Stable local dimensionality reduction approaches

Dimensionality reduction is a big challenge in many areas. A large number of local approaches, stemming from statistics or geometry, have been developed. However, in practice these local approaches are often in lack of robustness, since in contrast to maximum variance unfolding (MVU), which explicitly unfolds the manifold, they merely characterize local geometry structure. Moreover, the eigenproblems that they encounter, are hard to solve. We propose a unified framework that explicitly unfolds the manifold and reformulate local approaches as the semi-definite programs instead of the above-mentioned eigenproblems. Three well-known algorithms, locally linear embedding (LLE), laplacian eigenmaps (LE) and local tangent space alignment (LTSA) are reinterpreted and improved within this framework. Several experiments are presented to demonstrate the potential of our framework and the improvements of these local algorithms.     

Semi-supervised local Fisher discriminant analysis for dimensionality reduction

When only a small number of labeled samples are available, supervised dimensionality reduction methods tend to perform poorly because of overfitting. In such cases, unlabeled samples could be useful in improving the performance. In this paper, we propose a semi-supervised dimensionality reduction method which preserves the global structure of unlabeled samples in addition to separating labeled samples in different classes from each other. The proposed method, which we call SEmi-supervised Local Fisher discriminant analysis (SELF), has an analytic form of the globally optimal solution and it can be computed based on eigen-decomposition. We show the usefulness of SELF through experiments with benchmark and real-world document classification datasets.     

Trace ratio criterion based generalized discriminative learning for semi-supervised dimensionality reduction

Dealing with high-dimensional data has always been a major problem in many pattern recognition and machine learning applications. Trace ratio criterion is a criterion that can be applicable to many dimensionality reduction methods as it directly reflects Euclidean distance between data points of within or between classes. In this paper, we analyze the trace ratio problem and propose a new efficient algorithm to find the optimal solution. Based on the proposed algorithm, we are able to derive an orthogonal constrained semi-supervised learning framework. The new algorithm incorporates unlabeled data into training procedure so that it is able to preserve the discriminative structure as well as geometrical structure embedded in the original dataset. Under such a framework, many existing semi-supervised dimensionality reduction methods such as SDA, Lap-LDA, SSDR, SSMMC, can be improved using our proposed framework, which can also be used to formulate a corresponding kernel framework for handling nonlinear problems. Theoretical analysis indicates that there are certain relationships between linear and nonlinear methods. Finally, extensive simulations on synthetic dataset and real world dataset are presented to show the effectiveness of our algorithms. The results demonstrate that our proposed algorithm can achieve great superiority to other state-of-art algorithms.     

基于最大间隔准则的鲁棒多流形判别局部图嵌入算法

针对现有的多流形人脸识别算法大多直接使用带有噪声的原始数据进行处理，而带有噪声的数据往往会对算法的准确率产生负面影响的问题，提出了一种基于最大间距准则的鲁棒多流形判别局部图嵌入算法（RMMDLGE/MMC）。首先，通过引入一个降噪投影对原始数据进行迭代降噪处理，提取出更加纯净的数据；其次，对数据图像进行分块，建立多流形模型；再次，结合最大间隔准则的思想，寻求最优的投影矩阵使得不同流形上的样本距离尽可能大，同时相同流形上的样本距离尽可能小；最后，计算待识样本流形到训练样本流形的距离进行分类识别。实验结果表明，与表现较好的最大间距准则框架下的多流形局部图嵌入算法（MLGE/MMC）相比，所提算法在添加噪声的ORL、Yale和FERET库上的分类识别率分别提高了1.04、1.28和2.13个百分点，分类效果明显提高。     

Orthogonal margin discriminant projection for dimensionality reduction

Dimensionality reduction aims to represent high-dimensional data with much smaller number of features, which plays as a preprocessing step to remove the insignificant and irrelevant features in many machine learning applications, resulting in lower computational cost and better performance of classifiers. In most cases, the data points can be well classified with margin samples which are defined as furthest intra-class samples and nearest inter-class samples. Motivated by this observation, this paper proposes a linear supervised dimensionality reduction method called orthogonal margin discriminant projection (OMDP). After OMDP projection, intra-class data points become more compact and inter-class data points become more separated. Extensive experiments have been conducted to evaluate the proposed OMDP algorithm using several benchmark face data sets. The experimental results confirm the effectiveness of the proposed method.     

Trace ratio criterion for multi-view discriminant analysis

Applied Intelligence - Learning from heterogeneous views, termed multi-view learning (MvL), is a significant yet challenging problem in computer vision. Many existing MvL methods apply the two-view...     

Feature extraction using orthogonal discriminant local tangent space alignment

A novel algorithm called orthogonal discriminant local tangent space alignment (O-DLTSA) is proposed for supervised feature extraction. Derived from local tangent space alignment (LTSA), O-DLTSA not only inherits the advantages of LTSA which uses local tangent space as a representation of the local geometry so as to preserve the local structure, but also makes full use of class information and orthogonal subspace to improve discriminant power. The experimental results of applying O-DLTSA to standard face databases demonstrate the effectiveness of the proposed method.     

Local uncorrelated local discriminant embedding for face recognition

The feature extraction algorithm plays an important role in face recognition. However, the extracted features also have overlapping discriminant information. A property of the statistical uncorrelated criterion is that it eliminates the redundancy among the extracted discriminant features, while many algorithms generally ignore this property. In this paper, we introduce a novel feature extraction method called local uncorrelated local discriminant embedding (LULDE). The proposed approach can be seen as an extension of a local discriminant embedding (LDE) framework in three ways. First, a new local statistical uncorrelated criterion is proposed, which effectively captures the local information of interclass and intraclass. Second, we reconstruct the affinity matrices of an intrinsic graph and a penalty graph, which are mentioned in LDE to enhance the discriminant property. Finally, it overcomes the small-sample-size problem without using principal component analysis to preprocess the original data, which avoids losing some discriminant information. Experimental results on Yale, ORL, Extended Yale B, and FERET databases demonstrate that LULDE outperforms LDE and other representative uncorrelated feature extraction methods.     

Principal components selection for dimensionality reduction using discriminant information applied to fault diagnosis

The Principal Component Analysis is one of most applied dimensionality reduction techniques for process monitoring and fault diagnosis in industrial process. This work proposes a procedure based on the discriminant information contained in the principal components to determine the most significant ones in fault separability. The Tennessee Eastman Process industrial benchmark is used to illustrate the effectiveness of the proposal. The use of statistical hypothesis tests as a separability measure between multiple failures is proposed for the selection of the principal components. The classifier profile concept has been introduced for comparison purposes. Results show an improvement in the classification process when compared with traditional techniques and the StepWise selection. This has resulted in a better classification for a fixed number of components, or a smaller number of required components to obtain a prefixed error rate. In addition, the computational advantage is demonstrated.     

Extracting the optimal dimensionality for local tensor discriminant analysis

Supervised dimensionality reduction with tensor representation has attracted great interest in recent years. It has been successfully applied to problems with tensor data, such as image and video recognition tasks. However, in the tensor-based methods, how to select the suitable dimensions is a very important problem. Since the number of possible dimension combinations exponentially increases with respect to the order of tensor, manually selecting the suitable dimensions becomes an impossible task in the case of high-order tensor. In this paper, we aim at solving this important problem and propose an algorithm to extract the optimal dimensionality for local tensor discriminant analysis. Experimental results on a toy example and real-world data validate the effectiveness of the proposed method.     

Random walk-based fuzzy linear discriminant analysis for dimensionality reduction

Dealing with high-dimensional data has always been a major problem with the research of pattern recognition and machine learning, and linear discriminant analysis (LDA) is one of the most popular methods for dimensionality reduction. However, it suffers from the problem of being too sensitive to outliers. Hence to solve this problem, fuzzy membership can be introduced to enhance the performance of algorithms by reducing the effects of outliers. In this paper, we analyze the existing fuzzy strategies and propose a new effective one based on Markov random walks. The new fuzzy strategy can maintain high consistency of local and global discriminative information and preserve statistical properties of dataset. In addition, based on the proposed fuzzy strategy, we then derive an efficient fuzzy LDA algorithm by incorporating the fuzzy membership into learning. Theoretical analysis and extensive simulations show the effectiveness of our algorithm. The presented results demonstrate that our proposed algorithm can achieve significantly improved results compared with other existing algorithms.     

Unsupervised double weight graphs based discriminant analysis for dimensionality reduction

ABSTRACT

Dimensionality reduction plays an important role in pattern recognition tasks. Locality preserving projection and neighbourhood preserving embedding are popular unsupervised feature extraction methods, which try to preserve a certain local structure in the low-dimensional subspace. However, only considering the local neighbour information will limit the methods to achieve higher recognition accuracy. In this paper, an unsupervised double weight graphs based discriminant analysis method (uDWG-DA) is proposed. First, uDWG-DA considers both similar and dissimilar relationships among samples by using double weight graphs. In order to explore the dissimilar information, a new partitioning strategy is proposed to divide the data set into different clusters, where samples of different clusters are dissimilar. Then, based on L_2,1 norm, uDWG-DA finds the optimal projection to not only preserve the similar local structure but also increase the separability among different clusters of the data set. Experiments on four hyperspectral images validate the advantage and feasibility of the proposed method compared with other dimensionality reduction methods.     

Graph embedding and extensions: a general framework for dimensionality reduction

A large family of algorithms - supervised or unsupervised; stemming from statistics or geometry theory - has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework. In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided. Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms. By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called marginal Fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability. We show that MFA effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions. Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions     

A new quality assessment criterion for nonlinear dimensionality reduction

A new quality assessment criterion for evaluating the performance of the nonlinear dimensionality reduction (NLDR) methods is proposed in this paper. Differing from the current quality assessment criteria focusing on the local-neighborhood-preserving performance of the NLDR methods, the proposed criterion capitalizes on a new aspect, the global-structure-holding performance, of the NLDR methods. By taking both properties into consideration, the intrinsic capability of the NLDR methods can be more faithfully reflected, and hence more rational measurement for the proper selection of NLDR methods in real-life applications can be offered. The theoretical argument is supported by experiment results implemented on a series of benchmark data sets.     

完备正交邻域保持判别嵌入的人脸识别

为解决邻域保持判别嵌入算法所面临的小样本问题,并充分利用类内邻域散度矩阵零空间和非零空间中的判别信息进行人脸识别,提出一种完备正交邻域保持判别嵌入的人脸识别算法。首先间接地利用特征分解方法去除总体邻域散度矩阵的零空间;然后分别在类内邻域散度矩阵零空间和非零空间中提取最优判别矢量。此外,为进一步提高算法的识别性能,给出了基于瘦QR分解的正交投影矩阵的求解方法。在ORL和Yale人脸库上验证了以上算法的有效性。     

Generalized null space uncorrelated Fisher discriminant analysis for linear dimensionality reduction

We propose a generalized null space uncorrelated Fisher discriminant analysis (GNUFDA) technique integrating the uncorrelated discriminant analysis and weighted pairwise Fisher criterion. The GNUFDA can effectively deal with the small sample-size problem and perform satisfactorily when the dimensionality of the null space decreases with increase in the number of training samples per class and/or classes, C. The proposed GNUFDA can extract at most C-1 optimal uncorrelated discriminative vectors without being influenced by the null-space dimensionality.     

Uncorrelated trace ratio linear discriminant analysis for undersampled problems

For linear discriminant analysis (LDA), the ratio trace and trace ratio are two basic criteria generalized from the classical Fisher criterion function, while the orthogonal and uncorrelated constraints are two common conditions imposed on the optimal linear transformation. The ratio trace criterion with both the orthogonal and uncorrelated constraints have been extensively studied in the literature, whereas the trace ratio criterion receives less interest mainly due to the lack of a closed-form solution and efficient algorithms. In this paper, we make an extensive study on the uncorrelated trace ratio linear discriminant analysis, with particular emphasis on the application on the undersampled problem. Two regularization uncorrelated trace ratio LDA models are discussed for which the global solutions are characterized and efficient algorithms are established. Experimental comparison on several LDA approaches are conducted on several real world datasets, and the results show that the uncorrelated trace ratio LDA is competitive with the orthogonal trace ratio LDA, but is better than the results based on ratio trace criteria in terms of the classification performance.     

Discriminative unsupervised 2D dimensionality reduction with graph embedding

Dimensionality reduction is a great challenge in high dimensional unlabelled data processing. The existing dimensionality reduction methods are prone to employing similarity matrix and spectral clustering algorithm. However, the noises in original data always make the similarity matrix unreliable and degrade the clustering performance. Besides, existing spectral clustering methods just focus on the local structures and ignore the global discriminative information, which may lead to overfitting in some cases. To address these issues, a novel unsupervised 2-dimensional dimensionality reduction method is proposed in this paper, which incorporates the similarity matrix learning and global discriminant information into the procedure of dimensionality reduction. Particularly, the number of the connected components in the learned similarity matrix is equal to cluster number. We compare the proposed method with several 2-dimensional unsupervised dimensionality reduction methods and evaluate the clustering performance by K-means on several benchmark data sets. The experimental results show that the proposed method outperforms the state-of-the-art methods.     

Discover latent discriminant information for dimensionality reduction: Non-negative Sparseness Preserving Embedding

How to define sparse affinity weight matrices is still an open problem in existing manifold learning algorithms. In this paper, we propose a novel unsupervised learning method called Non-negative Sparseness Preserving Embedding (NSPE) for linear dimensionality reduction. Differing from the manifold learning-based subspace learning methods such as Locality Preserving Projections (LPP), Neighbor Preserving Embedding (NPE) and the recently proposed sparse representation based Sparsity Preserving Projections (SPP); NSPE preserves the non-negative sparse reconstruction relationships in low-dimensional subspace. Another novelty of NSPE is the sparseness constraint, which is directly added to control the non-negative sparse representation coefficients. This gives a more ground truth model to imitate the actions of the active neuron cells of V1 of the primate visual cortex on information processing. Although labels are not used in the training steps, the non-negative sparse representation can still discover the latent discriminant information and thus provides better measure coefficients and significant discriminant abilities for feature extraction. Moreover, NSPE is more efficient than the recently proposed sparse representation based SPP algorithm. Comprehensive comparison and extensive experiments show that NSPE has the competitive performance against the unsupervised learning algorithms such as classical PCA and the state-of-the-art techniques: LPP, NPE and SPP.     

Integrating local and global topological structures for semi-supervised dimensionality reduction

Dimensionality reduction plays an important role in many machine learning tasks. This paper studies semi-supervised dimensionality reduction using pairwise constraints. In this setting, domain knowledge is given in the form of pairwise constraint, which specifies whether a pair of instances belongs to the same class (must-link constraint) or different classes (cannot-link constraint). In this paper, a novel semi-supervised dimensionality reduction method called LGS3DR is proposed, which can integrate both local and global topological structures of the data as well as pairwise constraints. The LGS3DR method is effective and has a closed form solution. Experiments on data visualization and face recognition show that LGS3DR is superior to many existing dimensionality reduction methods.