraint is proposed for more flexibility. * Due to the existing of new block diagonal regularizer, an alternating optimization method to solve proposed model is developed. * The proposed DSC-BDC is evaluated on four databases, and the results demonstrate the effectiveness of our model. The rest of this paper is organized as follows. Section 2 gives the proposed DSC-BDC model and optimization algorithm. Section 3 provides experimental results and analysis based on four public databases. Finally, the conclusions and future works are summarized in Section 4.