Structure-constrained low-rank representation

Low-Rank Representation with Empirical Kernel Space Embedding of Manifolds

Low-Rank Representation (LRR) methods integrate low-rank constraints and projection operators to model the mapping from the sample space to low-dimensional manifolds. Nonetheless, existing approaches typically apply Euclidean algorithms directly to manifold data in the original input space, leading to suboptimal classification accuracy. To mitigate this limitation, we introduce an unsupervised low-rank projection learning method named Low-Rank Representation with Empirical Kernel Space Embedding of Manifolds (LRR-EKM). LRR-EKM leverages an empirical kernel mapping to project samples into the Reproduced Kernel Hilbert Space (RKHS), enabling the linear separability of non-linearly structured samples and facilitating improved low-dimensional manifold representations through Euclidean distance metrics. By incorporating a row sparsity constraint on the projection matrix, LRR-EKM not only identifies discriminative features and removes redundancies but also enhances the interpretability of the learned subspace. Additionally, we introduce a manifold structure preserving constraint to retain the original representation and distance information of the samples during projection. Comprehensive experimental evaluations across various real-world datasets validate the superior performance of our proposed method compared to the state-of-the-art methods. The code is publicly available at https://github.com/ff-raw-war/LRR-EKM.

Sparse Low-Rank Multi-View Subspace Clustering With Consensus Anchors and Unified Bipartite Graph

Anchor technology is popularly employed in multi-view subspace clustering (MVSC) to reduce the complexity cost. However, due to the sampling operation being performed on each individual view independently and not considering the distribution of samples in all views, the produced anchors are usually slightly distinguishable, failing to characterize the whole data. Moreover, it is necessary to fuse multiple separated graphs into one, which leads to the final clustering performance heavily subject to the fusion algorithm adopted. What is worse, existing MVSC methods generate dense bipartite graphs, where each sample is associated with all anchor candidates. We argue that this dense-connected mechanism will fail to capture the essential local structures and degrade the discrimination of samples belonging to the respective near anchor clusters. To alleviate these issues, we devise a clustering framework named SL-CAUBG. Specifically, we do not utilize sampling strategy but optimize to generate the consensus anchors within all views so as to explore the information between different views. Based on the consensus anchors, we skip the fusion stage and directly construct the unified bipartite graph across views. Most importantly, norm and Laplacian-rank constraints employed on the unified bipartite graph make it capture both local and global structures simultaneously. norm helps eliminate the scatters between anchors and samples by constructing sparse links and guarantees our graph to be with clear anchor-sample affinity relationship. Laplacian-rank helps extract the global characteristics by measuring the connectivity of unified bipartite graph. To deal with the nondifferentiable objective function caused by norm, we adopt an iterative re-weighted method and the Newton's method. To handle the nonconvex Laplacian-rank, we equivalently transform it as a convex trace constraint. We also devise a four-step alternate method with linear complexity to solve the resultant problem. Substantial experiments show the superiority of our SL-CAUBG.

Segmentation and Completion of Human Motion Sequence via Temporal Learning of Subspace Variety Model

Subspace-based models have been extensively employed in unsupervised segmentation and completion of human motion sequence (HMS). However, existing approaches often neglect the incorporation of temporal priors embedded in HMS, resulting in suboptimal results. This paper presents a subspace variety model for HMS, along with an innovative Temporal Learning of Subspace Variety Model (TL-SVM) method for enhanced segmentation and completion in HMS. The key idea is to segment incomplete HMS into motion clusters and extracting the subspace features of each motion through the temporal learning of the subspace variety model. Subsequently, the HMS is completed based on the extracted subspace features. Thus, the main challenge is to learn the subspace variety model with temporal priors when confronted with missing entries. To tackle this, the paper develops a spatio-temporal assignment consistency (STAC) constraint for the subspace variety model, leveraging temporal priors embedded in HMS. In addition, a subspace clustering approach under the STAC constraint is proposed to learn the subspace variety model by extracting subspace features from HMS and segmenting HMS into motion clusters alternatively. The proposed subspace clustering model can also handle missing entries with theoretical guarantees. Furthermore, the missing entries of HMS are completed by minimizing the distance between each human motion frame and its corresponding subspace. Extensive experimental results, along with comparisons to state-of-the-art methods on four benchmark datasets, underscore the advantages of the proposed method.

Automatical Spike Sorting With Low-Rank and Sparse Representation

Spikesorting is crucial in studying neural individually and synergistically encoding and decoding behaviors. However, existent spike sorting algorithms perform unsatisfactorily in real scenarios where heavy noises and overlapping samples are commonly in the spikes, and the spikes from different neurons are similar. To address such challenging scenarios, we propose an automatic spike sporting method in this paper, which integrally combines low-rank and sparse representation (LRSR) into a unified model. In particular, LRSR models spikes through low-rank optimization, uncovering global data structure for handling similar and overlapped samples. To eliminate the influence of the embedded noises, LRSR uses a sparse constraint, effectively separating spikes from noise. The optimization is solved using alternate augmented Lagrange multipliers methods. Moreover, we conclude with an automatic spike-sorting framework that employs the spectral clustering theorem to estimate the number of neurons. Extensive experiments over various simulated and real-world datasets demonstrate that our proposed method, LRSR, can handle spike sorting effectively and efficiently.

Subspace Clustering via Structured Sparse Relation Representation

Due to the corruptions or noises that existed in real-world data sets, the affinity graphs constructed by the classical spectral clustering-based subspace clustering algorithms may not be able to reveal the intrinsic subspace structures of data sets faithfully. In this article, we reconsidered the data reconstruction problem in spectral clustering-based algorithms and proposed the idea of "relation reconstruction." We pointed out that a data sample could be represented by the neighborhood relation computed between its neighbors and itself. The neighborhood relation could indicate the true membership of its corresponding original data sample to the subspaces of a data set. We also claimed that a data sample's neighborhood relation could be reconstructed by the neighborhood relations of other data samples; then, we suggested a much different way to define affinity graphs consequently. Based on these propositions, a sparse relation representation (SRR) method was proposed for solving subspace clustering problems. Moreover, by introducing the local structure information of original data sets into SRR, an extension of SRR, namely structured sparse relation representation (SSRR) was presented. We gave an optimization algorithm for solving SRR and SSRR problems and analyzed its computation burden and convergence. Finally, plentiful experiments conducted on different types of databases showed the superiorities of SRR and SSRR.

Semisupervised Affinity Matrix Learning via Dual-Channel Information Recovery

This article explores the problem of semisupervised affinity matrix learning, that is, learning an affinity matrix of data samples under the supervision of a small number of pairwise constraints (PCs). By observing that both the matrix encoding PCs, called pairwise constraint matrix (PCM) and the empirically constructed affinity matrix (EAM), express the similarity between samples, we assume that both of them are generated from a latent affinity matrix (LAM) that can depict the ideal pairwise relation between samples. Specifically, the PCM can be thought of as a partial observation of the LAM, while the EAM is a fully observed one but corrupted with noise/outliers. To this end, we innovatively cast the semisupervised affinity matrix learning as the recovery of the LAM guided by the PCM and EAM, which is technically formulated as a convex optimization problem. We also provide an efficient algorithm for solving the resulting model numerically. Extensive experiments on benchmark datasets demonstrate the significant superiority of our method over state-of-the-art ones when used for constrained clustering and dimensionality reduction. The code is publicly available at https://github.com/jyh-learning/LAM.

Multi-view manifold regularized compact low-rank representation for cancer samples clustering on multi-omics data

BACKGROUND

The identification of cancer types is of great significance for early diagnosis and clinical treatment of cancer. Clustering cancer samples is an important means to identify cancer types, which has been paid much attention in the field of bioinformatics. The purpose of cancer clustering is to find expression patterns of different cancer types, so that the samples with similar expression patterns can be gathered into the same type. In order to improve the accuracy and reliability of cancer clustering, many clustering methods begin to focus on the integration analysis of cancer multi-omics data. Obviously, the methods based on multi-omics data have more advantages than those using single omics data. However, the high heterogeneity and noise of cancer multi-omics data pose a great challenge to the multi-omics analysis method.

RESULTS

In this study, in order to extract more complementary information from cancer multi-omics data for cancer clustering, we propose a low-rank subspace clustering method called multi-view manifold regularized compact low-rank representation (MmCLRR). In MmCLRR, each omics data are regarded as a view, and it learns a consistent subspace representation by imposing a consistence constraint on the low-rank affinity matrix of each view to balance the agreement between different views. Moreover, the manifold regularization and concept factorization are introduced into our method. Relying on the concept factorization, the dictionary can be updated in the learning, which greatly improves the subspace learning ability of low-rank representation. We adopt linearized alternating direction method with adaptive penalty to solve the optimization problem of MmCLRR method.

CONCLUSIONS

Finally, we apply MmCLRR into the clustering of cancer samples based on multi-omics data, and the clustering results show that our method outperforms the existing multi-view methods.

Recursive Sample Scaling Low-Rank Representation

The low-rank representation (LRR) method has recently gained enormous popularity due to its robust approach in solving the subspace segmentation problem, particularly those concerning corrupted data. In this paper, the recursive sample scaling low-rank representation (RSS-LRR) method is proposed. The advantage of RSS-LRR over traditional LRR is that a cosine scaling factor is further introduced, which imposes a penalty on each sample to minimize noise and outlier influence better. Specifically, the cosine scaling factor is a similarity measure learned to extract each sample’s relationship with the low-rank representation’s principal components in the feature space. In order words, the smaller the angle between an individual data sample and the low-rank representation’s principal components, the more likely it is that the data sample is clean. Thus, the proposed method can then effectively obtain a good low-rank representation influenced mainly by clean data. Several experiments are performed with varying levels of corruption on ORL, CMU PIE, COIL20, COIL100, and LFW in order to evaluate RSS-LRR’s effectiveness over state-of-the-art low-rank methods. The experimental results show that RSS-LRR consistently performs better than the compared methods in image clustering and classification tasks.

A Robust Tensor-Based Submodule Clustering for Imaging Data Using l12 Regularization and Simultaneous Noise Recovery via Sparse and Low Rank Decomposition Approach

The massive generation of data, which includes images and videos, has made data management, analysis, information extraction difficult in recent years. To gather relevant information, this large amount of data needs to be grouped. Real-life data may be noise corrupted during data collection or transmission, and the majority of them are unlabeled, allowing for the use of robust unsupervised clustering techniques. Traditional clustering techniques, which vectorize the images, are unable to keep the geometrical structure of the images. Hence, a robust tensor-based submodule clustering method based on l12 regularization with improved clustering capability is formulated. The l12 induced tensor nuclear norm (TNN), integrated into the proposed method, offers better low rankness while retaining the self-expressiveness property of submodules. Unlike existing methods, the proposed method employs a simultaneous noise removal technique by twisting the lateral image slices of the input data tensor into frontal slices and eliminates the noise content in each image, using the principles of the sparse and low rank decomposition technique. Experiments are carried out over three datasets with varying amounts of sparse, Gaussian and salt and pepper noise. The experimental results demonstrate the superior performance of the proposed method over the existing state-of-the-art methods.

Multi‐view intrinsic low‐rank representation for robust face recognition and clustering

In the last years, subspace‐based multi‐view face recognition has attracted increasing attention and many related methods have been proposed. However, the most existing methods ignore the specific local structure of different views. This drawback can cause these methods' discriminating ability to degrade when many noisy samples exist in data. To tackle this problem, a multi‐view low‐rank representation method is proposed, which exploits both intrinsic relationships and specific local structures of different views simultaneously. It is achieved by hierarchical Bayesian methods that constrain the low‐rank representation of each view so that it matches a linear combination of an intrinsic representation matrix and a specific representation matrix to obtain common and specific characteristics of different views. The intrinsic representation matrix holds the consensus information between views, and the specific representation matrices indicate the diversity among views. Furthermore, the model injects a clustering structure into the low‐rank representation. This approach allows for adaptive adjustment of the clustering structure while pursuing the optimization of the low‐rank representation. Hence, the model can well capture both the relationship between data and the clustering structure explicitly. Extensive experiments on several datasets demonstrated the effectiveness of the proposed method compared to similar state‐of‐the‐art methods in classification and clustering.

Learning Dual Geometric Low-Rank Structure for Semisupervised Hyperspectral Image Classification

Most of the available graph-based semisupervised hyperspectral image classification methods adopt the cluster assumption to construct a Laplacian regularizer. However, they sometimes fail due to the existence of mixed pixels whose recorded spectra are a combination of several materials. In this paper, we propose a geometric low-rank Laplacian regularized semisupervised classifier, by exploring both the global spectral geometric structure and local spatial geometric structure of hyperspectral data. A new geometric regularized Laplacian low-rank representation (GLapLRR)-based graph is developed to evaluate spectral-spatial affinity of mixed pixels. By revealing the global low-rank and local spatial structure of images via GLapLRR, the constructed graph has the characteristics of spatial-spectral geometry description, robustness, and low sparsity, from which a more accurate classification of mixed pixels can be achieved. The proposed method is experimentally evaluated on three real hyperspectral datasets, and the results show that the proposed method outperforms its counterparts, when only a small number of labeled instances are available.

Survey on Multi-Output Learning

The aim of multi-output learning is to simultaneously predict multiple outputs given an input. It is an important learning problem for decision-making since making decisions in the real world often involves multiple complex factors and criteria. In recent times, an increasing number of research studies have focused on ways to predict multiple outputs at once. Such efforts have transpired in different forms according to the particular multi-output learning problem under study. Classic cases of multi-output learning include multi-label learning, multi-dimensional learning, multi-target regression, and others. From our survey of the topic, we were struck by a lack in studies that generalize the different forms of multi-output learning into a common framework. This article fills that gap with a comprehensive review and analysis of the multi-output learning paradigm. In particular, we characterize the four Vs of multi-output learning, i.e., volume, velocity, variety, and veracity, and the ways in which the four Vs both benefit and bring challenges to multi-output learning by taking inspiration from big data. We analyze the life cycle of output labeling, present the main mathematical definitions of multi-output learning, and examine the field's key challenges and corresponding solutions as found in the literature. Several model evaluation metrics and popular data repositories are also discussed. Last but not least, we highlight some emerging challenges with multi-output learning from the perspective of the four Vs as potential research directions worthy of further studies.

Low-rank discriminative regression learning for image classification

As a famous multivariable analysis technique, regression methods, such as ridge regression, are widely used for image representation and dimensionality reduction. However, the metric of ridge regression and its variants is always the Frobenius norm (F-norm), which is sensitive to outliers and noise in data. At the same time, the performance of the ridge regression and its extensions is limited by the class number of the data. To address these problems, we propose a novel regression learning method which named low-rank discriminative regression learning (LDRL) for image representation. LDRL assumes that the input data is corrupted and thus the L₁ norm can be used as a sparse constraint on the noised matrix to recover the clean data for regression, which can improve the robustness of the algorithm. Due to learn a novel project matrix that is not limited by the number of classes, LDRL is suitable for classifying the data set no matter whether there is a small or large number of classes. The performance of the proposed LDRL is evaluated on six public image databases. The experimental results prove that LDRL obtains better performance than existing regression 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.

Identifying Autism Spectrum Disorder With Multi-Site fMRI via Low-Rank Domain Adaptation

Autism spectrum disorder (ASD) is a neurodevelopmental disorder that is characterized by a wide range of symptoms. Identifying biomarkers for accurate diagnosis is crucial for early intervention of ASD. While multi-site data increase sample size and statistical power, they suffer from inter-site heterogeneity. To address this issue, we propose a multi-site adaption framework via low-rank representation decomposition (maLRR) for ASD identification based on functional MRI (fMRI). The main idea is to determine a common low-rank representation for data from the multiple sites, aiming to reduce differences in data distributions. Treating one site as a target domain and the remaining sites as source domains, data from these domains are transformed (i.e., adapted) to a common space using low-rank representation. To reduce data heterogeneity between the target and source domains, data from the source domains are linearly represented in the common space by those from the target domain. We evaluated the proposed method on both synthetic and real multi-site fMRI data for ASD identification. The results suggest that our method yields superior performance over several state-of-the-art domain adaptation methods.

Adaptive weighted locality-constrained sparse coding for glaucoma diagnosis

Glaucoma is a sight-threading disease which can lead to irreversible blindness. Currently, extracting the vertical cup-to-disc ratio (CDR) from 2D retinal fundus images is promising for automatic glaucoma diagnosis. In this paper, we present a novel sparse coding approach for glaucoma diagnosis called adaptive weighted locality-constrained sparse coding (AWLCSC). Different from the existing reconstruction-based glaucoma diagnosis approaches, the weighted matrix in AWLCSC is constructed by adaptively fusing multiple distance measurement information between the reference images and the testing image, making our approach more robust and effective to glaucoma diagnosis. In our approach, the disc image is firstly extracted and reconstructed according to the proposed AWLCSC technique. Then, with the usage of the obtained reconstruction coefficients and a series of reference disc images with known CDRs, the CDR of the testing disc image can be automated estimation for glaucoma diagnosis. The performance of the proposed AWLCSC is evaluated on two publicly available DRISHTI-GS1 and RIM-ONE r2 databases. The experimental results indicate that the proposed approach outperforms the state-of-the-art approaches. Graphical abstract The flowchart of the proposed approach for glaucoma diagnosis.

Constrained Discriminative Projection Learning for Image Classification

Projection learning is widely used in extracting discriminative features for classification. Although numerous methods have already been proposed for this goal, they barely explore the label information during projection learning and fail to obtain satisfactory performance. Besides, many existing methods can learn only a limited number of projections for feature extraction which may degrade the performance in recognition. To address these problems, we propose a novel constrained discriminative projection learning (CDPL) method for image classification. Specifically, CDPL can be formulated as a joint optimization problem over subspace learning and classification. The proposed method incorporates the low-rank constraint to learn a robust subspace which can be used as a bridge to seamlessly connect the original visual features and objective outputs. A regression function is adopted to explicitly exploit the class label information so as to enhance the discriminability of subspace. Unlike existing methods, we use two matrices to perform feature learning and regression, respectively, such that the proposed approach can obtain more projections and achieve superior performance in classification tasks. The experiments on several datasets show clearly the advantages of our method against other state-of-the-art methods.

Zero-Shot Learning via Robust Latent Representation and Manifold Regularization

Zero-shot learning (ZSL) for visual recognition aims to accurately recognize the objects of unseen classes through mapping the visual feature to an embedding space spanned by class semantic information. However, the semantic gap across visual features and their underlying semantics is still a big obstacle in ZSL. Conventional ZSL methods construct that the mapping typically focus on the original visual features that are independent of the ZSL tasks, thus degrading the prediction performance. In this paper, we propose an effective method to uncover an appropriate latent representation of data for the purpose of zero-shot classification. Specifically, we formulate a novel framework to jointly learn the latent subspace and cross-modal embedding to link visual features with their semantic representations. The proposed framework combines feature learning and semantics prediction, such that the learned data representation is more discriminative to predict the semantic vectors, hence improving the overall classification performance. To learn a robust latent subspace, we explicitly avoid the information loss by ensuring the reconstruction ability of the obtained data representation. An efficient algorithm is designed to solve the proposed optimization problem. To fully exploit the intrinsic geometric structure of data, we develop a manifold regularization strategy to refine the learned semantic representations, leading to further improvements of the classification performance. To validate the effectiveness of the proposed approach, extensive experiments are conducted on three ZSL benchmarks and encouraging results are achieved compared with the state-of-the-art ZSL methods.

Online Robust Low-Rank Tensor Modeling for Streaming Data Analysis

Tensor data (i.e., the data having multiple dimensions) are quickly growing in scale in many practical applications, which poses new challenges for data modeling and analysis approaches, such as high-order relations of large complexity, gross noise, and varying data scale. Existing low-rank data analysis methods, which are effective at analyzing matrix data, may fail in the regime of tensor data due to these challenges. A robust and scalable low-rank tensor modeling method is heavily desired. In this paper, we develop an online robust low-rank tensor modeling (ORLTM) method to address these challenges. The ORLTM method leverages the high-order correlations among all tensor modes to model an intrinsic low-rank structure of streaming tensor data online and can effectively analyze data residing in a mixture of multiple subspaces by virtue of dictionary learning. ORLTM consumes a very limited memory space that remains constant regardless of the increase of tensor data size, which facilitates processing tensor data at a large scale. More concretely, it models each mode unfolding of streaming tensor data using the bilinear formulation of tensor nuclear norms. With this reformulation, ORLTM employs a stochastic optimization algorithm to learn the tensor low-rank structure alternatively for online updating. To capture the final tensors, ORLTM uses an average pooling operation on folded tensors in all modes. We also provide the analysis regarding computational complexity, memory cost, and convergence. Moreover, we extend ORLTM to the image alignment scenario by incorporating the geometrical transformations and linearizing the constraints. Extensive empirical studies on synthetic database and three practical vision tasks, including video background subtraction, image alignment, and visual tracking, have demonstrated the superiority of the proposed method.

A Fast and Accurate Matrix Completion Method Based on QR Decomposition and L2,1 -Norm Minimization

Low-rank matrix completion aims to recover matrices with missing entries and has attracted considerable attention from machine learning researchers. Most of the existing methods, such as weighted nuclear-norm-minimization-based methods and Qatar Riyal (QR)-decomposition-based methods, cannot provide both convergence accuracy and convergence speed. To investigate a fast and accurate completion method, an iterative QR-decomposition-based method is proposed for computing an approximate singular value decomposition. This method can compute the largest singular values of a matrix by iterative QR decomposition. Then, under the framework of matrix trifactorization, a method for computing an approximate SVD based on QR decomposition (CSVD-QR)-based L_2,1 -norm minimization method (LNM-QR) is proposed for fast matrix completion. Theoretical analysis shows that this QR-decomposition-based method can obtain the same optimal solution as a nuclear norm minimization method, i.e., the L_2,1 -norm of a submatrix can converge to its nuclear norm. Consequently, an LNM-QR-based iteratively reweighted L_2,1 -norm minimization method (IRLNM-QR) is proposed to improve the accuracy of LNM-QR. Theoretical analysis shows that IRLNM-QR is as accurate as an iteratively reweighted nuclear norm minimization method, which is much more accurate than the traditional QR-decomposition-based matrix completion methods. Experimental results obtained on both synthetic and real-world visual data sets show that our methods are much faster and more accurate than the state-of-the-art methods.

Nonparametric Bayesian Correlated Group Regression With Applications to Image Classification

Sparse Bayesian learning has emerged as a powerful tool to tackle various image classification tasks. The existing sparse Bayesian models usually use independent Gaussian distribution as the prior knowledge for the noise. However, this assumption often contradicts to the practical observations in which the noise is long tail and pixels containing noise are spatially correlated. To handle the practical noise, this paper proposes to partition the noise image into several 2-D groups and adopt the long-tail distribution, i.e., the scale mixture of the matrix Gaussian distribution, to model each group to capture the intragroup correlation of the noise. Under the nonparametric Bayesian estimation, the low-rank-induced prior and the matrix Gamma distribution prior are imposed on the covariance matrix of each group, respectively, to induce two Bayesian correlated group regression (BCGR) methods. Moreover, the proposed methods are extended to the case with unknown group structure. Our BCGR method provides an effective way to automatically fit the noise distribution and integrates the long-tail attribute and structure information of the practical noise into model. Therefore, the estimated coefficients are better for reconstructing the desired data. We apply BCGR to address image classification task and utilize the learned covariance matrices to construct a grouped Mahalanobis distance to measure the reconstruction residual of each class in the design of a classifier. Experimental results demonstrate the effectiveness of our new BCGR model.

Approximate Low-Rank Projection Learning for Feature Extraction

Feature extraction plays a significant role in pattern recognition. Recently, many representation-based feature extraction methods have been proposed and achieved successes in many applications. As an excellent unsupervised feature extraction method, latent low-rank representation (LatLRR) has shown its power in extracting salient features. However, LatLRR has the following three disadvantages: 1) the dimension of features obtained using LatLRR cannot be reduced, which is not preferred in feature extraction; 2) two low-rank matrices are separately learned so that the overall optimality may not be guaranteed; and 3) LatLRR is an unsupervised method, which by far has not been extended to the supervised scenario. To this end, in this paper, we first propose to use two different matrices to approximate the low-rank projection in LatLRR so that the dimension of obtained features can be reduced, which is more flexible than original LatLRR. Then, we treat the two low-rank matrices in LatLRR as a whole in the process of learning. In this way, they can be boosted mutually so that the obtained projection can extract more discriminative features. Finally, we extend LatLRR to the supervised scenario by integrating feature extraction with the ridge regression. Thus, the process of feature extraction is closely related to the classification so that the extracted features are discriminative. Extensive experiments are conducted on different databases for unsupervised and supervised feature extraction, and very encouraging results are achieved in comparison with many state-of-the-arts methods.

Learning Low-Rank Decomposition for Pan-Sharpening With Spatial-Spectral Offsets

Finding accurate injection components is the key issue in pan-sharpening methods. In this paper, a low-rank pan-sharpening (LRP) model is developed from a new perspective of offset learning. Two offsets are defined to represent the spatial and spectral differences between low-resolution multispectral and high-resolution multispectral (HRMS) images, respectively. In order to reduce spatial and spectral distortions, spatial equalization and spectral proportion constraints are designed and cast on the offsets, to develop a spatial and spectral constrained stable low-rank decomposition algorithm via augmented Lagrange multiplier. By fine modeling and heuristic learning, our method can simultaneously reduce spatial and spectral distortions in the fused HRMS images. Moreover, our method can efficiently deal with noises and outliers in source images, for exploring low-rank and sparse characteristics of data. Extensive experiments are taken on several image data sets, and the results demonstrate the efficiency of the proposed LRP.

Subspace Clustering via Learning an Adaptive Low-Rank Graph

By using a sparse representation or low-rank representation of data, the graph-based subspace clustering has recently attracted considerable attention in computer vision, given its capability and efficiency in clustering data. However, the graph weights built using the representation coefficients are not the exact ones as the traditional definition is in a deterministic way. The two steps of representation and clustering are conducted in an independent manner, thus an overall optimal result cannot be guaranteed. Furthermore, it is unclear how the clustering performance will be affected by using this graph. For example, the graph parameters, i.e., the weights on edges, have to be artificially pre-specified while it is very difficult to choose the optimum. To this end, in this paper, a novel subspace clustering via learning an adaptive low-rank graph affinity matrix is proposed, where the affinity matrix and the representation coefficients are learned in a unified framework. As such, the pre-computed graph regularizer is effectively obviated and better performance can be achieved. Experimental results on several famous databases demonstrate that the proposed method performs better against the state-of-the-art approaches, in clustering.

Nonlinear Low-Rank Matrix Completion for Human Motion Recovery

Human motion capture data has been widely used in many areas, but it involves a complex capture process and the captured data inevitably contains missing data due to the occlusions caused by the actor's body or clothing. Motion recovery, which aims to recover the underlying complete motion sequence from its degraded observation, still remains as a challenging task due to the nonlinear structure and kinematics property embedded in motion data. Low-rank matrix completion based methods have shown promising performance in short-time-missing motion recovery problems. However, low-rank matrix completion, which is designed for linear data, lacks the theoretic guarantee when applied to the recovery of nonlinear motion data. To overcome this drawback, we propose a tailored nonlinear matrix completion model for human motion recovery. Within the model, we first learn a combined low-rank kernel via multiple kernel learning. By exploiting the learned kernel, we embed the motion data into a high dimensional Hilbert space where motion data is of desirable low-rank and we then use the low-rank matrix completion to recover motions. In addition, we add two kinematic constraints to the proposed model to preserve the kinematics property of human motion. Extensive experiment results and comparisons with five other state-of-the-art methods demonstrate the advantage of the proposed method.

Robust Latent Subspace Learning for Image Classification

This paper proposes a novel method, called robust latent subspace learning (RLSL), for image classification. We formulate an RLSL problem as a joint optimization problem over both the latent SL and classification model parameter predication, which simultaneously minimizes: 1) the regression loss between the learned data representation and objective outputs and 2) the reconstruction error between the learned data representation and original inputs. The latent subspace can be used as a bridge that is expected to seamlessly connect the origin visual features and their class labels and hence improve the overall prediction performance. RLSL combines feature learning with classification so that the learned data representation in the latent subspace is more discriminative for classification. To learn a robust latent subspace, we use a sparse item to compensate error, which helps suppress the interference of noise via weakening its response during regression. An efficient optimization algorithm is designed to solve the proposed optimization problem. To validate the effectiveness of the proposed RLSL method, we conduct experiments on diverse databases and encouraging recognition results are achieved compared with many state-of-the-arts methods.

Deep Domain Generalization With Structured Low-Rank Constraint

Domain adaptation nowadays attracts increasing interests in pattern recognition and computer vision field, since it is an appealing technique in fighting off weakly labeled or even totally unlabeled target data by leveraging knowledge from external well-learned sources. Conventional domain adaptation assumes that target data are still accessible in the training stage. However, we would always confront such cases in reality that the target data are totally blind in the training stage. This is extremely challenging since we have no prior knowledge of the target. In this paper, we develop a deep domain generalization framework with structured low-rank constraint to facilitate the unseen target domain evaluation by capturing consistent knowledge across multiple related source domains. Specifically, multiple domain-specific deep neural networks are built to capture the rich information within multiple sources. Meanwhile, a domain-invariant deep neural network is jointly designed to uncover most consistent and common knowledge across multiple sources so that we can generalize it to unseen target domains in the test stage. Moreover, structured low-rank constraint is exploited to align multiple domain-specific networks and the domain-invariant one in order to better transfer knowledge from multiple sources to boost the learning problem in unseen target domains. Extensive experiments are conducted on several cross-domain benchmarks and the experimental results show the superiority of our algorithm by comparing it with state-of-the-art domain generalization approaches.

An Efficient Representation-Based Method for Boundary Point and Outlier Detection

Detecting boundary points (including outliers) is often more interesting than detecting normal observations, since they represent valid, interesting, and potentially valuable patterns. Since data representation can uncover the intrinsic data structure, we present an efficient representation-based method for detecting such points, which are generally located around the margin of densely distributed data, such as a cluster. For each point, the negative components in its representation generally correspond to the boundary points among its affine combination of points. In the presented method, the reverse unreachability of a point is proposed to evaluate to what degree this observation is a boundary point. The reverse unreachability can be calculated by counting the number of zero and negative components in the representation. The reverse unreachability explicitly takes into account the global data structure and reveals the disconnectivity between a data point and other points. This paper reveals that the reverse unreachability of points with lower density has a higher score. Note that the score of reverse unreachability of an outlier is greater than that of a boundary point. The top- ranked points can thus be identified as outliers. The greater the value of the reverse unreachability, the more likely the point is a boundary point. Compared with related methods, our method better reflects the characteristics of the data, and simultaneously detects outliers and boundary points regardless of their distribution and the dimensionality of the space. Experimental results obtained for a number of synthetic and real-world data sets demonstrate the effectiveness and efficiency of our method.

Descriptor Learning via Supervised Manifold Regularization for Multioutput Regression

Multioutput regression has recently shown great ability to solve challenging problems in both computer vision and medical image analysis. However, due to the huge image variability and ambiguity, it is fundamentally challenging to handle the highly complex input-target relationship of multioutput regression, especially with indiscriminate high-dimensional representations. In this paper, we propose a novel supervised descriptor learning (SDL) algorithm for multioutput regression, which can establish discriminative and compact feature representations to improve the multivariate estimation performance. The SDL is formulated as generalized low-rank approximations of matrices with a supervised manifold regularization. The SDL is able to simultaneously extract discriminative features closely related to multivariate targets and remove irrelevant and redundant information by transforming raw features into a new low-dimensional space aligned to targets. The achieved discriminative while compact descriptor largely reduces the variability and ambiguity for multioutput regression, which enables more accurate and efficient multivariate estimation. We conduct extensive evaluation of the proposed SDL on both synthetic data and real-world multioutput regression tasks for both computer vision and medical image analysis. Experimental results have shown that the proposed SDL can achieve high multivariate estimation accuracy on all tasks and largely outperforms the algorithms in the state of the arts. Our method establishes a novel SDL framework for multioutput regression, which can be widely used to boost the performance in different applications.

Block-Diagonal Constrained Low-Rank and Sparse Graph for Discriminant Analysis of Image Data

Recently, low-rank and sparse model-based dimensionality reduction (DR) methods have aroused lots of interest. In this paper, we propose an effective supervised DR technique named block-diagonal constrained low-rank and sparse-based embedding (BLSE). BLSE has two steps, i.e., block-diagonal constrained low-rank and sparse representation (BLSR) and block-diagonal constrained low-rank and sparse graph embedding (BLSGE). Firstly, the BLSR model is developed to reveal the intrinsic intra-class and inter-class adjacent relationships as well as the local neighborhood relations and global structure of data. Particularly, there are mainly three items considered in BLSR. First, a sparse constraint is required to discover the local data structure. Second, a low-rank criterion is incorporated to capture the global structure in data. Third, a block-diagonal regularization is imposed on the representation to promote discrimination between different classes. Based on BLSR, informative and discriminative intra-class and inter-class graphs are constructed. With the graphs, BLSGE seeks a low-dimensional embedding subspace by simultaneously minimizing the intra-class scatter and maximizing the inter-class scatter. Experiments on public benchmark face and object image datasets demonstrate the effectiveness of the proposed approach.

Low-Rank Embedding for Robust Image Feature Extraction

Robustness to noises, outliers, and corruptions is an important issue in linear dimensionality reduction. Since the sample-specific corruptions and outliers exist, the class-special structure or the local geometric structure is destroyed, and thus, many existing methods, including the popular manifold learning- based linear dimensionality methods, fail to achieve good performance in recognition tasks. In this paper, we focus on the unsupervised robust linear dimensionality reduction on corrupted data by introducing the robust low-rank representation (LRR). Thus, a robust linear dimensionality reduction technique termed low-rank embedding (LRE) is proposed in this paper, which provides a robust image representation to uncover the potential relationship among the images to reduce the negative influence from the occlusion and corruption so as to enhance the algorithm's robustness in image feature extraction. LRE searches the optimal LRR and optimal subspace simultaneously. The model of LRE can be solved by alternatively iterating the argument Lagrangian multiplier method and the eigendecomposition. The theoretical analysis, including convergence analysis and computational complexity, of the algorithms is presented. Experiments on some well-known databases with different corruptions show that LRE is superior to the previous methods of feature extraction, and therefore, it indicates the robustness of the proposed method. The code of this paper can be downloaded from http://www.scholat.com/laizhihui.

Multi-stage classification method oriented to aerial image based on low-rank recovery and multi-feature fusion sparse representation

Automatic classification of terrain surfaces from an aerial image is essential for an autonomous unmanned aerial vehicle (UAV) landing at an unprepared site by using vision. Diverse terrain surfaces may show similar spectral properties due to the illumination and noise that easily cause poor classification performance. To address this issue, a multi-stage classification algorithm based on low-rank recovery and multi-feature fusion sparse representation is proposed. First, color moments and Gabor texture feature are extracted from training data and stacked as column vectors of a dictionary. Then we perform low-rank matrix recovery for the dictionary by using augmented Lagrange multipliers and construct a multi-stage terrain classifier. Experimental results on an aerial map database that we prepared verify the classification accuracy and robustness of the proposed method.