Attention reweighted sparse subspace clustering

Subspace clustering methods with sparsity prior, such as Sparse Subspace Clustering (SSC) [1], are effective in partitioning the data that lie in a union of subspaces. Most of those methods require certain assumptions, e.g. independence or disjointness, on the subspaces. These assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods. In this paper, we propose \(\ell ^{0}\)-induced sparse subspace clustering (\(\ell ^{0}\)-SSC). In contrast to the required assumptions, such as independence or disjointness, on subspaces for most existing sparse subspace clustering methods, we prove that subspace-sparse representation, a key element in subspace clustering, can be obtained by \(\ell ^{0}\)-SSC for arbitrary distinct underlying subspaces almost surely under the mild i.i.d. assumption on the data generation. We also present the “no free lunch” theorem that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding \(\ell ^{0}\) problem of \(\ell ^{0}\)-SSC. We develop a novel approximate algorithm named Approximate \(\ell ^{0}\)-SSC (\(\hbox {A}\ell ^{0}\)-SSC) that employs proximal gradient descent to obtain a sub-optimal solution to the optimization problem of \(\ell ^{0}\)-SSC with theoretical guarantee, and the sub-optimal solution is used to build a sparse similarity matrix for clustering. Extensive experimental results on various data sets demonstrate the superiority of \(\hbox {A}\ell ^{0}\)-SSC compared to other competing clustering methods.

Subspace clustering has important and wide applications in computer vision and pattern recognition. It is a challenging task to learn low-dimensional subspace structures due to the possible errors (e.g., noise and corruptions) existing in high-dimensional data. Recent subspace clustering methods usually assume a sparse representation of corrupted errors and correct the errors iteratively. However large corruptions in real-world applications can not be well addressed by these methods. A novel optimization model for robust subspace clustering is proposed in this paper. The objective function of our model mainly includes two parts. The first part aims to achieve a sparse representation of each high-dimensional data point with other data points. The second part aims to maximize the correntropy between a given data point and its low-dimensional representation with other points. Correntropy is a robust measure so that the influence of large corruptions on subspace clustering can be greatly suppressed. An extension of our method with explicit introduction of representation error terms into the model is also proposed. Half-quadratic minimization is provided as an efficient solution to the proposed robust subspace clustering formulations. Experimental results on Hopkins 155 dataset and Extended Yale Database B demonstrate that our method outperforms state-of-the-art subspace clustering methods.

Data clustering is an important research topic in data mining and signal processing communications. In all the data clustering methods, the subspace spectral clustering methods based on self expression model, e.g., the Sparse Subspace Clustering (SSC) and the Low Rank Representation (LRR) methods, have attracted a lot of attention and shown good performance. The key step of SSC and LRR is to construct a proper affinity or similarity matrix of data for spectral clustering. Recently, Laplacian graph constraint was introduced into the basic SSC and LRR and obtained considerable improvement. However, the current graph construction methods do not well exploit and reveal the non-linear properties of the clustering data, which is common for high dimensional data. In this paper, we introduce the classic manifold learning method, the Local Linear Embedding (LLE), to learn the non-linear structure underlying the data and use the learned local geometry of manifold as a regularization for SSC and LRR, which results the proposed LLE-SSC and LLE-LRR clustering methods. Additionally, to solve the complex optimization problem involved in the proposed models, an efficient algorithm is also proposed. We test the proposed data clustering methods on several types of public databases. The experimental results show that our methods outperform typical subspace clustering methods with Laplacian graph constraint.

Subspace clustering based on latent low rank representation with Frobenius norm minimization

We propose Ordered Subspace Clustering (OSC) to segment data drawn from a sequentially ordered union of subspaces. Current subspace clustering techniques learn the relationships within a set of data and then use a separate clustering algorithm such as NCut for final segmentation. In contrast our technique, under certain conditions, is capable of segmenting clusters intrinsically without providing the number of clusters as a parameter. Similar to Sparse Subspace Clustering (SSC) we formulate the problem as one of finding a sparse representation but include a new penalty term to take care of sequential data. We test our method on data drawn from infrared hyper spectral data, video sequences and face images. Our experiments show that our method, OSC, outperforms the state of the art methods: Spatial Subspace Clustering (SpatSC), Low-Rank Representation (LRR) and SSC.

Gene expression data is a kind of high dimension and small sample size data. The clustering accuracy of conventional clustering techniques is lower on gene expression data due to its high dimension. Because some subspace segmentation approaches can be better applied in the high-dimensional space, three new subspace clustering models for gene expression data sets are proposed in this work. The proposed projection subspace clustering models have projection sparse subspace clustering, projection low-rank representation subspace clustering and projection least-squares regression subspace clustering which combine projection technique with sparse subspace clustering, low-rank representation and least-square regression, respectively. In order to compute the inner product in the high-dimensional space, the kernel function is used to the projection subspace clustering models. The experimental results on six gene expression data sets show these models are effective.

Multi-view data clustering often aims to utilize various representations or views of original data to improve the clustering performance compared to the single-view clustering approach. Most multi-view subspace clustering methods are proposed to construct the affinity matrix of each view individually and then implement with spectral clustering for multi-view data clustering. The multi-view low-rank sparse subspace clustering (MLRSSC) is an effective and popular clustering algorithm among multi-view subspace clustering. This method can explore the joint subspace representation through creating an affinity matrix integrated of all views of input data. In addition, the low-rank and sparsity constraints are introduced into this method to enhance the clustering results. However, the original MLRSSC uses the mean square error as the fidelity term while not consider the complex noise pollution in real situations. Therefore, we introduce a complex noise modeling approach, i.e., independent and piecewise identically distributed (IPID) noise model, for MLRSSC to improve its performance. The related experimental results confirm that this proposed algorithm surpasses many state-of-the-art subspace clustering methods on several real-world datasets.

This paper studies the subspace segmentation problem. Given a set of data points drawn from a union of subspaces, the goal is to partition them into their underlying subspaces they were drawn from. The spectral clustering method is used as the framework. It requires to find an affinity matrix which is close to block diagonal, with nonzero entries corresponding to the data point pairs from the same subspace. In this work, we argue that both sparsity and the grouping effect are important for subspace segmentation. A sparse affinity matrix tends to be block diagonal, with less connections between data points from different subspaces. The grouping effect ensures that the highly corrected data which are usually from the same subspace can be grouped together. Sparse Subspace Clustering (SSC), by using $\ell^1$-minimization, encourages sparsity for data selection, but it lacks of the grouping effect. On the contrary, Low-Rank Representation (LRR), by rank minimization, and Least Squares Regression (LSR), by $\ell^2$-regularization, exhibit strong grouping effect, but they are short in subset selection. Thus the obtained affinity matrix is usually very sparse by SSC, yet very dense by LRR and LSR. In this work, we propose the Correlation Adaptive Subspace Segmentation (CASS) method by using trace Lasso. CASS is a data correlation dependent method which simultaneously performs automatic data selection and groups correlated data together. It can be regarded as a method which adaptively balances SSC and LSR. Both theoretical and experimental results show the effectiveness of CASS.

Multi-kernel sparse subspace clustering on the Riemannian manifold of symmetric positive definite matrices

Sparse subspace clustering (SSC) has been widely applied in remote sensing demonstrating excellent performance. Recent extensions incorporate spatial information, typically via smoothness-enforcing regularization. We propose an alternative approach: a joint sparsity SSC model, where pixels within a local region are enforced to select a common set of samples in the subspace-sparse representation. The corresponding optimization problem is solved by the alternating direction method of multipliers (ADMM). Experimental results on real data show a significant improvement over SSC and related state-of-the-art methods.

Community detection method based on mixed-norm sparse subspace clustering

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

Subspace clustering is an effective algorithm refer to the problem which separate the data lying on a union of subspaces. Usually the algorithm consists of two steps. First, an affinity matrix is calculated from the self-representation of the data. Second, spectral clustering is used to cluster the data by the affinity matrix. The paper introduces a new method based on the sparse subspace clustering. The new method used a new objective which considered both the sparseness and grouping effect. And the affinity matrix which is calculated from the self-representation of the data is furthered optimized. It proved to be more efficient than existing subspace clustering methods.

Video co-segmentation refers to the task of jointly segmenting common objects appearing in a given group of videos. In practice, high-dimensional data such as videos can be conceptually thought as being drawn from a union of subspaces corresponding to categories rather than from a smooth manifold. Therefore, segmenting data into respective subspaces --- subspace clustering --- finds widespread applications in computer vision, including co-segmentation. State-of-the-art methods via subspace clustering seek to solve the problem in two steps: First, an affinity matrix is built from data, with appearance features or motion patterns. Second, the data are segmented by applying spectral clustering to the affinity matrix. However, this process is insufficient to obtain an optimal solution since it does not take into account the {\em interdependence} of the affinity matrix with the segmentation. In this work, we present a novel unified video co-segmentation framework inspired by the recent Structured Sparse Subspace Clustering ($\mathrm{S^{3}C}$) based on the {\em self-expressiveness} model. Our method yields more consistent segmentation results. In order to improve the detectability of motion features with missing trajectories due to occlusion or tracked points moving out of frames, we add an extra-dimensional signature to the motion trajectories. Moreover, we reformulate the $\mathrm{S^{3}C}$ algorithm by adding the affine subspace constraint in order to make it more suitable to segment rigid motions lying in affine subspaces of dimension at most $3$. Our experiments on MOViCS dataset show that our framework achieves the highest overall performance among baseline algorithms and demonstrate its robustness to heavy noise.

Nonlinear subspace clustering by functional link neural networks