Subspace clustering methods based on data self-expression have become very popular for learning from data that lie in a union of low-dimensional linear subspaces. However, the applicability of subspace clustering has been limited because practical visual data in raw form do not necessarily lie in such linear subspaces. On the other hand, while Convolutional Neural Network (ConvNet) has been demonstrated to be a powerful tool for extracting discriminative features from visual data, training such a ConvNet usually requires a large amount of labeled data, which are unavailable in subspace clustering applications. To achieve simultaneous feature learning and subspace clustering, we propose an end-to-end trainable framework, called Self-Supervised Convolutional Subspace Clustering Network (S @math ConvSCN), that combines a ConvNet module (for feature learning), a self-expression module (for subspace clustering) and a spectral clustering module (for self-supervision) into a joint optimization framework. Particularly, we introduce a dual self-supervision that exploits the output of spectral clustering to supervise the training of the feature learning module (via a classification loss) and the self-expression module (via a spectral clustering loss). Our experiments on four benchmark datasets show the effectiveness of the dual self-supervision and demonstrate superior performance of our proposed approach.
For subspace clustering in feature space, we further divide the existing methods into two types. The first type uses , which is induced via a Mercer kernel, , @cite_52 @cite_22 @cite_20 @cite_49 , or constructed via matrix decomposition, , @cite_21 , @cite_17 . The second type use , which is designed by manual feature extraction, , @cite_32 , or is learned from data, , @cite_8 @cite_18 .
@cite_18 @cite_22 @cite_8 @cite_21 @cite_52 @cite_32 @cite_49 @cite_20 @cite_17
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Given a data set from a union of multiple linear sub-spaces, 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 sub-spaces. 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 sub-space clustering by eliminating the effect of the errors from the projection space (representation) rather than from the input space. We firstly prove that l1-, l2-, and l∞ -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 sub-space clustering algorithm, called Thresholding Ridge Regression (TRR). TRR calculates the ∞2 -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. We propose a new subspace clustering method that integrates feature selection into subspace clustering. Rather than using all features to construct a low-rank representation of the data, we find such a representation using only relevant features, which helps in revealing more accurate data relationships. Two variants are proposed by using both convex and nonconvex rank approximations. Extensive experimental results confirm the effectiveness of the proposed method and models. Subspace clustering mines clusters hidden in subspaces of high-dimensional data sets. Density-based approaches have been shown to successfully mine clusters of arbitrary shape even in the presence of noise in full space clustering. Exhaustive search of all density-based subspace clusters, however, results in infeasible runtimes for large high-dimensional data sets. This is due to the exponential number of possible subspace projections in addition to the high computational cost of density-based clustering. In this paper, we propose lossless efficient detection of density-based subspace clusters. In our EDSC (efficient density-based subspace clustering) algorithm we reduce the high computational cost of density-based subspace clustering by a complete multistep filter-and-refine algorithm. Our first hypercube filter step avoids exhaustive search of all regions in all subspaces by enclosing potentially density-based clusters in hypercubes. Our second filter step provides additional pruning based on a density monotonicity property. In the final refinement step, the exact unbiased density-based subspace clustering result is detected. As we prove that pruning is lossless in both filter steps, we guarantee completeness of the result. In thorough experiments on synthetic and real world data sets, we demonstrate substantial efficiency gains. Our lossless EDSC approach outperforms existing density-based subspace clustering algorithms by orders of magnitude. Subspace clustering methods based on l1, l2 or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, l1 regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, l2 and nuclear norm regularization provide efficient closed form solutions, but require very strong assumptions to guarantee a subspace-preserving affinity, e.g., independent subspaces and uncorrupted data. In this paper we study a subspace clustering method based on orthogonal matching pursuit. We show that the method is both computationally efficient and guaranteed to give a subspace-preserving affinity under broad conditions. Experiments on synthetic data verify our theoretical analysis, and applications in handwritten digit and face clustering show that our approach achieves the best trade off between accuracy and efficiency. Moreover, our approach is the first one to handle 100,000 data points. To uncover an appropriate latent subspace for data representation, in this paper we propose a novel Robust Structured Subspace Learning (RSSL) algorithm by integrating image understanding and feature learning into a joint learning framework. The learned subspace is adopted as an intermediate space to reduce the semantic gap between the low-level visual features and the high-level semantics. To guarantee the subspace to be compact and discriminative, the intrinsic geometric structure of data, and the local and global structural consistencies over labels are exploited simultaneously in the proposed algorithm. Besides, we adopt the @math -norm for the formulations of loss function and regularization respectively to make our algorithm robust to the outliers and noise. An efficient algorithm is designed to solve the proposed optimization problem. It is noted that the proposed framework is a general one which can leverage several well-known algorithms as special cases and elucidate their intrinsic relationships. To validate the effectiveness of the proposed method, extensive experiments are conducted on diversity datasets for different image understanding tasks, i.e., image tagging, clustering, and classification, and the more encouraging results are achieved compared with some state-of-the-art approaches. Subspace clustering refers to the problem of segmenting data drawn from a union of subspaces. State of the art approaches for solving this problem follow a two-stage approach. In the first step, an affinity matrix is learned from the data using sparse or low-rank minimization techniques. In the second step, the segmentation is found by applying spectral clustering to this affinity. While this approach has led to state of the art results in many applications, it is sub-optimal because it does not exploit the fact that the affinity and the segmentation depend on each other. In this paper, we propose a unified optimization framework for learning both the affinity and the segmentation. Our framework is based on expressing each data point as a structured sparse linear combination of all other data points, where the structure is induced by a norm that depends on the unknown segmentation. We show that both the segmentation and the structured sparse representation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Hopkins 155 motion segmentation database, and the Extended Yale B data set demonstrate the effectiveness of our approach. In subspace clustering, a group of data points belonging to a union of subspaces are assigned membership to their respective subspaces. This paper presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of subspace clustering using a new geometrical idea whereby subspaces are identified based on their relative novelties. We present two frameworks in which the idea of innovation pursuit is used to distinguish the subspaces. Underlying the first framework is an iterative method that finds the subspaces consecutively by solving a series of simple linear optimization problems, each searching for a direction of innovation in the span of the data potentially orthogonal to all subspaces except for the one to be identified in one step of the algorithm. A detailed mathematical analysis is provided establishing sufficient conditions for iPursuit to correctly cluster the data. The proposed approach can provably yield exact clustering even when the subspaces have significant intersections. It is shown that the complexity of the iterative approach scales only linearly in the number of data points and subspaces, and quadratically in the dimension of the subspaces. The second framework integrates iPursuit with spectral clustering to yield a new variant of spectral-clustering-based algorithms. The numerical simulations with both real and synthetic data demonstrate that iPursuit can often outperform the state-of-the-art subspace clustering algorithms, more so for subspaces with significant intersections, and that it significantly improves the state-of-the-art result for subspace-segmentation-based face clustering. Many clustering algorithms tend to break down in high-dimensional feature spaces, because the clusters often exist only in specific subspaces (attribute subsets) of the original feature space. Therefore, the task of projected clustering (or subspace clustering) has been defined recently. As a solution to tackle this problem, we propose the concept of local subspace preferences, which captures the main directions of high point density. Using this concept, we adopt density-based clustering to cope with high-dimensional data. In particular, we achieve the following advantages over existing approaches: Our proposed method has a determinate result, does not depend on the order of processing, is robust against noise, performs only one single scan over the database, and is linear in the number of dimensions. A broad experimental evaluation shows that our approach yields results of significantly better quality than recent work on clustering high-dimensional data. In this paper, we present a deep extension of Sparse Subspace Clustering, termed Deep Sparse Subspace Clustering (DSSC). Regularized by the unit sphere distribution assumption for the learned deep features, DSSC can infer a new data affinity matrix by simultaneously satisfying the sparsity principle of SSC and the nonlinearity given by neural networks. One of the appealing advantages brought by DSSC is: when original real-world data do not meet the class-specific linear subspace distribution assumption, DSSC can employ neural networks to make the assumption valid with its hierarchical nonlinear transformations. To the best of our knowledge, this is among the first deep learning based subspace clustering methods. Extensive experiments are conducted on four real-world datasets to show the proposed DSSC is significantly superior to 12 existing methods for subspace clustering. Spectral clustering based subspace clustering methods have emerged recently. When the inputs are 2-dimensional (2D) data, most existing clustering methods convert such data to vectors as preprocessing, which severely damages spatial information of the data. In this paper, we propose a novel subspace clustering method for 2D data with enhanced capability of retaining spatial information for clustering. It seeks two projection matrices and simultaneously constructs a linear representation of the projected data, such that the sought projections help construct the most expressive representation with the most variational information. We regularize our method based on covariance matrices directly obtained from 2D data, which have much smaller size and are more computationally amiable. Moreover, to exploit nonlinear structures of the data, a nonlinear version is proposed, which constructs an adaptive manifold according to updated projections. The learning processes of projections, representation, and manifold thus mutually enhance each other, leading to a powerful data representation. Efficient optimization procedures are proposed, which generate non-increasing objective value sequence with theoretical convergence guarantee. Extensive experimental results confirm the effectiveness of proposed method. Subspace clustering (also called projected clustering) addresses the problem that different sets of attributes may be relevant for different clusters in high dimensional feature spaces. In this paper, we propose the algorithm DiSH (Detecting Subspace cluster Hierarchies) that improves in the following points over existing approaches: First, DiSH can detect clusters in subspaces of significantly different dimensionality. Second, DiSH uncovers complex hierarchies of nested subspace clusters, i.e. clusters in lower-dimensional subspaces that are embedded within higher-dimensional subspace clusters. These hierarchies do not only consist of single inclusions, but may also exhibit multiple inclusions and thus, can only be modeled using graphs rather than trees. Third, DiSH is able to detect clusters of different size, shape, and density. Furthermore, we propose to visualize the complex hierarchies by means of an appropriate visualization model, the so-called subspace clustering graph, such that the relationships between the subspace clusters can be explored at a glance. Several comparative experiments show the performance and the effectivity of DiSH. In this paper we propose a discriminant learning framework for problems in which data consist of linear subspaces instead of vectors. By treating subspaces as basic elements, we can make learning algorithms adapt naturally to the problems with linear invariant structures. We propose a unifying view on the subspace-based learning method by formulating the problems on the Grassmann manifold, which is the set of fixed-dimensional linear subspaces of a Euclidean space. Previous methods on the problem typically adopt an inconsistent strategy: feature extraction is performed in the Euclidean space while non-Euclidean distances are used. In our approach, we treat each sub-space as a point in the Grassmann space, and perform feature extraction and classification in the same space. We show feasibility of the approach by using the Grassmann kernel functions such as the Projection kernel and the Binet-Cauchy kernel. Experiments with real image databases show that the proposed method performs well compared with state-of-the-art algorithms. Many clustering algorithms are not applicable to high-dimensional feature spaces, because the clusters often exist only in specific subspaces of the original feature space. Those clusters are also called subspace clusters. In this paper, we propose the algorithm HiSC (Hierarchical Subspace Clustering) that can detect hierarchies of nested subspace clusters, i.e. the relationships of lower-dimensional subspace clusters that are embedded within higher-dimensional subspace clusters. Several comparative experiments using synthetic and real data sets show the performance and the effectivity of HiSC. Subspace clustering aims to cluster unlabeled samples into multiple groups by implicitly seeking a subspace to fit each group. Most of existing methods are based on a shallow linear model, which may fail in handling data with nonlinear structure. In this paper, we propose a novel subspace clustering method -- deeP subspAce clusteRing with sparsiTY prior (PARTY) -- based on a new deep learning architecture. PARTY explicitly learns to progressively transform input data into nonlinear latent space and to be adaptive to the local and global subspace structure simultaneously. In particular, considering local structure, PARTY learns representation for the input data with minimal reconstruction error. Moreover, PARTY incorporates a prior sparsity information into the hidden representation learning to preserve the sparse reconstruction relation over the whole data set. To the best of our knowledge, PARTY is the first deep learning based subspace clustering method. Extensive experiments verify the effectiveness of our method. Subspace clustering is an extension of traditional clustering that seeks to find clusters in different subspaces within a dataset. Often in high dimensional data, many dimensions are irrelevant and can mask existing clusters in noisy data. Feature selection removes irrelevant and redundant dimensions by analyzing the entire dataset. Subspace clustering algorithms localize the search for relevant dimensions allowing them to find clusters that exist in multiple, possibly overlapping subspaces. There are two major branches of subspace clustering based on their search strategy. Top-down algorithms find an initial clustering in the full set of dimensions and evaluate the subspaces of each cluster, iteratively improving the results. Bottom-up approaches find dense regions in low dimensional spaces and combine them to form clusters. This paper presents a survey of the various subspace clustering algorithms along with a hierarchy organizing the algorithms by their defining characteristics. We then compare the two main approaches to subspace clustering using empirical scalability and accuracy tests and discuss some potential applications where subspace clustering could be particularly useful. A method called spatial subspace clustering (SpatSC) is proposed for the hyperspectral data segmentation problem focusing on the hyperspectral data taken from a drill hole, which can be seen as one-dimensional image data compared with hyperspectral multispectral image data. Addressing this problem has several practical uses, such as improving interpretability of the data, and, especially, obtaining a better understanding of the mineralogy. SpatSC is a combination of subspace learning and the fused least absolute shrinkage and selection operator. As a result, it is able to produce spatially smooth clusters. From this point of view, it can be simply interpreted as a spatial information guided subspace learning algorithm. SpatSC has flexible structures that embrace the cases with and without library of pure spectra. It can be further extended, for example, using different error structures, such as including rank operator. We test this method on both simulated data and real-world hyperspectral data. SpatSC produces stable and continuous segments, which are more interpretable than those obtained from other state-of-the-art subspace learning algorithms. Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. Existing algebraic subspace clustering methods are based on fitting the data with an algebraic variety and decomposing this variety into its constituent subspaces. Such methods are well suited to the case of a known number of subspaces of known and equal dimensions, where a single polynomial vanishing in the variety is sufficient to identify the subspaces. While subspaces of unknown and arbitrary dimensions can be handled using multiple vanishing polynomials, current approaches are not robust to corrupted data due to the difficulty of estimating the number of polynomials. As a consequence, the current practice is to use a single polynomial to fit the data with a union of hyperplanes containing the union of subspaces, an approach that works well only when the dimensions of the subspaces are high enough. In this paper, we propose a new algebraic subspace clustering algorithm, which can identify the subspace ... In this paper, we tackle the problem of clustering data points drawn from a union of linear (or affine) subspaces. To this end, we introduce an efficient subspace clustering algorithm that estimates dense connections between the points lying in the same subspace. In particular, instead of following the standard compressive sensing approach, we formulate subspace clustering as a Frobenius norm minimization problem, which inherently yields denser con- nections between the data points. While in the noise-free case we rely on the self-expressiveness of the observations, in the presence of noise we simultaneously learn a clean dictionary to represent the data. Our formulation lets us address the subspace clustering problem efficiently. More specifically, the solution can be obtained in closed-form for outlier-free observations, and by performing a series of linear operations in the presence of outliers. Interestingly, we show that our Frobenius norm formulation shares the same solution as the popular nuclear norm minimization approach when the data is free of any noise, or, in the case of corrupted data, when a clean dictionary is learned. Our experimental evaluation on motion segmentation and face clustering demonstrates the benefits of our algorithm in terms of clustering accuracy and efficiency. Subspace clustering is an unsupervised learning problem that aims at grouping data points into multiple "clusters" so that data points in a single cluster lie approximately on a low-dimensional linear subspace. It is originally motivated by 3D motion segmentation in computer vision, but has recently been generically applied to a wide range of statistical machine learning problems, which often involves sensitive datasets about human subjects. This raises a dire concern for data privacy. In this work, we build on the framework of differential privacy and present two provably private subspace clustering algorithms. We demonstrate via both theory and experiments that one of the presented methods enjoys formal privacy and utility guarantees; the other one asymptotically preserves differential privacy while having good performance in practice. Along the course of the proof, we also obtain two new provable guarantees for the agnostic subspace clustering and the graph connectivity problem which might be of independent interests. Subspace clustering is the task of identifying clusters in subspaces of the input dimensions of a given dataset. Noisy data in certain attributes cause difficulties for traditional clustering algorithms, because the high discrepancies within them can make objects appear too different to be grouped in the same cluster. This requires methods specially designed for subspace clustering. This paper presents our second approach to subspace and projected clustering based on self-organizing maps (SOMs), which is a local adaptive receptive field dimension selective SOM. By introducing a time-variant topology, our method is an improvement in terms of clustering quality, computational cost, and parameterization. This enables the method to identify the correct number of clusters and their respective relevant dimensions, and thus it presents nearly perfect results in synthetic datasets and surpasses our previous method in most of the real-world datasets considered.