In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.

Low-Rank Representation (LRR) [16,17] is an effective method for exploring the multiple subspace structures of data. Usually, the observed data matrix itself is chosen as the dictionary, which is a key aspect of LRR. However, such a strategy may depress the performance, especially when the observations are insufficient and/or grossly corrupted. In this paper we therefore propose to construct the dictionary by using both observed and unobserved, hidden data. We show that the effects of the hidden data can be approximately recovered by solving a nuclear norm minimization problem, which is convex and can be solved efficiently. The formulation of the proposed method, called Latent Low-Rank Representation (LatLRR), seamlessly integrates subspace segmentation and feature extraction into a unified framework, and thus provides us with a solution for both subspace segmentation and feature extraction. As a subspace segmentation algorithm, LatLRR is an enhanced version of LRR and outperforms the state-of-the-art algorithms. Being an unsupervised feature extraction algorithm, LatLRR is able to robustly extract salient features from corrupted data, and thus can work much better than the benchmark that utilizes the original data vectors as features for classification. Compared to dimension reduction based methods, LatLRR is more robust to noise.

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This paper addresses the problem of detecting salient areas within natural images. We shall mainly study the problem under unsupervised setting, i.e., saliency detection without learning from labeled images. A solution of multitask sparsity pursuit is proposed to integrate multiple types of features for detecting saliency collaboratively. Given an image described by multiple features, its saliency map is inferred by seeking the consistently sparse elements from the joint decompositions of multiple-feature matrices into pairs of low-rank and sparse matrices. The inference process is formulated as a constrained nuclear norm and as an l(2, 1)-norm minimization problem, which is convex and can be solved efficiently with an augmented Lagrange multiplier method. Compared with previous methods, which usually make use of multiple features by combining the saliency maps obtained from individual features, the proposed method seamlessly integrates multiple features to produce jointly the saliency map with a single inference step and thus produces more accurate and reliable results. In addition to the unsupervised setting, the proposed method can be also generalized to incorporate the top-down priors obtained from supervised environment. Extensive experiments well validate its superiority over other state-of-the-art methods.

This paper investigates how to boost region-based im age segmentation by pursuing a new solution to fuse multi ple types of image features . A collaborative image segmen tation framework, called multi-task low-rank affinity pur suit, is presented for such a purpose . Given an image de scribed with multiple types of features, we aim at inferring a unified affinity matrix that implicitly encodes the segmen tation of the image . This is achieved by seeking the sparsity consistent low-rank affinities from the joint decompositions of multiple feature matrices into pairs of sparse and low rank matrices, the latter of which is expressed as the pro duction of the image feature matrix and its corresponding image affinity matrix . The inference process is formulated as a constrained nuclear norm and t'2,1-norm minimization problem, which is convex and can be solved efficiently with the Augmented Lagrange Multiplier method . Compared to previous methods, which are usually based on a single type of features, the proposed method seamlessly integrates mul tiple types of features to jointly produce the affinity matrix within a single inference step, and produces more accu rate and reliable segmentation results . Experiments on the MSRC dataset and Berkeley segmentation dataset well val idate the superiority of using multiple features over single feature and also the superiority of our method over conven tional methods for feature fusion . Moreover, our method is shown to be very competitive while comparing to other state-of-the-art methods .