It is a key to construct a similarity graph in graph-oriented subspace learning and clustering. In a similarity graph, each vertex denotes a data point and the edge weight represents the similarity between two points. There are two popular schemes to construct a similarity graph, i.e., pairwise distance based scheme and linear representation based scheme. Most existing works have only involved one of the above schemes and suffered from some limitations. Specifically, pairwise distance based methods are sensitive to the noises and outliers compared with linear representation based methods. On the other hand, there is the possibility that linear representation based algorithms wrongly select inter-subspaces points to represent a point, which will degrade the performance. In this paper, we propose an algorithm, called Locally Linear Representation (LLR), which integrates pairwise distance with linear representation together to address the problems. The proposed algorithm can automatically encode each data point over a set of points that not only could denote the objective point with less residual error, but also are close to the point in Euclidean space. The experimental results show that our approach is promising in subspace learning and subspace clustering.

In this paper, we recast the subspace clustering as a verification problem. Our idea comes from an assumption that the distribution between a given sample x and cluster centers Omega is invariant to different distance metrics on the manifold, where each distribution is defined as a probability map (i.e. soft-assignment) between x and Omega. To verify this so-called invariance of distribution, we propose a deep learning based subspace clustering method which simultaneously learns a compact representation using a neural network and a clustering assignment by minimizing the discrepancy between pair-wise sample-centers distributions. To the best of our knowledge, this is the first work to reformulate clustering as a verification problem. Moreover, the proposed method is also one of the first several cascade clustering models which jointly learn representation and clustering in end-to-end manner. Extensive experimental results show the effectiveness of our algorithm comparing with 11 state-of-the-art clustering approaches on four data sets regarding to four evaluation metrics.

Under the framework of spectral clustering, the key of subspace clustering is building a similarity graph which describes the neighborhood relations among data points. Some recent works build the graph using sparse, low-rank, and $\ell_2$-norm-based representation, and have achieved state-of-the-art performance. However, these methods have suffered from the following two limitations. First, the time complexities of these methods are at least proportional to the cube of the data size, which make those methods inefficient for solving large-scale problems. Second, they cannot cope with out-of-sample data that are not used to construct the similarity graph. To cluster each out-of-sample datum, the methods have to recalculate the similarity graph and the cluster membership of the whole data set. In this paper, we propose a unified framework which makes representation-based subspace clustering algorithms feasible to cluster both out-of-sample and large-scale data. Under our framework, the large-scale problem is tackled by converting it as out-of-sample problem in the manner of "sampling, clustering, coding, and classifying". Furthermore, we give an estimation for the error bounds by treating each subspace as a point in a hyperspace. Extensive experimental results on various benchmark data sets show that our methods outperform several recently-proposed scalable methods in clustering large-scale data set.

Given a data set from a union of multiple linear subspaces, a robust subspace clustering algorithm fits each group of data points with a low-dimensional subspace and then clusters these data even though they are grossly corrupted or sampled from the union of dependent subspaces. Under the framework of spectral clustering, recent works using sparse representation, low rank representation and their extensions achieve robust clustering results by formulating the errors (e.g., corruptions) into their objective functions so that the errors can be removed from the inputs. However, these approaches have suffered from the limitation that the structure of the errors should be known as the prior knowledge. In this paper, we present a new method of robust subspace clustering by eliminating the effect of the errors from the projection space (representation) rather than from the input space. We firstly prove that ell_1-, ell_2-, and ell_infty-norm-based linear projection spaces share the property of intra-subspace projection dominance, i.e., the coefficients over intra-subspace data points are larger than those over inter-subspace data points. Based on this property, we propose a robust and efficient subspace clustering algorithm, called Thresholding Ridge Regression (TRR). TRR calculates the ell2-norm-based coefficients of a given data set and performs a hard thresholding operator; and then the coefficients are used to build a similarity graph for clustering. Experimental studies show that TRR outperforms the state-of-the-art methods with respect to clustering quality, robustness, and time-saving.