Hyper-Laplacian regularized multi-view subspace clustering with low-rank tensor constraint

Data clustering is a fundamental topic in machine learning and data mining areas. In recent years, researchers have proposed a series of effective methods based on Low Rank Representation (LRR) which could explore low-dimension subspace structure embedded in original data effectively. The traditional LRR methods are usually designed for vectorial data from linear spaces with Euclidean distance. However, high-dimension data (such as video clip or imageset) are always considered as non-linear manifold data such as Grassmann manifold with non-linear metric. In addition, traditional LRR clustering method always adopt single nuclear norm as low rank constraint which would lead to suboptimal solution and decrease the clustering accuracy. In this paper, we proposed a new low rank method on Grassmann manifold for video or imageset data clustering task. In the proposed method, video or imageset data are formulated as sample data on Grassmann manifold first. And then a double low rank constraint is proposed by combining the nuclear norm and bilinear representation to better construct the representation matrix. The experimental results on several public datasets show that the proposed method outperforms the state-of-the-art clustering methods.

Similarity graph learning is the most key technique for multiview spectral clustering. However, existing methods fail when applied to uncertain data contaminated with various types of noise in an open environment. Due to the damaged structure by noise, unreliable similar relationships are learned, which extends similarity inconsistency among views. Moreover, the high-order correlation hidden in graphs are ignored generally. To address these problems, we propose a reliable similarity learning scheme for multiview clustering on uncertain data. This method can significantly improve spectral clustering performance in a noisy environment, and the contributions of our scheme include the following three aspects: 1) Uncertain data subspace reconstruction and adaptive graph learning are combined to construct a view-specific graph from high-quality recovered data, thus improving robustness. 2) A low-rank tensor constraint is utilized to facilitate multiview fusion, where the latent high-order correlation among view graphs will be fully explored when learning the consensus graph structure. 3) Data recovery, view-specific graphs, and latent consensus tensor structure are assembled into a unified framework, to be optimized jointly for mutual benefit. Our study also develops an efficient algorithm for obtaining overall solutions. The experimental results on several datasets demonstrate that our proposed approach shows significant improvements in robustness and evaluation metrics over the comparison methods.

For an object classification system, the most critical obstacles toward real-world applications are often caused by large intra-class variability, arising from different lightings, occlusion, and corruption, in limited sample sets. Most methods in the literature would fail when the training samples are heavily occluded, corrupted or have significant illumination or viewpoint variations. Besides, most of the existing methods and especially deep learning-based methods, need large training sets to achieve a satisfactory recognition performance. Although using the pre-trained network on a generic large-scale data set and fine-tune it to the small-sized target data set is a widely used technique, this would not help when the content of base and target data sets are very different. To address these issues simultaneously, we propose a joint projection and low-rank dictionary learning method using dual graph constraints. Specifically, a structured class-specific dictionary is learned in the low-dimensional space, and the discrimination is further improved by imposing a graph constraint on the coding coefficients, that maximizes the intra-class compactness and inter-class separability. We enforce structural incoherence and low-rank constraints on sub-dictionaries to reduce the redundancy among them, and also make them robust to variations and outliers. To preserve the intrinsic structure of data, we introduce a supervised neighborhood graph into the framework to make the proposed method robust to small-sized and high-dimensional data sets. Experimental results on several benchmark data sets verify the superior performance of our method for object classification of small-sized data sets, which include a considerable amount of different kinds of variation, and may have high-dimensional feature vectors.

Discriminative feature extraction based on sparse and low-rank representation

In this paper, we address the multiview nonlinear subspace representation problem. Traditional multiview subspace learning methods assume that the heterogeneous features of the data usually lie within the union of multiple linear subspaces. However, instead of linear subspaces, the data feature actually resides in multiple nonlinear subspaces in many real-world applications, resulting in unsatisfactory clustering performance. To overcome this, we propose a hyper-Laplacian regularized multilinear multiview self-representation model, which is referred to as HLR-M2VS, to jointly learn multiple views correlation and a local geometrical structure in a unified tensor space and view-specific self-representation feature spaces, respectively. In unified tensor space, a well-founded tensor low-rank regularization is adopted to impose on the self-representation coefficient tensor to ensure global consensus among different views. In view-specific feature space, hypergraph-induced hyper-Laplacian regularization is utilized to preserve the local geometrical structure embedded in a high-dimensional ambient space. An efficient algorithm is then derived to solve the optimization problem of the established model with theoretical convergence guarantee. Furthermore, the proposed model can be extended to semisupervised classification without introducing any additional parameters. An extensive experiment of our method is conducted on many challenging datasets, where a clear advance over state-of-the-art multiview clustering and multiview semisupervised classification approaches is achieved.

Graph matching is a widely researched topic which has been utilized in various applications of computer vision. Due to the combinatorial nature of graph matching, it is NP-hard to find an exact solution. So exact graph matching is always relaxed to inexact graph matching which seeks to find an approximate solution for the original problem. For a matching problem in quadratic form, semidefinite programming (SDP) relaxation is proven to be effective. However, previous SDP relaxation methods discard the constraint that the solution matrix is rank one, because the rank of a matrix is non-convex. In this paper, we explore some good properties of the solution matrix. By relaxing the rank into convex form using the properties, we propose to reformulate the graph matching with low rank constraint into a standard SDP, which can be easily solved. We test our method on both synthetic and real world data. The experimental results demonstrate that our method effectively handles low rank constraint and achieves competitive performance on robustness test against state-of-the-art counterparts.

Hyperspectral images (HSIs) have a wealth of applications in many areas, due to their fine spectral discrimination ability. However, in the practical imaging process, HSIs are often degraded by a mixture of various types of noise, for example, Gaussian noise, impulse noise, dead pixels, dead lines, and stripe noise. Low-rank matrix decomposition theory has been widely used in HSI denoising, and has achieved competitive results by modeling the impulse noise, dead pixels, dead lines, and stripe noise as sparse components. However, the existing low-rank-based methods for HSI denoising cannot completely remove stripe noise when the stripe noise is no longer sparse. In this article, we extend the HSI observation model and propose a double low-rank (DLR) matrix decomposition method for HSI denoising and destriping. By simultaneously exploring the low-rank characteristic of the lexicographically ordered noise-free HSI and the low-rank structure of the stripe noise on each band of the HSI, the two low-rank constraints are formulated into one unified framework, to achieve separation of the noise-free HSI, stripe noise, and other mixed noise. The proposed DLR model is then solved by the augmented Lagrange multiplier (ALM) algorithm efficiently. Both simulation and real HSI data experiments were carried out to verify the superiority of the proposed DLR method.

We present an efficient global optimization algorithm for exponential family principal component analysis (PCA) and associated low-rank matrix factorization problems. Exponential family PCA has been shown to improve the results of standard PCA on non-Gaussian data. Unfortunately, the widespread use of exponential family PCA has been hampered by the existence of only local optimization procedures. The prevailing assumption has been that the non-convexity of the problem prevents an efficient global optimization approach from being developed. Fortunately, this pessimism is unfounded. We present a reformulation of the underlying optimization problem that preserves the identity of the global solution while admitting an efficient optimization procedure. The algorithm we develop involves only a sub-gradient optimization of a convex objective plus associated eigenvector computations. (No general purpose semidefinite programming solver is required.) The low-rank constraint is exactly preserved, while the method can be kernelized through a consistent approximation to admit a fixed non-linearity. We demonstrate improved solution quality with the global solver, and also add to the evidence that exponential family PCA produces superior results to standard PCA on non-Gaussian data.

Benefiting from the joint consideration of geometric structures and low-rank constraint, graph low-rank representation (GLRR) method has led to the state-of-the-art results in many applications. However, it faces the limitations that the structure of errors should be known a prior, the isolated construction of graph Laplacian matrix, and the over shrinkage of the leading rank components. To improve GLRR in these regards, this paper proposes a new LRR model, namely iterative reconstrained LRR via weighted nonconvex regularization, using three distinguished properties on the concerned representation matrix. The first characterizes various distributions of the errors into an adaptively learned weight factor for more flexibility of noise suppression. The second generates an accurate graph matrix from weighted observations for less afflicted by noisy features. The third employs a parameterized rational function to reveal the importance of different rank components for better approximation to the intrinsic subspace structure. Following a deep exploration of automatic thresholding, parallel update, and partial SVD operation, we derive a computationally efficient low-rank representation algorithm using an iterative reconstrained framework and accelerated proximal gradient method. Comprehensive experiments are conducted on synthetic data, image clustering, and background subtraction to achieve several quantitative benchmarks as clustering accuracy, normalized mutual information, and execution time. Results demonstrate the robustness and efficiency of IRWNR compared with other state-of-the-art models.

CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance.

Spectral clustering has become one of the most popular clustering algorithms in recent years. In real-world clustering problems, the data points for clustering may have considerable noise. To the best of our knowledge, no single clustering algorithm is able to identify all different types of cluster structures. In the existing spectral clustering methods, little effort has been made to explicitly handle both the possibly considerable noise in data points and the robustness of clustering methods, which often degrades the clustering performance. In this paper, motivated by resampling and matrix aggregation, we propose a method for robust spectral clustering. In our method, we first construct multiple transition probability matrices, each is constructed by a subset of randomly selected features. Then, these matrices can be used to recover a shared low-rank similarity matrix, which is the input to the spectral clustering, and several sparse matrices, which represent the noise. The corresponding optimization problem has a low-rank constraint on the transition probability matrix. To solve the corresponding optimization problem, an optimization procedure based on the scheme of Augmented Lagrangian Method of Multipliers is designed. Experimental results on several real-world datasets show that our method has superior performance over several state-of-the-art clustering methods.

Hyperspectral image (HSI) classification is an important technology for hyperspectral data analysis, which has the challenge of high dimensionality and limited training samples. Moreover, due to the spatial continuity of material distribution, it is reasonable to improve the performance of HSI classification from the local similarity viewpoint, i.e., pixels in the same local region have a high probability to share similar features. Therefore, in this paper, a novel method based on local similarity and kernel low-rank representation is proposed to improve the classification accuracy for HSI. First, taking into consideration that pixels in the same homogeneous region share similar spectral features (termed local similarity), the HSI is segmented by the superpixel segmentation algorithm to obtain the homogeneous regions. Then, a model of kernel low-rank representation based on local similarity (KLRRLS) is proposed for HSI classification. In the model of KLRRLS, the nonlinear structure characteristics of HSI are extracted by employing the kernel trick, and the low-rank constraint is used as a prior to restrict the recovery coefficient matrix in each segmentation region, thus efficiently solving the linear non-separable problem and exploiting the contextual information of HSI. Finally, the classification results are generated by fusing the result of each segmented map via the maximum voting mechanism. Experimental results on three real HSI datasets demonstrate that the proposed method can effectively improve the classification accuracy.

Dimensionality reduction is an essential task in hyperspectral image processing. How to preserve the original intrinsic structure information and enhance the discriminant ability is still a challenge in this area. Recently, with the advantage of preserving global intrinsic structure information, low-rank representation has been applied to dimensionality reduction and achieved promising performance. By exploiting the submanifold information of the original data set, multimanifold learning is effective in enhancing the discriminant ability of the processed data set. In addition, due to the ability of preserving the spatial neighborhood structure information, the tensor analysis has become a popular technique for hyperspectral image processing. Motivated by the above-mentioned analysis, a novel tensor-based low-rank graph with multimanifold regularization (T-LGMR) for dimensionality reduction of hyperspectral images is proposed in this paper. In the T-LGMR, a low-rank constraint is employed to preserve the global data structure while multimanifold information is utilized to enhance the discriminant ability, and tensor representation is used to preserve the spatial neighborhood information. Finally, dimensionality reduction is achieved in the graph embedding framework. Experimental results on three real hyperspectral data sets demonstrate the superiority of the proposed method over several state-of-the-art approaches.

Multi-target regression is the problem where each instance is associated with multiple continuous target outputs simultaneously. Its major challenges arise with jointly exploring the complex input-output relationships and inter-target correlations. One representative approach is to build many independent single-target Support Vector Regression (SVR) models for each output target which can capture complex input-output relationships via the kernel trick. However, it does not involve inter-target correlations to improve the performance. Meanwhile, there are also many regularization-based methods which mainly explore the linear inter-target correlations, e.g., a low-rank constraint on the parameter matrix. However, in practice, it might be restrictive to assume the targets to be linearly related, and allowing for nonlinear relationships is a challenge. Motivated by Learning Using Privileged Information (LUPI), we propose a novel privileged multi-target support vector regression (MT-PSVR) model which can jointly explore the complex input-output relationships and nonlinear inter-target correlations. It explicitly explores inter-target correlations by viewing other targets as privileged information when training each target model. Besides, it can naturally use the kernel trick to explore both the complex input-output relationships and nonlinear inter-target correlations. Experimental results on many benchmark datasets validate the effectiveness of our approach.

ABSTRACTIn the area of hyperspectral image (HSI) classification, graph-based semi-supervised learning (SSL) has been proved to be highly effective. Constructing a proper graph is critical for graph-based SSL tasks. In HSI, spectral distance is widely used to calculate the weight of graph edge, though it can be influenced by noise and outliers. Meanwhile, links among all the data points are incorporated in the graph, including those from different subspaces. Thus the constructed graph might contain incorrect information. In this letter, a novel semi-supervised HSI classification method using local low-rank representation (SL2R) is proposed. Edge weight calculation will not be affected by noise or outliers thanks to the robustness of low-rank representation (LRR). Since each graph is constructed at local level, where pixels are basically embedded in the same subspace, links among uncorrelated pixels can be removed. Moreover, spatial context is naturally characterized by low-rank constraint on adjacent pixels. Experimental results on two data sets (Indian Pines and Botswana) confirm the effectiveness of the proposed method.