In this paper, we study the robust subspace clustering problem, which aims to cluster the given possibly noisy data points into their underlying subspaces. A large pool of previous subspace clustering methods focus on the graph construction by different regularization of the representation coefficient. We instead focus on the robustness of the model to non-Gaussian noises. We propose a new robust clustering method by using the correntropy induced metric, which is robust for handling the non-Gaussian and impulsive noises. Also we further extend the method for handling the data with outlier rows features. The multiplicative form of half-quadratic optimization is used to optimize the non-convex correntropy objective function of the proposed models. Extensive experiments on face datasets well demonstrate that the proposed methods are more robust to corruptions and occlusions.
Many subspace clustering methods have been proposed @cite_10 @cite_14 @cite_5 @cite_13 . In this work, we focus on the recent graph based subspace clustering methods @cite_17 @cite_16 @cite_14 @cite_8 @cite_19 . These methods are based on the spectral clustering, and its first step aims to construct an affinity (or graph) matrix which is close to be block diagonal, with zero elements corresponding to data pair from different subspaces. After the affinity matrix is learned, the Normalized Cut @cite_12 is employed to segment the data into multiple clusters. For a given data matrix @math , where @math denotes the feature dimension and @math is the number of data points, the most recent methods, including L1-graph @cite_17 or Sparse Subspace Clustering (SSC) @cite_16 , Low-Rank Representation (LRR) @cite_14 @cite_8 , Multi-Subspace Representation (MSR) @cite_15 and Least Squares Representation (LSR) @cite_19 learn the affinity matrix @math by solving the following common problem
@cite_14 @cite_8 @cite_10 @cite_19 @cite_5 @cite_15 @cite_16 @cite_13 @cite_12 @cite_17
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We propose low-rank representation (LRR) to segment data drawn from a union of multiple linear (or affine) subspaces. Given a set of data vectors, LRR seeks the lowest-rank representation among all the candidates that represent all vectors as the linear combination of the bases in a dictionary. Unlike the well-known sparse representation (SR), which computes the sparsest representation of each data vector individually, LRR aims at finding the lowest-rank representation of a collection of vectors jointly. LRR better captures the global structure of data, giving a more effective tool for robust subspace segmentation from corrupted data. Both theoretical and experimental results show that LRR is a promising tool for subspace segmentation. 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. Over the past few years, several methods for segmenting a scene containing multiple rigidly moving objects have been proposed. However, most existing methods have been tested on a handful of sequences only, and each method has been often tested on a different set of sequences. Therefore, the comparison of different methods has been fairly limited. In this paper, we compare four 3D motion segmentation algorithms for affine cameras on a benchmark of 155 motion sequences of checkerboard, traffic, and articulated scenes. The key task in graph-oriented learning is con-structing an informative graph to model the geometrical and discriminant structure of a data manifold. Since traditional graph construction methods are sensitive to noise and less datum-adaptive to changes in density, a new graph construction method so-calledl1-Graph has been proposed [1] recently. A graph construction method needs to have two important properties: sparsity and locality. However, the l1-Graph is strong in sparsity property, but weak in locality. In order to overcome such limitation, we propose a new method of constructing an informative graph using automatic group sparse regularization based on the work ofl1-Graph, which is called as group sparse graph (GroupSp-Graph). The newly developed GroupSp-Graph has the same noise-insensitive property asl1-Graph, and also can successively preserve the group and local information in the graph. In other words, the proposed group sparse graph has both properties of sparsity and locality simultaneously. Furthermore, we integrate the proposed graph with several graph-oriented learning algorithms: spectral em-bedding, spectral clustering, subspace learning and manifold regularized non-negative matrix factorization. The empirical studies on benchmark data sets show that the proposed algo-rithms achieve considerable improvement over classic graph constructing methods and the l1-Graph method in various learning task. This paper presents the multi-subspace discovery problem and provides a theoretical solution which is guaranteed to recover the number of subspaces, the dimensions of each subspace, and the members of data points of each subspace simultaneously. We further propose a data representation model to handle noisy real world data. We develop a novel optimization approach to learn the presented model which is guaranteed to converge to global optimizers. As applications of our models, we first apply our solutions as preprocessing in a series of machine learning problems, including clustering, classification, and semisupervised learning. We found that our method automatically obtains robust data presentation which preserves the affine subspace structures of high dimensional data and generate more accurate results in the learning tasks. We also establish a robust standalone classifier which directly utilizes our sparse and low rank representation model. Experimental results indicate our methods improve the quality of data by preprocessing and the standalone classifier outperforms some state-of-the-art learning approaches. We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained exactly' by using l1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms state-of-the-art methods. Sparse subspace learning has drawn more and more attentions recently. However, most of the sparse subspace learning methods are unsupervised and unsuitable for classification tasks. In this paper, a new sparse subspace learning algorithm called discriminant sparse neighborhood preserving embedding (DSNPE) is proposed by adding the discriminant information into sparse neighborhood preserving embedding (SNPE). DSNPE not only preserves the sparse reconstructive relationship of SNPE, but also sufficiently utilizes the global discriminant structures from the following two aspects: (1) maximum margin criterion (MMC) is added into the objective function of DSNPE; (2) only the training samples with the same label as the current sample are used to compute the sparse reconstructive relationship. Extensive experiments on three face image datasets (Yale, Extended Yale B and AR) demonstrate the effectiveness of the proposed DSNPE method. We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.