In recent years, low-rank representation (LRR) has received increasing attention on subspace clustering. Due to inevitable matrix inversion and singular value decomposition in each iteration, however, most of existing LRR algorithms may suffer from high computational complexity, and hence can not cope with the large-scale sample data commendably. To overcome this problem, in this paper, we propose a bilateral fast low-rank representation (BFLRR), which has a linear time complexity with respect to the number of samples. Specifically, we introduce the equivalent transformation method to remove the null spaces of both the columns and rows of the coefficient matrix so that a hypercompact coefficient matrix can be learned. Furthermore, the proposed BFLRR is embedded into a distributed framework as DFC-BFLRR to make it more efficient, which utilizes a combination of the global and local projection matrices. Extensive experiments are carried out on real datasets, and the results testify that the proposed methods not only perform faster-computing speed but also obtain favorable clustering accuracy in comparison with the competing methods among large-scale sample data.

As well known, low rank representation method (LRR) has obtained promising performance for subspace clustering, and many LRR variants have been developed, which mainly solve the three problems existing in LRR: 1) Problem of mean calculation; 2) Problem of deviating from the real low rank solution; 3) Problem of high computation cost on the large-scale data. In this paper, we first propose a universal LRR method referring to the first two problems. More specifically, we introduce the ability of removing the optimal mean automatically into LRR and extend it to a Schatten$p$-norm minimization problem. Then, referring to the third problem, we reformulate the universal LRR version as an equivalent fast optimization version by removing the null space of data. More importantly, the effective theory proof is proposed to guarantee that the fast optimization method can dramatically improve the algorithmic computation efficiency on the large-scale data but without any loss of information. Finally, experimental results demonstrate the effectiveness and efficiency of the proposed method.

Inspired by the mean calculation of RPCA_OM and inductiveness of IRPCA, we first propose an inductive robust principal component analysis method with removing the optimal mean automatically, which is shorted as IRPCA_OM. Furthermore, IRPCA_OM is extended to Schatten-$p$norm and a more general framework (i.e., EIRPCA_OM) is presented. The objective function of EIRPCA_OM includes two terms, the first term is a robust reconstruction error term constrained by an$\ell _{2,1}$-norm and the second term is a regularization term constrained by a Schatten-$p$norm. The proposed EIRPCA_OM method is robust, inductive and accurate. However, on the high-dimensional data, it would spend a large computation cost in training stage. To this end, a fast version of EIRPCA_OM called as FEIRPCA_OM is proposed, and its basic idea is to eliminate the zero eigenvalues of data matrix. More importantly, an effective theoretical proof is presented to ensure that FEIRPCA_OM has faster processing speed than EIRPCA_OM when processing high-dimensional data, but without any performance loss. Based on it, we also can exchange the less performance loss for the higher computation efficiency by removing the small eigenvalues of data matrix. Experimental results on the public datasets demonstrate that FEIRPCA_OM works efficiently on the high-dimensional data.

Current unsupervised feature selection methods cannot well select the effective features from the corrupted data. To this end, we propose a robust unsupervised feature selection method under the robust principal component analysis (PCA) reconstruction criterion, which is named the adaptive weighted sparse PCA (AW-SPCA). In the proposed method, both the regularization term and the reconstruction error term are constrained by the ℓ2,1-norm: the ℓ2,1-norm regularization term plays a role in the feature selection, while the ℓ2,1-norm reconstruction error term plays a role in the robust reconstruction. The proposed method is in a convex formulation, and the selected features by it can be used for robust reconstruction and clustering. Experimental results demonstrate that the proposed method can obtain better reconstruction and clustering performance, especially for the corrupted data.

In general, low-rank representation (LRR) aims to find the lowest rank representation with respect to a dictionary. In fact, the dictionary is a key aspect of low-rank representation. However, a lot of low-rank representation methods usually use the data itself as a dictionary (i.e., a fixed dictionary), which may degrade their performances due to the lack of clustering ability of a fixed dictionary. To this end, we propose learning a locality-preserving dictionary instead of the fixed dictionary for low-rank representation, where the locality-preserving dictionary is constructed by using a graph regularization technique to capture the intrinsic geometric structure of the dictionary and, hence, the locality-preserving dictionary has an underlying clustering ability. In this way, the obtained low-rank representation via the locality-preserving dictionary has a better grouping-effect representation. Inversely, a better grouping-effect representation can help to learn a good dictionary. The locality-preserving dictionary and the grouping-effect representation interact with each other, where dual pursuit is called. The proposed method, namely, Dual Pursuit for Subspace Learning, provides us with a robust method for clustering and classification simultaneously, and compares favorably with the other state-of-the-art methods.

In this paper, we propose an unsupervised feature selection method from the perspective of optimal reconstruction. The features selected by the proposed method can well represent the original data, and the effectiveness of the selected features is demonstrated by robust reconstruction and clustering. The proposed method emphasizes the joint l2, 1-norms minimization on both reconstruction term and regularization term to make them be column-sparse. Relying on the column-sparse property of reconstruction term and regularization term, the proposed method is able to improve the robustness to outliers and select the effective features. The proposed objective function is nonconvex. Fortunately, it can be equivalently reformulated as a convex form (with change of variables) to capture a global optimization solution. In fact, the proposed method is related to the optimal mean robust principal component analysis (OMRPCA) because the proposed method is a sparse self-contained regression type of OMRPCA. Since OMRPCA essentially adds the adaptive weights for data samples, we call the proposed method adaptive weighted sparse principal component analysis (AW-SPCA). Experimental results demonstrate the effectiveness of AW-SPCA.

In order to improve the interpretation of principal components, many sparse principal component analysis (PCA) methods have been proposed by in the form of self-contained regression-type. In this paper, we generalize the steps needed to move from PCA-like methods to its self-contained regression-type, and propose a joint sparse pixel weighted PCA method. More specifically, we generalize a self-contained regression-type framework of graph embedding. Unlike the regression-type of graph embedding relying on the regular low-dimensional data, the self-contained regression-type framework does not rely on the regular low-dimensional data of graph embedding. The learned low-dimensional data in the form of self-contained regression theoretically approximates to the regular low-dimensional data. Under this self-contained regression-type, sparse regularization term can be arbitrarily added, and hence, the learned sparse regression coefficients can interpret the low-dimensional data. By using the joint sparse ℓ2,1-norm regularizer, a sparse self-contained regression-type of pixel weighted PCA can be produced. Experiments on six data sets demonstrate that the proposed method is both feasible and effective.

Many conventional computer vision object tracking methods are sensitive to partial occlusion and background clutter. This is because the partial occlusion or little background information may exist in the bounding box, which tends to cause the drift. To this end, in this paper, we propose a robust tracker based on key patch sparse representation (KPSR) to reduce the disturbance of partial occlusion or unavoidable background information. Specifically, KPSR first uses patch sparse representations to get the patch score of each patch. Second, KPSR proposes a selection criterion of key patch to judge the patches within the bounding box and select the key patch according to its location and occlusion case. Third, KPSR designs the corresponding contribution factor for the sampled patches to emphasize the contribution of the selected key patches. Comparing the KPSR with eight other contemporary tracking methods on 13 benchmark video data sets, the experimental results show that the KPSR tracker outperforms classical or state-of-the-art tracking methods in the presence of partial occlusion, background clutter, and illumination change.

Low-Rank Representation (LRR) is an effective self-expressiveness method, which uses the observed data itself as the dictionary to reconstruct the original data. LRR focuses on representing the global low-dimensional information, but ignores the real fact that data often resides on low-dimensional manifolds embedded in a high-dimensional data. Therefore, LRR can not capture the non-linear geometric structures within data. As well known, locality preserving projections (LPP) is able to preserve the intrinsic geometry structure embedded in high-dimensional data. To this end, we treat the projected data by LPP as an adaptive dictionary, and such a dictionary can capture the intrinsic geometry structures of data. In this way, our method is in favor of the global low-rank representation. Specifically, the proposed method provides a way to reconstruct the original data from two views, and hence we call this proposed method as Simultaneous Dual-Views Reconstruction with Adaptive Dictionary and Low-Rank Representation. The proposed method can be used for unsupervised feature extraction and subspace clustering. Experiments on benchmark databases show the excellent performance of this proposed method in comparison with other state-of-the-art methods.