Subspace Representation Research Articles

Combination of spatially enhanced bag-of-visual-words model and genuine difference subspace for fake coin detection

Fake coins are harmful for society, the detection of which is of paramount importance. Due to the large quantities of fake coins in the real world, it is impossible to examine them manually. To address this issue, we present an intelligent system to automatically detect fake coins based on their images. The intelligent system consists of two components: coin image representation and classifier learning. To represent the coin image, a new spatially enhanced bag-of-visual-words model, called SEBOVW model, is proposed. Afterwards, we improve the representation by building a genuine difference subspace. The coin is finally represented based on its projection onto this subspace. In order to discriminate between genuine and fake coins, we train a classifier using the subspace representations. A thorough evaluation of the proposed intelligent system has been conducted on four coin datasets, consisting of thousands of coins of different denominations and from two countries. Promising experimental results in excess of 98% accuracy demonstrate its effectiveness and validity.

Hypergraph-Regularized Low-Rank Subspace Clustering Using Superpixels for Unsupervised Spatial–Spectral Hyperspectral Classification

Low-rank subspace representations have been observed to be well-suited to hyperspectral imagery, which tends to have a global structure composed of a small number of ground-cover signatures, and additional graph-based regularization can further incorporate local information. However, in the context of unsupervised classification, existing approaches typically limit consideration to simple graphs built on spectral information alone. In contrast, a hypergraph-based low-rank subspace clustering is proposed to capture a more complex manifold structure. In addition, basing the hypergraph on a superpixel segmentation of the image exploits structure that is meaningful both spatially as well as spectrally. The experimental results reveal performance for the proposed superpixel-hypergraph approach superior to that of competing techniques representative of several prominent classes of unsupervised classification for hyperspectral imagery.

Double-Factor-Regularized Low-Rank Tensor Factorization for Mixed Noise Removal in Hyperspectral Image

As a preprocessing step, hyperspectral image (HSI) restoration plays a critical role in many subsequent applications. Recently, based on the framework of subspace representation and low-rank matrix/tensor factorization (LRMF/LRTF), many single-factor-regularized methods add various regularizations on the spatial factor to characterize its spatial prior knowledge. However, these methods neglect the common characteristics among different bands and the spectral continuity of HSIs. To tackle this issue, this article establishes a bridge between the factor-based regularization and the HSI priors and proposes a double-factor-regularized LRTF model for HSI mixed noise removal. The proposed model employs LRTF to characterize the spectral global low rankness, introduces a weighted group sparsity constraint on the spatial difference images (SpatDIs) of the spatial factor to promote the group sparsity in the SpatDIs of HSIs, and suggests a continuity constraint on the spectral factor to promote the spectral continuity of HSIs. Moreover, we develop a proximal alternating minimization-based algorithm to solve the proposed model. Extensive experiments conducted on the simulated and real HSIs demonstrate that the proposed method has superior performance on mixed noise removal compared with the state-of-the-art methods based on subspace representation, noise modeling, and LRMF/LRTF.

Real-time reduced-order modeling of stochastic partial differential equations via time-dependent subspaces

We present a new methodology for the real-time reduced-order modeling of stochastic partial differential equations called the dynamically/bi-orthonormal (DBO) decomposition. In this method, the stochastic fields are approximated by a low-rank decomposition to spatial and stochastic subspaces. Each of these subspaces is represented by a set of orthonormal time-dependent modes. We derive exact evolution equations of these time-dependent modes and the evolution of the factorization of the reduced covariance matrix. We show that DBO is equivalent to the dynamically orthogonal (DO) [1] and bi-orthogonal (BO) [2] decompositions via linear and invertible transformation matrices that connect DBO to DO and BO. However, DBO shows several improvements compared to DO and BO: (i) DBO performs better than DO and BO for cases with ill-conditioned covariance matrix; (ii) In contrast to BO, the issue of eigenvalue crossing is not present in the DBO formulation; (iii) In contrast to DO, the stochastic modes are orthonormal, which leads to more accurate representation of the stochastic subspace. We study the convergence properties of the method and compare it to the DO and BO methods. For demonstration, we consider three cases: (i) stochastic linear advection equation, (ii) stochastic Burgers' equation, and (iii) stochastic incompressible flow over a bump in a channel. Overall we observe improvements in the numerical accuracy of DBO compared against DO and BO.

Subspace method for solving large-scale equivalent layer and density mapping problems

The equivalent layer technique explores an intrinsic property of potential fields that any observed field at the ground surface can be reconstituted from a fictitious continuous distribution at an arbitrary flat surface that is indistinguishable from the field of the true sources. This continuous distribution can be represented by a set of discrete cells of known position and size but unknown physical property by solving a linear system, with size proportional to the number of data points measured. The density distribution at the equivalent layer carries information about the true sources because it is a scaled and downward-continued version of the field generated by the true sources at the level that the geophysical survey was undertaken. The computation of this downward-continued field is unstable, and an equivalent source evaluation is constrained by the intense computational demand required to solve the associated large linear system. A new formulation is developed to directly solve large-scale gravity equivalent layer problems using a subspace representation for the unknown density distribution. This subspace basis is constructed by applying the singular value decomposition to the matrix containing the gridded data set. A procedure to diminish (by two orders of magnitude) the number of forward model evaluations is introduced by exploring the symmetry of the gravity kernel and its evaluation on a regular mesh. The density distribution at the equivalent layer is used to outline the spatial distribution of contrasting underlying sources, to discriminate regions with predominant positive or negative density contrast, and to estimate the mass excess or deficiency for sources with positive or negative density contrast. This technique is applied to determine density models for the Carajás airborne gravity survey and to analyze density distributions associated with banded iron formations and structures of the Carajás Mineral Province, Brazil, providing mass estimates for specific geologic unities.

Image Set Classification Using a Distance-Based Kernel Over Affine Grassmann Manifold.

Modeling image sets or videos as linear subspaces is quite popular for classification problems in machine learning. However, affine subspace modeling has not been explored much. In this article, we address the image sets classification problem by modeling them as affine subspaces. Affine subspaces are linear subspaces shifted from origin by an offset. The collection of the same dimensional affine subspaces of [Formula: see text] is known as affine Grassmann manifold (AGM) or affine Grassmannian that is a smooth and noncompact manifold. The non-Euclidean geometry of AGM and the nonunique representation of an affine subspace in AGM make the classification task in AGM difficult. In this article, we propose a novel affine subspace-based kernel that maps the points in AGM to a finite-dimensional Hilbert space. For this, we embed the AGM in a higher dimensional Grassmann manifold (GM) by embedding the offset vector in the Stiefel coordinates. The projection distance between two points in AGM is the measure of similarity obtained by the kernel function. The obtained kernel-gram matrix is further diagonalized to generate low-dimensional features in the Euclidean space corresponding to the points in AGM. Distance-preserving constraint along with sparsity constraint is used for minimum residual error classification by keeping the locally Euclidean structure of AGM in mind. Experimentation performed over four data sets for gait, object, hand, and body gesture recognition shows promising results compared with state-of-the-art techniques.

ECG biometric verification system: An i-vector to overcome variability factors

Dealing with signal and session variability is a common problem in biometric recognition system since biometric signal is frequently inconsistent over time. Health, aging, emotional conditions and different recording settings are some of the factors that contribute to the variability issue. This cause the two samples of the same subject tends to be different from each other hence giving a mismatch effect between the enrolments and testing condition. Over the years, solving the variability problem by subspace representation concept has become prevalent. Hence, it motivates us to validate a recognition algorithm based on factor analysis perspective and we use electrocardiogram (ECG) signal for our experimental data as it is subject to change over time and sensitive to different sensors. We first model each supervectors extracted from Gaussian Mixture Model (GMM) into two different factors which are subject and session independent supervectors based on Joint Factor Analysis (JFA) algorithm. For the second model which is based on i-vector approach, the supervectors extracted from GMM is first modelled to be a single total factor and a compensation method is then employed to compensate the variability effect. Three compensation methods for the i-vector are employed which are Probabilistic Linear Discriminate Analysis (PLDA), Linear Discriminate Analysis (LDA) and Within Class Covariance Normalization (WCCN). The ECG-ID database obtained from physionet database consists of 90 subjects with a total of 310 ECG recordings; each recorded for 20 seconds are used in this study. Experimental results reveal the robustness of the i-vectors PLDA approach by giving 2.156% and 2.155% of Equal Error Rate (EER) for protocol 1 and 2, respectively.

Latent Low-Rank Sparse Multi-view Subspace Clustering

To solve the problem of multi-view clustering,a latent low-rank sparse multi-view subspace clustering(LLSMSC)algorithm is proposed.A latent space shared by all views is constructed to explore the complementary information of multi-view data.The global and local structure of multi-view data can be captured to attain promising clustering results by imposing low-rank constraint and sparse constraint on the implicit latent subspace representation simultaneously.An algorithm based on augmented Lagrangian multiplier with alternating direction minimization strategy is employed to solve the optimization problem.Experiments on six benchmark datasets verify the effectiveness and superiority of LLSMSC.

Structured block diagonal representation for subspace clustering

The aim of the subspace clustering is to segment the high-dimensional data into the corresponding subspace. The structured sparse subspace clustering and the block diagonal representation clustering are quite advanced spectral-type subspace clustering algorithms when handling to the linear subspaces. In this paper, the respective advantages of these two algorithms are fully exploited, and the structured block diagonal representation (SBDR) subspace clustering is proposed. In many classical spectral-type subspace clustering algorithms, the affinity matrix which obeys the block diagonal property can not necessarily bring satisfying clustering results. However, the k-block diagonal regularizer of the SBDR algorithm directly pursues the block diagonal matrix, and this regularizer is obviously more effective. On the other hand, the general procedure of the spectral-type subspace clustering algorithm is to get the affinity matrix firstly and next perform the spectral clustering. The SBDR algorithm considers the intrinsic relationship of the two seemingly separate steps, the subspace structure matrix obtained by the spectral clustering is used iteratively to facilitate a better initialization for the representation matrix. The experimental results on the synthetic dataset and the real dataset have demonstrated the superior performance of the proposed algorithm over other prevalent subspace clustering algorithms.

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

In this paper, we propose a novel hyper-Laplacian regularized multiview subspace clustering with low-rank tensor constraint method, which is referred as HLR-MSCLRT. In the HLR-MSCLRT model, the subspace representation matrices of different views are stacked as a tensor, and then the high order correlations among data can be captured. To reduce the redundancy information of the learned subspace representations, a low-rank constraint is adopted to the constructed tensor. Since data in the real world often reside in multiple nonlinear subspaces, the HLR-MSCLRT model utilizes the hyper-Laplacian graph regularization to preserve the local geometry structure embedded in a high-dimensional ambient space. An efficient algorithm is also presented to solve the optimization problem of the HLR-MSCLRT model. The experimental results on some data sets show that the proposed HLR-MSCLRT model outperforms many state-of-the-art multi-view clustering approaches.

Tensorized Multi-view Subspace Representation Learning

Self-representation based subspace learning has shown its effectiveness in many applications. In this paper, we promote the traditional subspace representation learning by simultaneously taking advantages of multiple views and prior constraint. Accordingly, we establish a novel algorithm termed as Tensorized Multi-view Subspace Representation Learning. To exploit different views, the subspace representation matrices of different views are regarded as a low-rank tensor, which effectively models the high-order correlations of multi-view data. To incorporate prior information, a constraint matrix is devised to guide the subspace representation learning within a unified framework. The subspace representation tensor equipped with a low-rank constraint models elegantly the complementary information among different views, reduces redundancy of subspace representations, and then improves the accuracy of subsequent tasks. We formulate the model with a tensor nuclear norm minimization problem constrained with $$\ell _{2,1}$$ -norm and linear equalities. The minimization problem is efficiently solved by using an Augmented Lagrangian Alternating Direction Minimization method. Extensive experimental results on diverse multi-view datasets demonstrate the effectiveness of our algorithm.

Distributed Semi-Supervised Learning With Missing Data.

Data classification is usually challenged by the difficulty and/or high cost in collecting sufficient labeled data, and unavoidability of data missing. Besides, most of the existing algorithms belong to centralized processing, in which all of the training data must be stored and processed at a fusion center. But in many real applications, data are distributed over multiple nodes, and cannot be centralized to one node for processing due to various reasons. Considering this, in this article, we focus on the problem of distributed classification of missing data with a small proportion of labeled data samples, and develop a distributed semi-supervised missing-data classification (dS2MDC) algorithm. The proposed algorithm is a distributed joint subspace/classifier learning, that is, a latent subspace representation for missing feature imputation is learned jointly with the training of nonlinear classifiers modeled by the χ2 kernel using a semi-supervised learning strategy. Theoretical performance analysis and simulations on several datasets clearly validate the effectiveness of the proposed dS2MDC algorithm from different perspectives.

Hyperspectral image restoration by subspace representation with low‐rank constraint and spatial‐spectral total variation

Hyperspectral images (HSIs) restoration is an important preprocessing step. The spectral vectors in HSI can be separated into different classification based on the land-covers, which means the spectral space can be regarded as the union of several low-rank subspaces. Subspace low-rank representation (SLRR) is powerful in exploring the inner low-rank structure and has been applied for HSI restoration. However, the traditional SLRR only seek for the rank-minimum representation under a given dictionary, which may treat the structured sparse noise as inherent low-rank components. In addition, the SLRR framework cannot make full use of the spatial information. In this study, a framework named subspace representation with low-rank constraint and spatial-spectral total variation is proposed for HSI restoration. In which, an artificial rank constraint is introduced to control the rank of the representation result, which can improve the removal of the structured sparse noise and exploit the intrinsic structure of spectral space more effectively. Meanwhile, the spatial-spectral total variation regularisation is applied to enhance the spatial and spectral smoothness. Several experiments conducted in simulated and real HSI datasets demonstrate that the proposed method can achieve a state-of-the-art performance both in visual quality and quantitative assessments.

Study of Tangle on the Three Qubit Werner States by Using of Twirl Operation

In this paper we study the Tangle of three qubit Werner state using Twirl operation and association scheme. To do this, we introduce the invariants of Twirl operation using total spin representation. Then by using commutator property of this invariant with total spin, the general form of three qubit density matrix of Werner state is obtained. The restricted density matrix to the subspaces of the irreducible representations are diagonal4×4 and Block2×2. Then it provided that this given system has not Tangle. Finally, it is shown that common eigenvectors of this density matrix are equal to the idempotents of the group of association scheme S3.

Moreau-Enhanced Total Variation and Subspace Factorization for Hyperspectral Denoising

Hyperspectral images (HSIs) denoising aims at recovering noise-free images from noisy counterparts to improve image visualization. Recently, various prior knowledge has attracted much attention in HSI denoising, e.g., total variation (TV), low-rank, sparse representation, and so on. However, the computational cost of most existing algorithms increases exponentially with increasing spectral bands. In this paper, we fully take advantage of the global spectral correlation of HSI and design a unified framework named subspace-based Moreau-enhanced total variation and sparse factorization (SMTVSF) for multispectral image denoising. Specifically, SMTVSF decomposes an HSI image into the product of a projection matrix and abundance maps, followed by a ‘Moreau-enhanced’ TV (MTV) denoising step, i.e., a nonconvex regularizer involving the Moreau envelope mechnisam, to reconstruct all the abundance maps. Furthermore, the schemes of subspace representation penalizing the low-rank characteristic and ℓ 2 , 1 -norm modelling the structured sparse noise are embedded into our denoising framework to refine the abundance maps and projection matrix. We use the augmented Lagrange multiplier (ALM) algorithm to solve the resulting optimization problem. Extensive results under various noise levels of simulated and real hypspectral images demonstrate our superiority against other competing HSI recovery approaches in terms of quality metrics and visual effects. In addition, our method has a huge advantage in computational efficiency over many competitors, benefiting from its removal of most spectral dimensions during iterations.

Sentiment Analysis With Comparison Enhanced Deep Neural Network

Sentiment analysis is a significant task in Natural Language Processing. It refers to classification based on the emotional tendency in text by extracting text features. The existing results show that models based on RNN and CNN have good performance. In order to improve the performance of text sentiment analysis, we reformulate the classification task as a comparing problem, and propose Comparison Enhanced Bi-LSTM with Multi-Head Attention (CE-B-MHA). In fact, it is efficient to classify by comparison mechanism instead of doing complex calculation. In this model, bidirectional LSTM is used for initial feature extraction, and valuable information is extracted from different dimensions and representation subspaces by Multi-Head Attention. The comparison mechanism aims to score the feature vectors by comparing with the labeled vectors. The experimental results show that CE-B-MHA has better performance than many existing models on three sentiment analysis datasets.

Nonparametric Spectral-Spatial Anomaly Detection

Due to abundant spectral information contained in the hyperspectral images, they are suitable data for anomalous targets detection. The use of spatial features in addition to spectral ones can improve the anomaly detection performance. An anomaly detector, called nonparametric spectral-spatial detector (NSSD), is proposed in this work which utilizes the benefits of spatial features and local structures extracted by the morphological filters. The obtained spectral-spatial hypercube has high dimensionality. So, accurate estimates of the background statistics in small local windows may not be obtained. Applying conventional detectors such as Local Reed Xiaoli (RX) to the high dimensional data is not possible. To deal with this difficulty, a nonparametric distance, without any need to estimate the data statistics, is used instead of the Mahalanobis distance. According to the experimental results, the detection accuracy improvement of the proposed NSSD method compared to Global RX, Local RX, weighted RX, linear filtering based RX (LF-RX), background joint sparse representation detection (BJSRD), Kernel RX, subspace RX (SSRX) and RX and uniform target detector (RX-UTD) in average is 47.68%, 27.86%, 13.23%, 29.26%, 3.33%, 17.07%, 15.88%, and 44.25%, respectively.

Image Denoising with Adaptive Weighted Graph Filtering

Graph filtering, which is founded on the theory of graph signal processing, is proved as a useful tool for image denoising. Most graph filtering methods focus on learning an ideal lowpass filter to remove noise, where clean images are restored from noisy ones by retaining the image components in low graph frequency bands. However, this lowpass filter has limited ability to separate the low-frequency noise from clean images such that it makes the denoising procedure less effective. To address this issue, we propose an adaptive weighted graph filtering (AWGF) method to replace the design of traditional ideal lowpass filter. In detail, we reassess the existing low-rank denoising method with adaptive regularizer learning (ARLLR) from the view of graph filtering. A shrinkage approach subsequently is presented on the graph frequency domain, where the components of noisy image are adaptively decreased in each band by calculating their component significances. As a result, it makes the proposed graph filtering more explainable and suitable for denoising. Meanwhile, we demonstrate a graph filter under the constraint of subspace representation is employed in the ARLLR method. Therefore, ARLLR can be treated as a special form of graph filtering. It not only enriches the theory of graph filtering, but also builds a bridge from the low-rank methods to the graph filtering methods. In the experiments, we perform the AWGF method with a graph filter generated by the classical graph Laplacian matrix. The results show our method can achieve a comparable denoising performance with several state-of-the-art denoising methods.

A multi-feature bipartite graph ensemble for image segmentation

Image segmentation benefits from using multi-feature ensembles. In this paper, we propose a novel multi-layer bipartite graph model for more effective feature fusion. This model employs multiple graph layers, each representing a feature space. They share common vertices but have individual edge sets that are obtained from different feature spaces. The features are fused by the regularization defined on a Grassmann manifold, which compresses the graph layers into subspace representations and merges them into one. The merged graph is then fed into a graph-cut algorithm to generate the final segmentation. Experiments carried out using the Berkeley segmentation benchmark show that our model is effective in feature fusion and image segmentation, outperforming the state-of-the-art superpixel-based methods.

A Climate Classification of China through k-Nearest-Neighbor and Sparse Subspace Representation

AbstractClimate classification aims to divide a given region into distinct groups on the basis of meteorological variables, and it has important applications in fields such as agriculture and buildings. In this paper, we propose a novel spectral clustering–based method to partition 661 meteorological stations in China into several zones. First, the correlations are analyzed among five meteorological variables: daily average temperature, average relative humidity, sunshine hours, diurnal temperature range, and atmospheric pressure. The resulting weak linear correlation supports the classification under multiple views. Next, a similarity matrix/graph is constructed by combining the advantages of k-nearest-neighbor and sparse subspace representation. The blocking effect of the similarity matrix confirms the rationality of the classification. Then, we consider respectively the climate classification under a single view and multiple views. For the single view, atmospheric pressure has the highest imbalance degree, and sunshine hours and diurnal temperature range have the strongest classification consistency. The consistency rates are improved evidently in a multiple-view situation. Afterward, we propose a determining method for the number of clusters and make a sensitivity analysis on various parameters. Finally, we provide a further statistical analysis on the classification results and compare the consistency with another climate dataset. All experimental results show that the proposed classification method is feasible and effective in the climate zoning of China.