Constructing the L2-Graph for Robust Subspace Learning and Subspace Clustering

Reweighted Subspace Clustering Guided by Local and Global Structure Preservation

Subspace clustering has attracted significant interest for its capacity to partition high-dimensional data into multiple subspaces. The current approaches to subspace clustering predominantly revolve around two key aspects: 1) the construction of an effective similarity matrix and 2) the pursuit of sparsity within the projection matrix. However, assessing whether the dimensionality of the projected subspace is the true dimensionality of the data is challenging. Therefore, the clustering performance may decrease when dealing with scenarios such as subspace overlap, insufficient projected dimensions, data noise, etc., since the defined dimensionality of the projected lower-dimensional space may deviate significantly from its true value. In this research, we introduce a novel reweighting strategy, which is applied to projected coordinates for the first time and propose a reweighted subspace clustering model guided by the preservation of the both local and global structural characteristics (RWSC). The projected subspaces are reweighted to augment or suppress the importance of different coordinates, so that data with overlapping subspaces can be better distinguished and the redundant coordinates produced by the predefined number of projected dimensions can be further removed. By introducing reweighting strategies, the bias caused by imprecise dimensionalities in subspace clustering can be alleviated. Moreover, global scatter structure preservation and adaptive local structure learning are integrated into the proposed model, which helps RWSC capture more intrinsic structures and its robustness and applicability can then be improved. Through rigorous experiments on both synthetic and real-world datasets, the effectiveness and superiority of RWSC are empirically verified.

GRESS: Grouping Belief-Based Deep Contrastive Subspace Clustering

The self-expressive coefficient plays a crucial role in the self-expressiveness-based subspace clustering method. To enhance the precision of the self-expressive coefficient, we propose a novel deep subspace clustering method, named grouping belief-based deep contrastive subspace clustering (GRESS), which integrates the clustering information and higher-order relationship into the coefficient matrix. Specifically, we develop a deep contrastive subspace clustering module to enhance the learning of both self-expressive coefficients and cluster representations simultaneously. This approach enables the derivation of relatively noiseless self-expressive similarities and cluster-based similarities. To enable interaction between these two types of similarities, we propose a unique grouping belief-based affinity refinement module. This module leverages grouping belief to uncover the higher-order relationships within the similarity matrix, and integrates the well-designed noisy similarity suppression and similarity increment regularization to eliminate redundant connections while complete absent information. Extensive experimental results on four benchmark datasets validate the superiority of our proposed method GRESS over several state-of-the-art methods.

Semi-Supervised Mixture Learning for Graph Neural Networks With Neighbor Dependence

A graph neural network (GNN) is a powerful architecture for semi-supervised learning (SSL). However, the data-driven mode of GNNs raises some challenging problems. In particular, these models suffer from the limitations of incomplete attribute learning, insufficient structure capture, and the inability to distinguish between node attribute and graph structure, especially on label-scarce or attribute-missing data. In this article, we propose a novel framework, called graph coneighbor neural network (GCoNN), for node classification. It is composed of two modules: GCoNN Γ and GCoNN Γ° . GCoNN Γ is trained to establish the fundamental prototype for attribute learning on labeled data, while GCoNN Γ° learns neighbor dependence on transductive data through pseudolabels generated by GCoNN Γ . Next, GCoNN Γ is retrained to improve integration of node attribute and neighbor structure through feedback from GCoNN Γ° . GCoNN tends to convergence iteratively using such an approach. From a theoretical perspective, we analyze this iteration process from a generalized expectation-maximization (GEM) framework perspective which optimizes an evidence lower bound (ELBO) by amortized variational inference. Empirical evidence demonstrates that the state-of-the-art performance of the proposed approach outperforms other methods. We also apply GCoNN to brain functional networks, the results of which reveal response features across the brain which are physiologically plausible with respect to known language and visual functions.

Fast Discriminant Analysis With Adaptive Reconstruction Structure Preserving

Neighborhood reconstruction methods have been widely applied to feature engineering. Existing reconstruction-based discriminant analysis methods normally project high-dimensional data into a low-dimensional space while preserving the reconstruction relationships among samples. However, there are three limitations: 1) the reconstruction coefficients are learned based on the collaborative representation of all sample pairs, which requires the training time to be the cube of the number of samples; 2) these coefficients are learned in the original space, ignoring the interference of the noise and redundant features; and 3) there is a reconstruction relationship between heterogeneous samples; this will enlarge the similarity of heterogeneous samples in the subspace. In this article, we propose a fast and adaptive discriminant neighborhood projection model to tackle the above drawbacks. First, the local manifold structure is captured by bipartite graphs in which each sample is reconstructed by anchor points derived from the same class as that sample; this can avoid the reconstruction between heterogeneous samples. Second, the number of anchor points is far less than the number of samples; this strategy can reduce the time complexity substantially. Third, anchor points and reconstruction coefficients of bipartite graphs are updated adaptively in the process of dimensionality reduction, which can enhance the quality of bipartite graphs and extract discriminative features simultaneously. An iterative algorithm is designed to solve this model. Extensive results on toy data and benchmark datasets show the effectiveness and superiority of our model.

Hierarchical Neighbors Embedding

Manifold learning now plays an important role in machine learning and many relevant applications. In spite of the superior performance of manifold learning techniques in dealing with nonlinear data distribution, their performance would drop when facing the problem of data sparsity. It is hard to obtain satisfactory embeddings when sparsely sampled high-dimensional data are mapped into the observation space. To address this issue, in this article, we propose hierarchical neighbors embedding (HNE), which enhances the local connections through hierarchical combination of neighbors. And three different HNE-based implementations are derived by further analyzing the topological connection and reconstruction performance. The experimental results on both the synthetic and real-world datasets illustrate that our HNE-based methods could obtain more faithful embeddings with better topological and geometrical properties. From the view of embedding quality, HNE develops the outstanding advantages in dealing with data of general distributions. Furthermore, comparing with other state-of-the-art manifold learning methods, HNE shows its superiority in dealing with sparsely sampled data and weak-connected manifolds.

A study on a recommendation algorithm based on spectral clustering and GRU

With the development of e-commerce, the importance of recommendation algorithms has significantly increased. However, traditional recommendation systems struggle to address issues such as data sparsity and cold start. This article proposes an optimization method for a recommendation system based on spectral clustering (SC) and gated recurrent unit (GRU), named the GRU-KSC algorithm. Firstly, this paper improves the original spectral clustering algorithm by introducing Kmc2, proposing a novel spectral clustering recommendation algorithm (K-means++ SC, KSC) based on the existing SC algorithm. Secondly, building upon the original GRU model, the paper presents a hybrid recommendation algorithm (Hybrid GRU, HGRU) capable of capturing long-term user interests for a more personalized recommendation. Experiments conducted on real datasets demonstrate that our method outperforms existing benchmark methods in terms of accuracy and robustness.

iCVI-ARTMAP: Using Incremental Cluster Validity Indices and Adaptive Resonance Theory Reset Mechanism to Accelerate Validation and Achieve Multiprototype Unsupervised Representations

This article presents an adaptive resonance theory predictive mapping (ARTMAP) model, which uses incremental cluster validity indices (iCVIs) to perform unsupervised learning, namely, iCVI-ARTMAP. Incorporating iCVIs to the decision-making and many-to-one mapping capabilities of this adaptive resonance theory (ART)-based model can improve the choices of clusters to which samples are incrementally assigned. These improvements are accomplished by intelligently performing the operations of swapping sample assignments between clusters, splitting and merging clusters, and caching the values of variables when iCVI values need to be recomputed. Using recursive formulations enables iCVI-ARTMAP to considerably reduce the computational burden associated with cluster validity index (CVI)-based offline clustering. In this work, six iCVI-ARTMAP variants were realized via the integration of one information-theoretic and five sum-of-squares-based iCVIs into fuzzy ARTMAP. With proper choice of iCVI, iCVI-ARTMAP either outperformed or performed comparably to three ART-based and four non-ART-based clustering algorithms in experiments using benchmark datasets of different natures. Naturally, the performance of iCVI-ARTMAP is subject to the selected iCVI and its suitability to the data at hand; fortunately, it is a general model in which other iCVIs can be easily embedded.

Generalized Nonconvex Nonsmooth Low-Rank Matrix Recovery Framework With Feasible Algorithm Designs and Convergence Analysis

Decomposing data matrix into low-rank plus additive matrices is a commonly used strategy in pattern recognition and machine learning. This article mainly studies the alternating direction method of multiplier (ADMM) with two dual variables, which is used to optimize the generalized nonconvex nonsmooth low-rank matrix recovery problems. Furthermore, the minimization framework with a feasible optimization procedure is designed along with the theoretical analysis, where the variable sequences generated by the proposed ADMM can be proved to be bounded. Most importantly, it can be concluded from the Bolzano-Weierstrass theorem that there must exist a subsequence converging to a critical point, which satisfies the Karush-Kuhn-Tucher (KKT) conditions. Meanwhile, we further ensure the local and global convergence properties of the generated sequence relying on constructing the potential objective function. Particularly, the detailed convergence analysis would be regarded as one of the core contributions besides the algorithm designs and the model generality. Finally, the numerical simulations and the real-world applications are both provided to verify the consistence of the theoretical results, and we also validate the superiority in performance over several mostly related solvers to the tasks of image inpainting and subspace clustering.

Clustering of Multilayer Networks Using Joint Learning Algorithm With Orthogonality and Specificity of Features

Complex systems in nature and society consist of various types of interactions, where each type of interaction belongs to a layer, resulting in the so-called multilayer networks. Identifying specific modules for each layer is of great significance for revealing the structure-function relations in multilayer networks. However, the available approaches are criticized undesirable because they fail to explicitly the specificity of modules, and balance the specificity and connectivity of modules. To overcome these drawbacks, we propose an accurate and flexible algorithm by joint learning matrix factorization and sparse representation (jMFSR) for specific modules in multilayer networks, where matrix factorization extracts features of vertices and sparse representation discovers specific modules. To exploit the discriminative latent features of vertices in multilayer networks, jMFSR incorporates linear discriminant analysis (LDA) into non-negative matrix factorization (NMF) to learn features of vertices that distinguish the categories. To explicitly measure the specificity of features, jMFSR decomposes features of vertices into common and specific parts, thereby enhancing the quality of features. Then, jMFSR jointly learns feature extraction, common-specific feature factorization, and clustering of multilayer networks. The experiments on 11 datasets indicate that jMFSR significantly outperforms state-of-the-art baselines in terms of various measurements.

Robust Discriminant Subspace Clustering With Adaptive Local Structure Embedding

Unsupervised dimension reduction and clustering are frequently used as two separate steps to conduct clustering tasks in subspace. However, the two-step clustering methods may not necessarily reflect the cluster structure in the subspace. In addition, the existing subspace clustering methods do not consider the relationship between the low-dimensional representation and local structure in the input space. To address the above issues, we propose a robust discriminant subspace (RDS) clustering model with adaptive local structure embedding. Specifically, unlike the existing methods which incorporate dimension reduction and clustering via regularizer, thereby introducing extra parameters, RDS first integrates them into a unified matrix factorization (MF) model through theoretical proof. Furthermore, a similarity graph is constructed to learn the local structure. A constraint is imposed on the graph to guarantee that it has the same connected components with low-dimensional representation. In this spirit, the similarity graph serves as a tradeoff that adaptively balances the learning process between the low-dimensional space and the original space. Finally, RDS adopts the l_2,1 -norm to measure the residual error, which enhances the robustness to noise. Using the property of the l_2,1 -norm, RDS can be optimized efficiently without introducing more penalty terms. Experimental results on real-world benchmark datasets show that RDS can provide more interpretable clustering results and also outperform other state-of-the-art alternatives.

Fuzzy Sparse Subspace Clustering for Infrared Image Segmentation

Infrared image segmentation is a challenging task, due to interference of complex background and appearance inhomogeneity of foreground objects. A critical defect of fuzzy clustering for infrared image segmentation is that the method treats image pixels or fragments in isolation. In this paper, we propose to adopt self-representation from sparse subspace clustering in fuzzy clustering, aiming to introduce global correlation information into fuzzy clustering. Meanwhile, to apply sparse subspace clustering for non-linear samples from an infrared image, we leverage membership from fuzzy clustering to improve conventional sparse subspace clustering. The contributions of this paper are fourfold. First, by introducing self-representation coefficients modeled in sparse subspace clustering based on high-dimensional features, fuzzy clustering is capable of utilizing global information to resist complex background as well as intensity inhomogeneity of objects, so as to improve clustering accuracy. Second, fuzzy membership is tactfully exploited in the sparse subspace clustering framework. Thereby, the bottleneck of conventional sparse subspace clustering methods, that they could be barely applied to nonlinear samples, can be surmounted. Third, as we integrate fuzzy clustering and subspace clustering in a unified framework, features from two different aspects are employed, contributing to precise clustering results. Finally, we further incorporate neighbor information into clustering, thus effectively solving the uneven intensity problem in infrared image segmentation. Experiments examine the feasibility of proposed methods on various infrared images. Segmentation results demonstrate the effectiveness and efficiency of the proposed methods, which proves the superiority compared to other fuzzy clustering methods and sparse space clustering methods.

Maximum Block Energy Guided Robust Subspace Clustering

Subspace clustering is useful for clustering data points according to the underlying subspaces. Many methods have been presented in recent years, among which Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Least Squares Regression clustering (LSR) are three representative methods. These approaches achieve good results by assuming the structure of errors as a prior and removing errors in the original input space by modeling them in their objective functions. In this paper, we propose a novel method from an energy perspective to eliminate errors in the projected space rather than the input space. Since the block diagonal property can lead to correct clustering, we measure the correctness in terms of a block in the projected space with an energy function. A correct block corresponds to the subset of columns with the maximal energy. The energy of a block is defined based on the unary column, pairwise and high-order similarity of columns for each block. We relax the energy function of a block and approximate it by a constrained homogenous function. Moreover, we propose an efficient iterative algorithm to remove errors in the projected space. Both theoretical analysis and experiments show the superiority of our method over existing solutions to the clustering problem, especially when noise exists.

Multiple Kernel Clustering With Compressed Subspace Alignment

Multiple kernel clustering (MKC) has recently achieved remarkable progress in fusing multisource information to boost the clustering performance. However, the O(n²) memory consumption and O(n³) computational complexity prohibit these methods from being applied into median- or large-scale applications, where n denotes the number of samples. To address these issues, we carefully redesign the formulation of subspace segmentation-based MKC, which reduces the memory and computational complexity to O(n) and O(n²) , respectively. The proposed algorithm adopts a novel sampling strategy to enhance the performance and accelerate the speed of MKC. Specifically, we first mathematically model the sampling process and then learn it simultaneously during the procedure of information fusion. By this way, the generated anchor point set can better serve data reconstruction across different views, leading to improved discriminative capability of the reconstruction matrix and boosted clustering performance. Although the integrated sampling process makes the proposed algorithm less efficient than the linear complexity algorithms, the elaborate formulation makes our algorithm straightforward for parallelization. Through the acceleration of GPU and multicore techniques, our algorithm achieves superior performance against the compared state-of-the-art methods on six datasets with comparable time cost to the linear complexity algorithms.

Multi-Attribute Subspace Clustering via Auto-Weighted Tensor Nuclear Norm Minimization

Self-expressiveness based subspace clustering methods have received wide attention for unsupervised learning tasks. However, most existing subspace clustering methods consider data features as a whole and then focus only on one single self-representation. These approaches ignore the intrinsic multi-attribute information embedded in the original data feature and result in one-attribute self-representation. This paper proposes a novel multi-attribute subspace clustering (MASC) model that understands data from multiple attributes. MASC simultaneously learns multiple subspace representations corresponding to each specific attribute by exploiting the intrinsic multi-attribute features drawn from original data. In order to better capture the high-order correlation among multi-attribute representations, we represent them as a tensor in low-rank structure and propose the auto-weighted tensor nuclear norm (AWTNN) as a superior low-rank tensor approximation. Especially, the non-convex AWTNN fully considers the difference between singular values through the implicit and adaptive weights splitting during the AWTNN optimization procedure. We further develop an efficient algorithm to optimize the non-convex and multi-block MASC model and establish the convergence guarantees. A more comprehensive subspace representation can be obtained via aggregating these multi-attribute representations, which can be used to construct a clustering-friendly affinity matrix. Extensive experiments on eight real-world databases reveal that the proposed MASC exhibits superior performance over other subspace clustering methods.

Unsupervised Adaptive Embedding for Dimensionality Reduction

High-dimensional data are highly correlative and redundant, making it difficult to explore and analyze. Amount of unsupervised dimensionality reduction (DR) methods has been proposed, in which constructing a neighborhood graph is the primary step of DR methods. However, there exist two problems: 1) the construction of graph is usually separate from the selection of projection direction and 2) the original data are inevitably noisy. In this article, we propose an unsupervised adaptive embedding (UAE) method for DR to solve these challenges, which is a linear graph-embedding method. First, an adaptive allocation method of neighbors is proposed to construct the affinity graph. Second, the construction of affinity graph and calculation of projection matrix are integrated together. It considers the local relationship between samples and global characteristic of high-dimensional data, in which the cleaned data matrix is originally proposed to remove noise in subspace. The relationship between our method and local preserving projections (LPPs) is also explored. Finally, an alternative iteration optimization algorithm is derived to solve our model, the convergence and computational complexity of which are also analyzed. Comprehensive experiments on synthetic and benchmark datasets illustrate the superiority of our method.

Adaptive Data Structure Regularized Multiclass Discriminative Feature Selection

Feature selection (FS), which aims to identify the most informative subset of input features, is an important approach to dimensionality reduction. In this article, a novel FS framework is proposed for both unsupervised and semisupervised scenarios. To make efficient use of data distribution to evaluate features, the framework combines data structure learning (as referred to as data distribution modeling) and FS in a unified formulation such that the data structure learning improves the results of FS and vice versa. Moreover, two types of data structures, namely the soft and hard data structures, are learned and used in the proposed FS framework. The soft data structure refers to the pairwise weights among data samples, and the hard data structure refers to the estimated labels obtained from clustering or semisupervised classification. Both of these data structures are naturally formulated as regularization terms in the proposed framework. In the optimization process, the soft and hard data structures are learned from data represented by the selected features, and then, the most informative features are reselected by referring to the data structures. In this way, the framework uses the interactions between data structure learning and FS to select the most discriminative and informative features. Following the proposed framework, a new semisupervised FS (SSFS) method is derived and studied in depth. Experiments on real-world data sets demonstrate the effectiveness of the proposed method.

Structured Graph Learning for Scalable Subspace Clustering: From Single View to Multiview

Graph-based subspace clustering methods have exhibited promising performance. However, they still suffer some of these drawbacks: they encounter the expensive time overhead, they fail to explore the explicit clusters, and cannot generalize to unseen data points. In this work, we propose a scalable graph learning framework, seeking to address the above three challenges simultaneously. Specifically, it is based on the ideas of anchor points and bipartite graph. Rather than building an n×n graph, where n is the number of samples, we construct a bipartite graph to depict the relationship between samples and anchor points. Meanwhile, a connectivity constraint is employed to ensure that the connected components indicate clusters directly. We further establish the connection between our method and the K -means clustering. Moreover, a model to process multiview data is also proposed, which is linearly scaled with respect to n . Extensive experiments demonstrate the efficiency and effectiveness of our approach with respect to many state-of-the-art clustering methods.

Large-Scale Subspace Clustering by Independent Distributed and Parallel Coding

Subspace clustering is a popular method to discover underlying low-dimensional structures of high-dimensional multimedia data (e.g., images, videos, and texts). In this article, we consider a large-scale subspace clustering (LS²C) problem, that is, partitioning million data points with a millon dimensions. To address this, we explore an independent distributed and parallel framework by dividing big data/variable matrices and regularization by both columns and rows. Specifically, LS²C is independently decomposed into many subproblems by distributing those matrices into different machines by columns since the regularization of the code matrix is equal to a sum of that of its submatrices (e.g., square-of-Frobenius/ l₁ -norm). Consensus optimization is designed to solve these subproblems in a parallel way for saving communication costs. Moreover, we provide theoretical guarantees that LS²C can recover consensus subspace representations of high-dimensional data points under broad conditions. Compared with the state-of-the-art LS²C methods, our approach achieves better clustering results in public datasets, including a million images and videos.

Weighted Error Entropy-Based Information Theoretic Learning for Robust Subspace Representation

In most of the existing representation learning frameworks, the noise contaminating the data points is often assumed to be independent and identically distributed (i.i.d.), where the Gaussian distribution is often imposed. This assumption, though greatly simplifies the resulting representation problems, may not hold in many practical scenarios. For example, the noise in face representation is usually attributable to local variation, random occlusion, and unconstrained illumination, which is essentially structural, and hence, does not satisfy the i.i.d. property or the Gaussianity. In this article, we devise a generic noise model, referred to as independent and piecewise identically distributed (i.p.i.d.) model for robust presentation learning, where the statistical behavior of the underlying noise is characterized using a union of distributions. We demonstrate that our proposed i.p.i.d. model can better describe the complex noise encountered in practical scenarios and accommodate the traditional i.i.d. one as a special case. Assisted by the proposed noise model, we then develop a new information-theoretic learning framework for robust subspace representation through a novel minimum weighted error entropy criterion. Thanks to the superior modeling capability of the i.p.i.d. model, our proposed learning method achieves superior robustness against various types of noise. When applying our scheme to the subspace clustering and image recognition problems, we observe significant performance gains over the existing approaches.

Autoencoder-Based Latent Block-Diagonal Representation for Subspace Clustering

Block-diagonal representation (BDR) is an effective subspace clustering method. The existing BDR methods usually obtain a self-expression coefficient matrix from the original features by a shallow linear model. However, the underlying structure of real-world data is often nonlinear, thus those methods cannot faithfully reflect the intrinsic relationship among samples. To address this problem, we propose a novel latent BDR (LBDR) model to perform the subspace clustering on a nonlinear structure, which jointly learns an autoencoder and a BDR matrix. The autoencoder, which consists of a nonlinear encoder and a linear decoder, plays an important role to learn features from the nonlinear samples. Meanwhile, the learned features are used as a new dictionary for a linear model with block-diagonal regularization, which can ensure good performances for spectral clustering. Moreover, we theoretically prove that the learned features are located in the linear space, thus ensuring the effectiveness of the linear model using self-expression. Extensive experiments on various real-world datasets verify the superiority of our LBDR over the state-of-the-art subspace clustering approaches.

Dynamic Sparse Subspace Clustering for Evolving High-Dimensional Data Streams

In an era of ubiquitous large-scale evolving data streams, data stream clustering (DSC) has received lots of attention because the scale of the data streams far exceeds the ability of expert human analysts. It has been observed that high-dimensional data are usually distributed in a union of low-dimensional subspaces. In this article, we propose a novel sparse representation-based DSC algorithm, called evolutionary dynamic sparse subspace clustering (EDSSC). It can cope with the time-varying nature of subspaces underlying the evolving data streams, such as subspace emergence, disappearance, and recurrence. The proposed EDSSC consists of two phases: 1) static learning and 2) online clustering. During the first phase, a data structure for storing the statistic summary of data streams, called EDSSC summary, is proposed which can better address the dilemma between the two conflicting goals: 1) saving more points for accuracy of subspace clustering (SC) and 2) discarding more points for the efficiency of DSC. By further proposing an algorithm to estimate the subspace number, the proposed EDSSC does not need to know the number of subspaces. In the second phase, a more suitable index, called the average sparsity concentration index (ASCI), is proposed, which dramatically promotes the clustering accuracy compared to the conventionally utilized SCI index. In addition, the subspace evolution detection model based on the Page-Hinkley test is proposed where the appearing, disappearing, and recurring subspaces can be detected and adapted. Extinct experiments on real-world data streams show that the EDSSC outperforms the state-of-the-art online SC approaches.

Joint Optimal Transport With Convex Regularization for Robust Image Classification

The critical step of learning the robust regression model from high-dimensional visual data is how to characterize the error term. The existing methods mainly employ the nuclear norm to describe the error term, which are robust against structure noises (e.g., illumination changes and occlusions). Although the nuclear norm can describe the structure property of the error term, global distribution information is ignored in most of these methods. It is known that optimal transport (OT) is a robust distribution metric scheme due to that it can handle correspondences between different elements in the two distributions. Leveraging this property, this article presents a novel robust regression scheme by integrating OT with convex regularization. The OT-based regression with L₂ norm regularization (OTR) is first proposed to perform image classification. The alternating direction method of multipliers is developed to handle the model. To further address the occlusion problem in image classification, the extended OTR (EOTR) model is then presented by integrating the nuclear norm error term with an OTR model. In addition, we apply the alternating direction method of multipliers with Gaussian back substitution to solve EOTR and also provide the complexity and convergence analysis of our algorithms. Experiments were conducted on five benchmark datasets, including illumination changes and various occlusions. The experimental results demonstrate the performance of our robust regression model on biometric image classification against several state-of-the-art regression-based classification methods.

Noise-insensitive discriminative subspace fuzzy clustering

Discriminative subspace clustering (DSC) can make full use of linear discriminant analysis (LDA) to reduce the dimension of data and achieve effective clustering high-dimension data by clustering low-dimension data in discriminant subspace. However, most existing DSC algorithms do not consider the noise and outliers that may be contained in data sets, and when they are applied to the data sets with noise or outliers, and they often obtain poor performance due to the influence of noise and outliers. In this paper, we address the problem of the sensitivity of DSC to noise and outlier. Replacing the Euclidean distance in the objective function of LDA by an exponential non-Euclidean distance, we first develop a noise-insensitive LDA (NILDA) algorithm. Then, combining the proposed NILDA and a noise-insensitive fuzzy clustering algorithm: AFKM, we propose a noise-insensitive discriminative subspace fuzzy clustering (NIDSFC) algorithm. Experiments on some benchmark data sets show the effectiveness of the proposed NIDSFC algorithm.

δ-Norm-Based Robust Regression With Applications to Image Analysis

Up to now, various matrix norms (e.g., l₁ -norm, l₂ -norm, l_2,1 -norm, etc.) have been widely leveraged to form the loss function of different regression models, and have played an important role in image analysis. However, the previous regression models adopting the existing norms are sensitive to outliers and, thus, often bring about unsatisfactory results on the heavily corrupted images. This is because their adopted norms for measuring the data residual can hardly suppress the negative influence of noisy data, which will probably mislead the regression process. To address this issue, this paper proposes a novel δ (delta)-norm to count the nonzero blocks around an element in a vector or matrix, which weakens the impacts of outliers and also takes the structure property of examples into account. After that, we present the δ -norm-based robust regression (DRR) in which the data examples are mapped to the kernel space and measured by the proposed δ -norm. By exploring an explicit kernel function, we show that DRR has a closed-form solution, which suggests that DRR can be efficiently solved. To further handle the influences from mixed noise, DRR is extended to a multiscale version. The experimental results on image classification and background modeling datasets validate the superiority of the proposed approach to the existing state-of-the-art robust regression models.

Pseudo-Supervised Deep Subspace Clustering

Auto-Encoder (AE)-based deep subspace clustering (DSC) methods have achieved impressive performance due to the powerful representation extracted using deep neural networks while prioritizing categorical separability. However, self-reconstruction loss of an AE ignores rich useful relation information and might lead to indiscriminative representation, which inevitably degrades the clustering performance. It is also challenging to learn high-level similarity without feeding semantic labels. Another unsolved problem facing DSC is the huge memory cost due to n×n similarity matrix, which is incurred by the self-expression layer between an encoder and decoder. To tackle these problems, we use pairwise similarity to weigh the reconstruction loss to capture local structure information, while a similarity is learned by the self-expression layer. Pseudo-graphs and pseudo-labels, which allow benefiting from uncertain knowledge acquired during network training, are further employed to supervise similarity learning. Joint learning and iterative training facilitate to obtain an overall optimal solution. Extensive experiments on benchmark datasets demonstrate the superiority of our approach. By combining with the k -nearest neighbors algorithm, we further show that our method can address the large-scale and out-of-sample problems. The source code of our method is available: https://github.com/sckangz/SelfsupervisedSC.

Multi-Dimensional Visual Data Completion via Low-Rank Tensor Representation Under Coupled Transform

This paper addresses the tensor completion problem, which aims to recover missing information of multi-dimensional images. How to represent a low-rank structure embedded in the underlying data is the key issue in tensor completion. In this work, we suggest a novel low-rank tensor representation based on coupled transform, which fully exploits the spatial multi-scale nature and redundancy in spatial and spectral/temporal dimensions, leading to a better low tensor multi-rank approximation. More precisely, this representation is achieved by using two-dimensional framelet transform for the two spatial dimensions, one/two-dimensional Fourier transform for the temporal/spectral dimension, and then Karhunen-Loéve transform (via singular value decomposition) for the transformed tensor. Based on this low-rank tensor representation, we formulate a novel low-rank tensor completion model for recovering missing information in multi-dimensional visual data, which leads to a convex optimization problem. To tackle the proposed model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the structured optimization problem. Numerical examples on color images, multispectral images, and videos illustrate that the proposed method outperforms many state-of-the-art methods in qualitative and quantitative aspects.

Scaled Simplex Representation for Subspace Clustering

The self-expressive property of data points, that is, each data point can be linearly represented by the other data points in the same subspace, has proven effective in leading subspace clustering (SC) methods. Most self-expressive methods usually construct a feasible affinity matrix from a coefficient matrix, obtained by solving an optimization problem. However, the negative entries in the coefficient matrix are forced to be positive when constructing the affinity matrix via exponentiation, absolute symmetrization, or squaring operations. This consequently damages the inherent correlations among the data. Besides, the affine constraint used in these methods is not flexible enough for practical applications. To overcome these problems, in this article, we introduce a scaled simplex representation (SSR) for the SC problem. Specifically, the non-negative constraint is used to make the coefficient matrix physically meaningful, and the coefficient vector is constrained to be summed up to a scalar to make it more discriminative. The proposed SSR-based SC (SSRSC) model is reformulated as a linear equality-constrained problem, which is solved efficiently under the alternating direction method of multipliers framework. Experiments on benchmark datasets demonstrate that the proposed SSRSC algorithm is very efficient and outperforms the state-of-the-art SC methods on accuracy. The code can be found at https://github.com/csjunxu/SSRSC.

Deep Subspace Clustering

In this article, we propose a deep extension of sparse subspace clustering, termed deep subspace clustering with L1-norm (DSC-L1). Regularized by the unit sphere distribution assumption for the learned deep features, DSC-L1 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 DSC-L1 is that when original real-world data do not meet the class-specific linear subspace distribution assumption, DSC-L1 can employ neural networks to make the assumption valid with its nonlinear transformations. Moreover, we prove that our neural network could sufficiently approximate the minimizer under mild conditions. To the best of our knowledge, this could be one of the first deep-learning-based subspace clustering methods. Extensive experiments are conducted on four real-world data sets to show that the proposed method is significantly superior to 17 existing methods for subspace clustering on handcrafted features and raw data.

Deep Clustering With Sample-Assignment Invariance Prior

Most popular clustering methods map raw image data into a projection space in which the clustering assignment is obtained with the vanilla k-means approach. In this article, we discovered a novel prior, namely, there exists a common invariance when assigning an image sample to clusters using different metrics. In short, different distance metrics will lead to similar soft clustering assignments on the manifold. Based on such a novel prior, we propose a novel clustering method by minimizing the discrepancy between pairwise sample assignments for each data point. To the best of our knowledge, this could be the first work to reveal the sample-assignment invariance prior based on the idea of treating labels as ideal representations. Furthermore, the proposed method is one of the first end-to-end clustering approaches, which jointly learns clustering assignment and representation. Extensive experimental results show that the proposed method is remarkably superior to 16 state-of-the-art clustering methods on five image data sets in terms of four evaluation metrics.

Self-Weighted Clustering With Adaptive Neighbors

Many modern clustering models can be divided into two separated steps, i.e., constructing a similarity graph (SG) upon samples and partitioning each sample into the corresponding cluster based on SG. Therefore, learning a reasonable SG has become a hot issue in the clustering field. Many previous works that focus on constructing better SG have been proposed. However, most of them follow an ideal assumption that the importance of different features is equal, which is not adapted in practical applications. To alleviate this problem, this article proposes a self-weighted clustering with adaptive neighbors (SWCAN) model that can assign weights for different features, learn an SG, and partition samples into clusters simultaneously. In experiments, we observe that the SWCAN can assign weights for different features reasonably and outperform than comparison clustering models on synthetic and practical data sets.

Dual Shared-Specific Multiview Subspace Clustering

Multiview subspace clustering has received significant attention as the availability of diverse of multidomain and multiview real-world data has rapidly increased in the recent years. Boosting the performance of multiview clustering algorithms is challenged by two major factors. First, since original features from multiview data are highly redundant, reconstruction based on these attributes inevitably results in inferior performance. Second, since each view of such multiview data may contain unique knowledge as against the others, it remains a challenge to exploit complimentary information across multiple views while simultaneously investigating the uniqueness of each view. In this paper, we present a novel dual shared-specific multiview subspace clustering (DSS-MSC) approach that simultaneously learns the correlations between shared information across multiple views and also utilizes view-specific information to depict specific property for each independent view. Further, we formulate a dual learning framework to capture shared-specific information into the dimensional reduction and self-representation processes, which strengthens the ability of our approach to exploit shared information while preserving view-specific property effectively. The experimental results on several benchmark datasets have demonstrated the effectiveness of the proposed approach against other state-of-the-art techniques.

Subspace Clustering via Good Neighbors

Finding the informative subspaces of high-dimensional datasets is at the core of numerous applications in computer vision, where spectral-based subspace clustering is arguably the most widely studied method due to its strong empirical performance. Such algorithms first compute an affinity matrix to construct a self-representation for each sample using other samples as a dictionary. Sparsity and connectivity of the self-representation play important roles in effective subspace clustering. However, simultaneous optimization of both factors is difficult due to their conflicting nature, and most existing methods are designed to address only one factor. In this paper, we propose a post-processing technique to optimize both sparsity and connectivity by finding good neighbors. Good neighbors induce key connections among samples within a subspace and not only have large affinity coefficients but are also strongly connected to each other. We reassign the coefficients of the good neighbors and eliminate other entries to generate a new coefficient matrix. We show that the few good neighbors can effectively recover the subspace, and the proposed post-processing step of finding good neighbors is complementary to most existing subspace clustering algorithms. Experiments on five benchmark datasets show that the proposed algorithm performs favorably against the state-of-the-art methods with negligible additional computation cost.

Recursive Discriminative Subspace Learning With l1 -Norm Distance Constraint

In feature learning tasks, one of the most enormous challenges is to generate an efficient discriminative subspace. In this paper, we propose a novel subspace learning method, named recursive discriminative subspace learning with an l₁ -norm distance constraint (RDSL). RDSL can robustly extract features from the contaminated images and learn a discriminative subspace. With the use of an inequation-based l₁ -norm distance metric constraint, the minimized l₁ -norm distance metric objective function with slack variables induces samples in the same class to cluster as close as possible, meanwhile samples from different classes can be separated from each other as far as possible. By utilizing l₁ -norm items in both the objective function and the constraint, RDSL can well handle the noisy data and outliers. In addition, the large margin formulation makes the proposed method insensitive to initializations. We describe two approaches to solve RDSL with a recursive strategy. Experimental results on six benchmark datasets, including the original data and the contaminated data, demonstrate that RDSL outperforms the state-of-the-art methods.

Robust Graph Learning From Noisy Data

Learning graphs from data automatically have shown encouraging performance on clustering and semisupervised learning tasks. However, real data are often corrupted, which may cause the learned graph to be inexact or unreliable. In this paper, we propose a novel robust graph learning scheme to learn reliable graphs from the real-world noisy data by adaptively removing noise and errors in the raw data. We show that our proposed model can also be viewed as a robust version of manifold regularized robust principle component analysis (RPCA), where the quality of the graph plays a critical role. The proposed model is able to boost the performance of data clustering, semisupervised classification, and data recovery significantly, primarily due to two key factors: 1) enhanced low-rank recovery by exploiting the graph smoothness assumption and 2) improved graph construction by exploiting clean data recovered by RPCA. Thus, it boosts the clustering, semisupervised classification, and data recovery performance overall. Extensive experiments on image/document clustering, object recognition, image shadow removal, and video background subtraction reveal that our model outperforms the previous state-of-the-art methods.

l0 -Motivated Low-Rank Sparse Subspace Clustering

In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank and sparseness constraints on the data representation matrix. In low-rank sparse subspace clustering (LRSSC), nuclear and l₁ -norms are used to measure rank and sparsity. However, the use of nuclear and l₁ -norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two l₀ quasi-norm-based regularizations. First, this paper presents regularization based on multivariate generalization of minimax-concave penalty (GMC-LRSSC), which contains the global minimizers of a l₀ quasi-norm regularized objective. Afterward, we introduce the Schatten-0 ( S₀ ) and l₀ -regularized objective and approximate the proximal map of the joint solution using a proximal average method ( S₀/l₀ -LRSSC). The resulting nonconvex optimization problems are solved using an alternating direction method of multipliers with established convergence conditions of both algorithms. Results obtained on synthetic and four real-world datasets show the effectiveness of GMC-LRSSC and S₀/l₀ -LRSSC when compared to state-of-the-art methods.