Approximate Orthogonal Sparse Embedding for Dimensionality Reduction

A Weighting Method for Feature Dimension by Semisupervised Learning With Entropy

In this article, a semisupervised weighting method for feature dimension based on entropy is proposed for classification, dimension reduction, and correlation analysis. For real-world data, different feature dimensions usually show different importance. Generally, data in the same class are supposed to be similar, so their entropy should be small; and those in different classes are supposed to be dissimilar, so their entropy should be large. According to this, we propose a way to construct the weights of feature dimensions with the whole entropy and the innerclass entropies. The weights indicate the contribution of their corresponding feature dimensions in classification. They can be used to improve the performance of classification by giving a weighted distance metric and can be applied to dimension reduction and correlation analysis as well. Some numerical experiments are given to test the proposed method by comparing it with some other representative methods. They demonstrate that the proposed method is feasible and efficient in classification, dimension reduction, and correlation analysis.

The Real Eigenpairs of Symmetric Tensors and Its Application to Independent Component Analysis

It has been proved that the determination of independent components (ICs) in the independent component analysis (ICA) can be attributed to calculating the eigenpairs of high-order statistical tensors of the data. However, previous works can only obtain approximate solutions, which may affect the accuracy of the ICs. In addition, the number of ICs would need to be set manually. Recently, an algorithm based on semidefinite programming (SDP) has been proposed, which utilizes the first-order gradient information of the Lagrangian function and can obtain all the accurate real eigenpairs. In this article, for the first time, we introduce this into the ICA field, which tends to further improve the accuracy of the ICs. Note that the number of eigenpairs of symmetric tensors is usually larger than the number of ICs, indicating that the results directly obtained by SDP are redundant. Thus, in practice, it is necessary to introduce second-order derivative information to identify local extremum solutions. Therefore, originating from the SDP method, we present a new modified version, called modified SDP (MSDP), which incorporates the concept of the projected Hessian matrix into SDP and, thus, can intellectually exclude redundant ICs and select true ICs. Some cases that have been tested in the experiments demonstrate its effectiveness. Experiments on the image/sound blind separation and real multi/hyperspectral image also show its superiority in improving the accuracy of ICs and automatically determining the number of ICs. In addition, the results on hyperspectral simulation and real data also demonstrate that MSDP is also capable of dealing with cases, where the number of features is less than the number of ICs.

Fast Unsupervised Projection for Large-Scale Data

Dimensionality reduction (DR) technique has been frequently used to alleviate information redundancy and reduce computational complexity. Traditional DR methods generally are inability to deal with nonlinear data and have high computational complexity. To cope with the problems, we propose a fast unsupervised projection (FUP) method. The simplified graph of FUP is constructed by samples and representative points, where the number of the representative points selected through iterative optimization is less than that of samples. By generating the presented graph, it is proved that large-scale data can be projected faster in numerous scenarios. Thereafter, the orthogonality FUP (OFUP) method is proposed to ensure the orthogonality of projection matrix. Specifically, the OFUP method is proved to be equivalent to PCA upon certain parameter setting. Experimental results on benchmark data sets show the effectiveness in retaining the essential information.

Generalized Embedding Regression: A Framework for Supervised Feature Extraction

Sparse discriminative projection learning has attracted much attention due to its good performance in recognition tasks. In this article, a framework called generalized embedding regression (GER) is proposed, which can simultaneously perform low-dimensional embedding and sparse projection learning in a joint objective function with a generalized orthogonal constraint. Moreover, the label information is integrated into the model to preserve the global structure of data, and a rank constraint is imposed on the regression matrix to explore the underlying correlation structure of classes. Theoretical analysis shows that GER can obtain the same or approximate solution as some related methods with special settings. By utilizing this framework as a general platform, we design a novel supervised feature extraction approach called jointly sparse embedding regression (JSER). In JSER, we construct an intrinsic graph to characterize the intraclass similarity and a penalty graph to indicate the interclass separability. Then, the penalty graph Laplacian is used as the constraint matrix in the generalized orthogonal constraint to deal with interclass marginal points. Moreover, the L_2,1 -norm is imposed on the regression terms for robustness to outliers and data's variations and the regularization term for jointly sparse projection learning, leading to interesting semantic interpretability. An effective iterative algorithm is elaborately designed to solve the optimization problem of JSER. Theoretically, we prove that the subproblem of JSER is essentially an unbalanced Procrustes problem and can be solved iteratively. The convergence of the designed algorithm is also proved. Experimental results on six well-known data sets indicate the competitive performance and latent properties of JSER.

Learning Robust Discriminant Subspace Based on Joint L₂,ₚ- and L₂,ₛ-Norm Distance Metrics

Recently, there are many works on discriminant analysis, which promote the robustness of models against outliers by using L₁- or L_2,1-norm as the distance metric. However, both of their robustness and discriminant power are limited. In this article, we present a new robust discriminant subspace (RDS) learning method for feature extraction, with an objective function formulated in a different form. To guarantee the subspace to be robust and discriminative, we measure the within-class distances based on [Formula: see text]-norm and use [Formula: see text]-norm to measure the between-class distances. This also makes our method include rotational invariance. Since the proposed model involves both [Formula: see text]-norm maximization and [Formula: see text]-norm minimization, it is very challenging to solve. To address this problem, we present an efficient nongreedy iterative algorithm. Besides, motivated by trace ratio criterion, a mechanism of automatically balancing the contributions of different terms in our objective is found. RDS is very flexible, as it can be extended to other existing feature extraction techniques. An in-depth theoretical analysis of the algorithm's convergence is presented in this article. Experiments are conducted on several typical databases for image classification, and the promising results indicate the effectiveness of RDS.

Local quasi-linear embedding based on kronecker product expansion of vectors

Locally Linear Embedding (LLE) is honored as the first algorithm of manifold learning. Generally speaking, the relation between a data and its nearest neighbors is nonlinear and LLE only extracts its linear part. Therefore, local nonlinear embedding is an important direction of improvement to LLE. However, any attempt in this direction may lead to a significant increase in computational complexity. In this paper, a novel algorithm called local quasi-linear embedding (LQLE) is proposed. In our LQLE, each high-dimensional data vector is first expanded by using Kronecker product. The expanded vector contains not only the components of the original vector, but also the polynomials of its components. Then, each expanded vector of high dimensional data is linearly approximated with the expanded vectors of its nearest neighbors. In this way, the proposed LQLE achieves a certain degree of local nonlinearity and learns the data dimensionality reduction results under the principle of keeping local nonlinearity unchanged. More importantly, LQLE does not increase computation complexity by only replacing the data vectors with their Kronecker product expansions in the original LLE program. Experimental results between our proposed methods and four comparison algorithms on various datasets demonstrate the well performance of the proposed methods.

Learning Local-Global Multiple Correlation Filters for Robust Visual Tracking with Kalman Filter Redetection

Visual object tracking is a significant technology for camera-based sensor networks applications. Multilayer convolutional features comprehensively used in correlation filter (CF)-based tracking algorithms have achieved excellent performance. However, there are tracking failures in some challenging situations because ordinary features are not able to well represent the object appearance variations and the correlation filters are updated irrationally. In this paper, we propose a local-global multiple correlation filters (LGCF) tracking algorithm for edge computing systems capturing moving targets, such as vehicles and pedestrians. First, we construct a global correlation filter model with deep convolutional features, and choose horizontal or vertical division according to the aspect ratio to build two local filters with hand-crafted features. Then, we propose a local-global collaborative strategy to exchange information between local and global correlation filters. This strategy can avoid the wrong learning of the object appearance model. Finally, we propose a time-space peak to sidelobe ratio (TSPSR) to evaluate the stability of the current CF. When the estimated results of the current CF are not reliable, the Kalman filter redetection (KFR) model would be enabled to recapture the object. The experimental results show that our presented algorithm achieves better performances on OTB-2013 and OTB-2015 compared with the other latest 12 tracking algorithms. Moreover, our algorithm handles various challenges in object tracking well.

The Portrait Depiction of the Market Members Based on Data Mining

Aiming at the problem of portrait of members in shopping malls, this paper analyzes the similarities and differences of consumption behaviors between member groups and nonmember groups, and constructs the LRFMC model with [Formula: see text]-means algorithm to analyze the value of membership. Second, active states of members are divided according to the consumption time interval, and KNN algorithm model is established to predict member states and used to predict the membership status. Finally, it discusses which types of goods are more suitable for promotional activities and can bring more profits to the shopping mall.

Adaptive Weighted Sparse Principal Component Analysis for Robust Unsupervised Feature Selection

Current unsupervised feature selection methods cannot well select the effective features from the corrupted data. To this end, we propose a robust unsupervised feature selection method under the robust principal component analysis (PCA) reconstruction criterion, which is named the adaptive weighted sparse PCA (AW-SPCA). In the proposed method, both the regularization term and the reconstruction error term are constrained by the l_2,1 -norm: the l_2,1 -norm regularization term plays a role in the feature selection, while the l_2,1 -norm reconstruction error term plays a role in the robust reconstruction. The proposed method is in a convex formulation, and the selected features by it can be used for robust reconstruction and clustering. Experimental results demonstrate that the proposed method can obtain better reconstruction and clustering performance, especially for the corrupted data.

NPSA: Nonorthogonal Principal Skewness Analysis

Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a thirdorder generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to so lve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.

The Identification and Evaluation Model for Test Paper’s Color and Substance Concentration

The colorimetric method is usually used to test the concentration of substances. However, this method has a big error since different people have different sensitivities to colors. In this paper, in order to solve the identification problem of the color and the concentration of the test paper, firstly, we found out that the concentration of substance is correlated with the color reading by using the Pearson’s Chi-squared test method. And by the concentration coefficient of Pearson correlation analysis, the concentration of substance and color reading is highly correlated. Secondly, according to the RGB value of the paper image, the color moments of the image are calculated as the characteristics of the image, and the Levenberg–Marquardt (LM) neural network is established to classify the concentration of the substance. The accuracy of the training set model is 94.5%, and the accuracy of the test set model is 87.5%. The model precision is high, and the model has stronger generalization ability. Therefore, according to the RGB value of the test paper image, it is effective to establish the LM neural network model to identify the substance concentration.

Nonpeaked Discriminant Analysis for Data Representation

Of late, there are many studies on the robust discriminant analysis, which adopt L₁-norm as the distance metric, but their results are not robust enough to gain universal acceptance. To overcome this problem, the authors of this article present a nonpeaked discriminant analysis (NPDA) technique, in which cutting L₁-norm is adopted as the distance metric. As this kind of norm can better eliminate heavy outliers in learning models, the proposed algorithm is expected to be stronger in performing feature extraction tasks for data representation than the existing robust discriminant analysis techniques, which are based on the L₁-norm distance metric. The authors also present a comprehensive analysis to show that cutting L₁-norm distance can be computed equally well, using the difference between two special convex functions. Against this background, an efficient iterative algorithm is designed for the optimization of the proposed objective. Theoretical proofs on the convergence of the algorithm are also presented. Theoretical insights and effectiveness of the proposed method are validated by experimental tests on several real data sets.

Image Object Recognition via Deep Feature-Based Adaptive Joint Sparse Representation

An image object recognition approach based on deep features and adaptive weighted joint sparse representation (D-AJSR) is proposed in this paper. D-AJSR is a data-lightweight classification framework, which can classify and recognize objects well with few training samples. In D-AJSR, the convolutional neural network (CNN) is used to extract the deep features of the training samples and test samples. Then, we use the adaptive weighted joint sparse representation to identify the objects, in which the eigenvectors are reconstructed by calculating the contribution weights of each eigenvector. Aiming at the high-dimensional problem of deep features, we use the principal component analysis (PCA) method to reduce the dimensions. Lastly, combined with the joint sparse model, the public features and private features of images are extracted from the training sample feature set so as to construct the joint feature dictionary. Based on the joint feature dictionary, sparse representation-based classifier (SRC) is used to recognize the objects. Experiments on face images and remote sensing images show that D-AJSR is superior to the traditional SRC method and some other advanced methods.

Two-Dimensional Quaternion PCA and Sparse PCA

Benefited from quaternion representation that is able to encode the cross-channel correlation of color images, quaternion principle component analysis (QPCA) was proposed to extract features from color images while reducing the feature dimension. A quaternion covariance matrix (QCM) of input samples was constructed, and its eigenvectors were derived to find the solution of QPCA. However, eigen-decomposition leads to the fixed solution for the same input. This solution is susceptible to outliers and cannot be further optimized. To solve this problem, this paper proposes a novel quaternion ridge regression (QRR) model for two-dimensional QPCA (2D-QPCA). We mathematically prove that this QRR model is equivalent to the QCM model of 2D-QPCA. The QRR model is a general framework and is flexible to combine 2D-QPCA with other technologies or constraints to adapt different requirements of real-world applications. Including sparsity constraints, we then propose a quaternion sparse regression model for 2D-QSPCA to improve its robustness for classification. An alternating minimization algorithm is developed to iteratively learn the solution of 2D-QSPCA in the equivalent complex domain. In addition, 2D-QPCA and 2D-QSPCA can preserve the spatial structure of color images and have a low computation cost. Experiments on several challenging databases demonstrate that 2D-QPCA and 2D-QSPCA are effective in color face recognition, and 2D-QSPCA outperforms the state of the arts.

Discriminative Sparsity Graph Embedding for Unconstrained Face Recognition

In this paper, we propose a new dimensionality reduction method named Discriminative Sparsity Graph Embedding (DSGE) which considers the local structure information and the global distribution information simultaneously. Firstly, we adopt the intra-class compactness constraint to automatically construct the intrinsic adjacent graph, which enhances the reconstruction relationship between the given sample and the non-neighbor samples with the same class. Meanwhile, the inter-class compactness constraint is exploited to construct the penalty adjacent graph, which reduces the reconstruction influence between the given sample and the pseudo-neighbor samples with the different classes. Then, the global distribution constraints are introduced to the projection objective function for seeking the optimal subspace which compacts intra-classes samples and alienates inter-classes samples at the same time. Extensive experiments are carried out on AR, Extended Yale B, LFW and PubFig databases which are four representative face datasets, and the corresponding experimental results illustrate the effectiveness of our proposed method.

Flexible Affinity Matrix Learning for Unsupervised and Semisupervised Classification

In this paper, we propose a unified model called flexible affinity matrix learning (FAML) for unsupervised and semisupervised classification by exploiting both the relationship among data and the clustering structure simultaneously. To capture the relationship among data, we exploit the self-expressiveness property of data to learn a structured matrix in which the structures are induced by different norms. A rank constraint is imposed on the Laplacian matrix of the desired affinity matrix, so that the connected components of data are exactly equal to the cluster number. Thus, the clustering structure is explicit in the learned affinity matrix. By making the estimated affinity matrix approximate the structured matrix during the learning procedure, FAML allows the affinity matrix itself to be adaptively adjusted such that the learned affinity matrix can well capture both the relationship among data and the clustering structure. Thus, FAML has the potential to perform better than other related methods. We derive optimization algorithms to solve the corresponding problems. Extensive unsupervised and semisupervised classification experiments on both synthetic data and real-world benchmark data sets show that the proposed FAML consistently outperforms the state-of-the-art methods.

DBSDA : Lowering the Bound of Misclassification Rate for Sparse Linear Discriminant Analysis via Model Debiasing

Linear discriminant analysis (LDA) is a well-known technique for linear classification, feature extraction, and dimension reduction. To improve the accuracy of LDA under the high dimension low sample size (HDLSS) settings, shrunken estimators, such as Graphical Lasso, can be used to strike a balance between biases and variances. Although the estimator with induced sparsity obtains a faster convergence rate, however, the introduced bias may also degrade the performance. In this paper, we theoretically analyze how the sparsity and the convergence rate of the precision matrix (also known as inverse covariance matrix) estimator would affect the classification accuracy by proposing an analytic model on the upper bound of an LDA misclassification rate. Guided by the model, we propose a novel classifier, DBSDA , which improves classification accuracy through debiasing. Theoretical analysis shows that DBSDA possesses a reduced upper bound of misclassification rate and better asymptotic properties than sparse LDA (SDA). We conduct experiments on both synthetic datasets and real application datasets to confirm the correctness of our theoretical analysis and demonstrate the superiority of DBSDA over LDA, SDA, and other downstream competitors under HDLSS settings.

Robust dimensionality reduction via feature space to feature space distance metric learning

Images are often represented as vectors with high dimensions when involved in classification. As a result, dimensionality reduction methods have to be developed to avoid the curse of dimensionality. Among them, Laplacian eigenmaps (LE) have attracted widespread concentrations. In the original LE, point to point (P2P) distance metric is often adopted for manifold learning. Unfortunately, they show few impacts on robustness to noises. In this paper, a novel supervised dimensionality reduction method, named feature space to feature space distance metric learning (FSDML), is presented. For any point, it can construct a feature space spanned by its k intra-class nearest neighbors, which results in a local projection on its nearest feature space. Thus feature space to feature space (S2S) distance metric will be defined to Euclidean distance between two corresponding projections. On one hand, the proposed S2S distance metric displays superiority on robustness by the local projection. On the other hand, the projection on the nearest feature space contributes to fully mining local geometry information hidden in the original data. Moreover, both class label similarity and dissimilarity are also measured, based on which an intra-class graph and an inter-class graph will be individually modeled. Finally, a subspace can be found for classification by maximizing S2S based manifold to manifold distance and preserving S2S based locality of manifolds, simultaneously. Compared to some state-of-art dimensionality reduction methods, experiments validate the proposed method's performance either on synthesized data sets or on benchmark data sets.

Neighborhood preserving neural network for fault detection

A novel statistical feature extraction method, called the neighborhood preserving neural network (NPNN), is proposed in this paper. NPNN can be viewed as a nonlinear data-driven fault detection technique through preserving the local geometrical structure of normal process data. The "local geometrical structure " means that each sample can be constructed as a linear combination of its neighbors. NPNN is characterized by adaptively training a nonlinear neural network which takes the local geometrical structure of the data into consideration. Moreover, in order to extract uncorrelated and faithful features, NPNN adopts orthogonal constraints in the objective function. Through backpropagation and eigen decomposition (ED) technique, NPNN is optimized to extract low-dimensional features from original high-dimensional process data. After nonlinear feature extraction, Hotelling T² statistic and the squared prediction error (SPE) statistic are utilized for the fault detection tasks. The advantages of the proposed NPNN method are demonstrated by both theoretical analysis and case studies on the Tennessee Eastman (TE) benchmark process. Extensive experimental results show the superiority of NPNN in terms of missed detection rate (MDR) and false alarm rate (FAR). The source code of NPNN can be found in https://github.com/htzhaoecust/npnn.

Low-Rank Sparse Preserving Projections for Dimensionality Reduction

Learning an efficient projection to map high-dimensional data into a lower dimensional space is a rather challenging task in the community of pattern recognition and computer vision. Manifold learning is widely applied because it can disclose the intrinsic geometric structure of data. However, it only concerns the geometric structure and may lose its effectiveness in case of corrupted data. To address this challenge, we propose a novel dimensionality reduction method by combining the manifold learning and low-rank sparse representation, termed low-rank sparse preserving projections (LSPP), which can simultaneously preserve the intrinsic geometric structure and learn a robust representation to reduce the negative effects of corruptions. Therefore, LSPP is advantageous to extract robust features. Because the formulated LSPP problem has no closed-form solution, we use the linearized alternating direction method with adaptive penalty and eigen-decomposition to obtain the optimal projection. The convergence of LSPP is proven, and we also analyze its complexity. To validate the effectiveness and robustness of LSPP in feature extraction and dimensionality reduction, we make a critical comparison between LSPP and a series of related dimensionality reduction methods. The experimental results demonstrate the effectiveness of LSPP.

End-to-End Feature-Aware Label Space Encoding for Multilabel Classification With Many Classes

To make the problem of multilabel classification with many classes more tractable, in recent years, academia has seen efforts devoted to performing label space dimension reduction (LSDR). Specifically, LSDR encodes high-dimensional label vectors into low-dimensional code vectors lying in a latent space, so as to train predictive models at much lower costs. With respect to the prediction, it performs classification for any unseen instance by recovering a label vector from its predicted code vector via a decoding process. In this paper, we propose a novel method, namely End-to-End Feature-aware label space Encoding (E²FE), to perform LSDR. Instead of requiring an encoding function like most previous works, E²FE directly learns a code matrix formed by code vectors of the training instances in an end-to-end manner. Another distinct property of E²FE is its feature awareness attributable to the fact that the code matrix is learned by jointly maximizing the recoverability of the label space and the predictability of the latent space. Based on the learned code matrix, E²FE further trains predictive models to map instance features into code vectors, and also learns a linear decoding matrix for efficiently recovering the label vector of any unseen instance from its predicted code vector. Theoretical analyses show that both the code matrix and the linear decoding matrix in E²FE can be efficiently learned. Moreover, similar to previous works, E²FE can be specified to learn an encoding function. And it can also be extended with kernel tricks to handle nonlinear correlations between the feature space and the latent space. Comprehensive experiments conducted on diverse benchmark data sets with many classes show consistent performance gains of E²FE over the state-of-the-art methods.

Regularized Label Relaxation Linear Regression

Linear regression (LR) and some of its variants have been widely used for classification problems. Most of these methods assume that during the learning phase, the training samples can be exactly transformed into a strict binary label matrix, which has too little freedom to fit the labels adequately. To address this problem, in this paper, we propose a novel regularized label relaxation LR method, which has the following notable characteristics. First, the proposed method relaxes the strict binary label matrix into a slack variable matrix by introducing a nonnegative label relaxation matrix into LR, which provides more freedom to fit the labels and simultaneously enlarges the margins between different classes as much as possible. Second, the proposed method constructs the class compactness graph based on manifold learning and uses it as the regularization item to avoid the problem of overfitting. The class compactness graph is used to ensure that the samples sharing the same labels can be kept close after they are transformed. Two different algorithms, which are, respectively, based on -norm and -norm loss functions are devised. These two algorithms have compact closed-form solutions in each iteration so that they are easily implemented. Extensive experiments show that these two algorithms outperform the state-of-the-art algorithms in terms of the classification accuracy and running time.

Reweighted Low-Rank Matrix Analysis With Structural Smoothness for Image Denoising

In this paper, we develop a new low-rank matrix recovery algorithm for image denoising. We incorporate the total variation (TV) norm and the pixel range constraint into the existing reweighted low-rank matrix analysis to achieve structural smoothness and to significantly improve quality in the recovered image. Our proposed mathematical formulation of the low-rank matrix recovery problem combines the nuclear norm, TV norm, and norm, thereby allowing us to exploit the low-rank property of natural images, enhance the structural smoothness, and detect and remove large sparse noise. Using the iterative alternating direction and fast gradient projection methods, we develop an algorithm to solve the proposed challenging non-convex optimization problem. We conduct extensive performance evaluations on single-image denoising, hyper-spectral image denoising, and video background modeling from corrupted images. Our experimental results demonstrate that the proposed method outperforms the state-of-the-art low-rank matrix recovery methods, particularly for large random noise. For example, when the density of random sparse noise is 30%, for single-image denoising, our proposed method is able to improve the quality of the restored image by up to 4.21 dB over existing methods.

Feature Genes Selection Using Supervised Locally Linear Embedding and Correlation Coefficient for Microarray Classification

The selection of feature genes with high recognition ability from the gene expression profiles has gained great significance in biology. However, most of the existing methods have a high time complexity and poor classification performance. Motivated by this, an effective feature selection method, called supervised locally linear embedding and Spearman's rank correlation coefficient (SLLE-SC²), is proposed which is based on the concept of locally linear embedding and correlation coefficient algorithms. Supervised locally linear embedding takes into account class label information and improves the classification performance. Furthermore, Spearman's rank correlation coefficient is used to remove the coexpression genes. The experiment results obtained on four public tumor microarray datasets illustrate that our method is valid and feasible.