Recently, multi-view subspace clustering has attracted extensive attention due to the rapid increase of multi-view data in many real-world applications. The main goal of this task is to learn a common representation of multiple subspaces from the given multi-view data, and most existing methods usually directly merge multiple groups of features by the single-step integration. However, there may exist large disparities among different views of the data, and thus the conventional single-step practice can hardly obtain a generally consistent feature representation for the multi-view data. To overcome this challenge, we present a novel approach dubbed “Asymptotics-Aware Multi-view Subspace Clustering (A$^{2}$MSC)” to pursue a consistent feature representation in a multi-step way, which iteratively conducts the data recovery to gradually reduce the differences between pairwise views. Specifically, we construct an asymptotic learning rule to update the feature representation, and the iteration result converges to a consistent feature vector for characterizing each instance of the original multi-view data. After that, we utilize such a new feature representation to learn a clustering-oriented similarity matrix via minimizing a self-expressive objective, and we also design the corresponding optimization algorithm to solve it with convergence guarantees. Theoretically, we prove that the learned asymptotic representation effectively integrates multiple views, thereby ensuring the effective handling of multi-view data. Empirically, extensive experimental results demonstrate the superiority of our proposed A$^{2}$MSC over the state-of-the-art multi-view subspace clustering approaches.

The self-expressive strategy has shown excellent capabilities in realizing low-dimensional representations of high-dimensional data for subspace clustering algorithms. The existing designs, however, are formulated on the linearization assumptions of the data, neglecting the precise characterization of linear relationships within samples. Considering that real-world data adheres to diverse distribution forms, it becomes impractical to first treat the samples as existing in a uniform linear space before finding an appropriate manifold space. To handle this challenge, we propose a novel self-expressive-based learning framework termed metric learning-based subspace clustering (MLSC). Particularly, we smoothly incorporate metric learning into the subspace clustering framework by introducing adaptive neighbors learning and defining a linearity-aware distance to discover the linear manifold space of the original data. We simultaneously utilize the generated representation of the linear structure as input for self-expressiveness to pursue an ideal similarity matrix, which establishes an essential connection with the linearization assumption of the self-expressive strategy. Furthermore, we theoretically demonstrate that our measure can accurately describe the level of linear correlation between instances. Finally, our tests demonstrate that the proposed MLSC attains competitive clustering results compared to state-of-the-art approaches on benchmark datasets.

Block diagonal representation, which aims to compel the desired representation coefficient to have a block diagonal structure directly, has extensive applications in the domains of computer vision and machine learning. However, single residual modeling in existing works is not robust enough when handling complex noise (i.e., sparse noise and impulsive noise) in reality. To overcome this challenge, a novel Sparseness and Correntropy-based Block Diagonal Representation (SC-BDR) model is proposed, which is able to pursue ideal block diagonal representation and effectively deal with various types of noise. Furthermore, the corresponding optimization algorithm is designed for the proposed problem, and we also conduct extensive experiments to demonstrate the robustness and effectiveness of the SC-BDR model on real-world data.

Obtaining a good similarity matrix is extremely important in subspace clustering. Current state-of-the-art methods learn the similarity matrix through self-expressive strategy. However, these methods directly adopt original samples as a set of basis to represent itself linearly. It is difficult to accurately describe the linear relation between samples in the real-world applications, and thus is hard to find an ideal similarity matrix. To better represent the linear relation of samples, we present a subspace clustering model, Linearity-Aware Subspace Clustering (LASC), which can consciously learn the similarity matrix by employing a linearity-aware metric. This is a new subspace clustering method that combines metric learning and subspace clustering into a joint learning framework. In our model, we first utilize the self-expressive strategy to obtain an initial subspace structure and discover a low-dimensional representation of the original data. Subsequently, we use the proposed metric to learn an intrinsic similarity matrix with linearity-aware on the obtained subspace. Based on such a learned similarity matrix, the inter-cluster distance becomes larger than the intra-cluster distances, and thus successfully obtaining a good subspace cluster result. In addition, to enrich the similarity matrix with more consistent knowledge, we adopt a collaborative learning strategy for self-expressive subspace learning and linearity-aware subspace learning. Moreover, we provide detailed mathematical analysis to show that the metric can properly characterize the linear correlation between samples.

Block-diagonal representation (BDR) is an effective subspace clustering method. The existing BDR methods usually obtain a self-expression coefficient matrix from the original features by a shallow linear model. However, the underlying structure of real-world data is often nonlinear, thus those methods cannot faithfully reflect the intrinsic relationship among samples. To address this problem, we propose a novel latent BDR (LBDR) model to perform the subspace clustering on a nonlinear structure, which jointly learns an autoencoder and a BDR matrix. The autoencoder, which consists of a nonlinear encoder and a linear decoder, plays an important role to learn features from the nonlinear samples. Meanwhile, the learned features are used as a new dictionary for a linear model with block-diagonal regularization, which can ensure good performances for spectral clustering. Moreover, we theoretically prove that the learned features are located in the linear space, thus ensuring the effectiveness of the linear model using self-expression. Extensive experiments on various real-world datasets verify the superiority of our LBDR over the state-of-the-art subspace clustering approaches.