Graph Embedding Framework Research Articles

Collaborative Graph Embedding: A Simple Way to Generally Enhance Subspace Learning Algorithms

Collaborative representation (CR), known as an effective way to address the signal representation (regression) problem, has achieved remarkable success in visual classification. According to our theoretical analysis, the subspace learning issue can also be deemed as a signal representation problem. Therefore, we extend the graph embedding (GE) framework as a CR model to improve the discriminating power of the subspace learning algorithm. The new GE framework, which is named collaborative GE (CGE) framework, enjoys many desirable properties of CR. From theoretical analysis, CGE is robust to the noise and has the same computational complexity as GE. From experimental analysis, CGE can generally enhance the subspace learning algorithms and a reasonable regularization parameter can be inferred from its intrinsic graph. Several state-of-the-art subspace learning algorithms are plugged into our framework to produce their collaborative versions. Meanwhile, by exploring the intrinsic relation among GE methods, we present a new collaborative method named collaborative class-scattering locality preserving projections (CCSLPPs). The results of extensive experiments on ORL, AR, Scene15, Caltech256, LFW-A, and OU-ISIR-A databases demonstrate that the collaborative versions consistently outperform their original algorithms with a remarkable improvement and CCSLPP gets the best performance compared with all used methods.

Semisupervised Sparse Manifold Discriminative Analysis for Feature Extraction of Hyperspectral Images

The graph embedding (GE) framework is very useful to extract the discriminative features of hyperspectral images (HSIs) for classification. However, a major challenge of GE is how to select a proper neighborhood size for graph construction. To overcome this drawback, a new semisupervised discriminative learning algorithm, which is called the semisupervised sparse manifold discriminative analysis (S 3 MDA) method, was proposed by using manifold-based sparse representation (MSR) and GE. The proposed algorithm utilizes MSR to obtain the sparse coefficients of labeled and unlabeled samples. Then, it constructs a within-class graph and a between-class graph using the sparse coefficients of labeled samples, as well as an unsupervised graph with the sparse coefficients of unlabeled samples. Finally, it uses these graphs to obtain a projection matrix for feature extraction (FE) of HSI in a low-dimensional space. The S 3 MDA method not only inherits the merits of MSR to reveal the sparse manifold properties of data but also enhances interclass separability and intraclass compactness to improve the discriminating power for classification. Extensive experiments on two real HSI data sets obtained with a reflective optics system imaging spectrometer and an airborne visible/infrared imaging spectrometer show that the proposed algorithm is significantly superior to other state-of-the-art FE methods in terms of classification accuracy.

Marginal discriminant analysis using support vectors for dimensionality reduction of hyperspectral data

ABSTRACTFeature extraction (FE) is an efficient pre-processing step in hyperspectral image (HSI) classification. This article proposes a novel supervised FE method based on graph embedding framework (GEF). This method, which is called marginal discriminant analysis using support vectors (MDSV), can be used as a linear dimensionality reduction approach. The proposed method constructs inner and support graphs to capture both global and local structures of data points. The global geometrical structure of data in each class is described by the inner graph. The support graph uses support vectors (SVs) to detect the local inter-class structure of different classes. Incorporating these graphs enables MDSV to maximize the margin between classes in the projected space. Implementation of MDSV on four benchmark hyperspectral datasets confirms its efficiency as an appropriate pre-processing method before classification of HSIs.

Locally Discriminant Diffusion Projection and Its Application in Speech Emotion Recognition

The existing Diffusion Maps method brings diffusion to data samples by Markov random walk. In this paper, to provide a general solution form of Diffusion Maps, first, we propose the generalized single-graph-diffusion embedding framework on the basis of graph embedding framework. Second, by designing the embedding graph of the framework, an algorithm, namely Locally Discriminant Diffusion Projection (LDDP), is proposed for speech emotion recognition. This algorithm is the projection form of the improved Diffusion Maps, which includes both discriminant information and local information. The linear or kernelized form of LDDP (i.e., LLDDP or KLDDP) is used to achieve the dimensionality reduction of original speech emotion features. We validate the proposed algorithm on two widely used speech emotion databases, EMO-DB and eNTERFACE'05. The experimental results show that the proposed LDDP methods, including LLDDP and KLDDP, outperform some other state-of-the-art dimensionality reduction methods which are based on graph embedding or discriminant analysis.

Low-Rank Discriminant Embedding for Multiview Learning.

This paper focuses on the specific problem of multiview learning where samples have the same feature set but different probability distributions, e.g., different viewpoints or different modalities. Since samples lying in different distributions cannot be compared directly, this paper aims to learn a latent subspace shared by multiple views assuming that the input views are generated from this latent subspace. Previous approaches usually learn the common subspace by either maximizing the empirical likelihood, or preserving the geometric structure. However, considering the complementarity between the two objectives, this paper proposes a novel approach, named low-rank discriminant embedding (LRDE), for multiview learning by taking full advantage of both sides. By further considering the duality between data points and features of multiview scene, i.e., data points can be grouped based on their distribution on features, while features can be grouped based on their distribution on the data points, LRDE not only deploys low-rank constraints on both sample level and feature level to dig out the shared factors across different views, but also preserves geometric information in both the ambient sample space and the embedding feature space by designing a novel graph structure under the framework of graph embedding. Finally, LRDE jointly optimizes low-rank representation and graph embedding in a unified framework. Comprehensive experiments in both multiview manner and pairwise manner demonstrate that LRDE performs much better than previous approaches proposed in recent literatures.

A Graph-Embedding Approach to Hierarchical Visual Word Mergence.

Appropriately merging visual words are an effective dimension reduction method for the bag-of-visual-words model in image classification. The approach of hierarchically merging visual words has been extensively employed, because it gives a fully determined merging hierarchy. Existing supervised hierarchical merging methods take different approaches and realize the merging process with various formulations. In this paper, we propose a unified hierarchical merging approach built upon the graph-embedding framework. Our approach is able to merge visual words for any scenario, where a preferred structure and an undesired structure are defined, and, therefore, can effectively attend to all kinds of requirements for the word-merging process. In terms of computational efficiency, we show that our algorithm can seamlessly integrate a fast search strategy developed in our previous work and, thus, well maintain the state-of-the-art merging speed. To the best of our survey, the proposed approach is the first one that addresses the hierarchical visual word mergence in such a flexible and unified manner. As demonstrated, it can maintain excellent image classification performance even after a significant dimension reduction, and outperform all the existing comparable visual word-merging methods. In a broad sense, our work provides an open platform for applying, evaluating, and developing new criteria for hierarchical word-merging tasks.

Regularized Embedded Multiple Kernel Dimensionality Reduction for Mine Signal Processing.

Traditional multiple kernel dimensionality reduction models are generally based on graph embedding and manifold assumption. But such assumption might be invalid for some high-dimensional or sparse data due to the curse of dimensionality, which has a negative influence on the performance of multiple kernel learning. In addition, some models might be ill-posed if the rank of matrices in their objective functions was not high enough. To address these issues, we extend the traditional graph embedding framework and propose a novel regularized embedded multiple kernel dimensionality reduction method. Different from the conventional convex relaxation technique, the proposed algorithm directly takes advantage of a binary search and an alternative optimization scheme to obtain optimal solutions efficiently. The experimental results demonstrate the effectiveness of the proposed method for supervised, unsupervised, and semisupervised scenarios.

Locally Discriminant Diffusion Projection and Its Application in Speech Emotion Recognition

The existing Diffusion Maps method brings diffusion to data samples by Markov random walk. In this paper, to provide a general solution form of Diffusion Maps, first, we propose the generalized single-graph-diffusion embedding framework on the basis of graph embedding framework. Second, by designing the embedding graph of the framework, an algorithm, namely Locally Discriminant Diffusion Projection (LDDP), is proposed for speech emotion recognition. This algorithm is the projection form of the improved Diffusion Maps, which includes both discriminant information and local information. The linear or kernelized form of LDDP (i.e., LLDDP or KLDDP) is used to achieve the dimensionality reduction of original speech emotion features. We validate the proposed algorithm on two widely used speech emotion databases, EMO-DB and eNTERFACE'05. The experimental results show that the proposed LDDP methods, including LLDDP and KLDDP, outperform some other state-of-the-art dimensionality reduction methods which are based on graph embedding or discriminant analysis.

A regularized least square based discriminative projections for feature extraction

In this paper, we present a regularized least square based discriminative projections (RLSDP) method for feature extraction. First, we show that both sparse representation based classifier (SRC) and collaborative representation based classification (CRC) are regularized least square in nature. Second, a regularized least square based graph embedding framework (RLSGE) is constructed. Third, a RLSGE based feature extraction method is given, named regularized least square based discriminant projections (RLSDP). In RLSDP, the within-class compactness information is characterized by the reconstruction residual from the same class, which is consistent with the idea of reconstruction; the between-class separability information is characterized by the between-class scatter matrix like Fisher LDA. RLSDP is much faster than SPP since RLSDP adopts the L2 norm constraint while SPP adopts the L1 norm constraint. The experimental results on AR face database, FERET face database, and the PolyU FKP database demonstrate that RLSDP works well in feature extraction and has a great recognition performance.

A novel dimensionality reduction technique based on kernel optimization through graph embedding

In this paper, we propose a new method for kernel optimization in kernel-based dimensionality reduction techniques such as kernel principal component analysis and kernel discriminant analysis. The main idea is to use the graph embedding framework for these techniques and, therefore, by formulating a new minimization problem to simultaneously optimize the kernel parameters and the projection vectors of the chosen dimensionality reduction method. Experimental results are conducted in various datasets, varying from real-world publicly available databases for classification benchmarking to facial expressions and face recognition databases. Our proposed method outperforms other competing ones in classification performance. Moreover, our method provides a systematic way to deal with kernel parameters whose calculation was treated rather superficially so far and/or experimentally, in most of the cases.

Discriminative Analysis for Symmetric Positive Definite Matrices on Lie Groups

In this paper, we study discriminative analysis of symmetric positive definite (SPD) matrices on Lie groups (LGs), namely, transforming an LG into a dimension-reduced one by optimizing data separability. In particular, we take the space of SPD matrices, e.g., covariance matrices, as a concrete example of LGs, which has proved to be a powerful tool for high-order image feature representation. The discriminative transformation of an LG is achieved by optimizing the within-class compactness as well as the between-class separability based on the popular graph embedding framework. A new kernel based on the geodesic distance between two samples in the dimension-reduced LG is then defined and fed into classical kernel-based classifiers, e.g., support vector machine, for various visual classification tasks. Extensive experiments on five public datasets, i.e., Scene-15, Caltech101, UIUC-Sport, MIT-Indoor, and VOC07, well demonstrate the effectiveness of discriminative analysis for SPD matrices on LGs, and the state-of-the-art performances are reported.

Graph Regularized Sparsity Discriminant Analysis for face recognition

Manifold learning and Sparse Representation Classifier are two popular techniques for face recognition. Because manifold learning can find low-dimensional representations for high-dimensional data, it is widely applied in computer vision and pattern recognition. Most of the manifold learning algorithms can be unified in the graph embedding framework, where the first step is to determine the adjacent graphs. Traditional methods use k nearest neighbor or the ε-ball schemes. However, they are parametric and sensitive to noises. Moreover, it is hard to determine the size of appropriate neighborhoods. To deal with these problems, in this paper, Graph Regularized Sparsity Discriminant Analysis, GRSDA, for short, is proposed. Based on graph embedding and sparsity preserving projection, the weight matrices for intrinsic and penalty graphs are obtained through sparse representation. GRSDA seeks a subspace in which samples in intra-classes are as compact as possible while samples in inter-classes are as separable as possible. Specifically, samples in the low-dimensional space can preserve the sparse locality relationship in the same class, while enhancing the separability for samples in different classes. Hence, GRSDA can achieve better performance. Extensive experiments were carried out on ORL, YALE-B and AR face databases, and the results confirmed that the proposed algorithm outperformed LPP, UDP, SPP and DSNPE.

Graph Maximum Margin Criterion for Face Recognition

In this paper, we propose a graph maximum margin criterion (GMMC) which provides a unified method for overcoming the small sample size problem encountered by the algorithms interpreted in a general graph embedding framework. The proposed GMMC-based feature extraction algorithms compute the discriminant vectors by maximizing the difference between the graph between-class scatter matrix and the graph within-class scatter matrix and then the singularity problem is avoided. An efficient and stable algorithm for implementing GMMC is also proposed. We also reveal the eigenvalue distribution of GMMC. Experiments on the ORL, PIE and AR face databases show the effectiveness of the proposed GMMC-based feature extraction methods.

Linear dimensionality reduction based on Hybrid structure preserving projections

Recent advances have shown the methods based on local structure preserving projections can effectively learn discriminative features. The two attractive approaches for characterizing such data structure are: the classical nearest neighbor strategy for neighborhood structure and the sparse coding algorithm for sparsity structure. Motivated by the intuitive analysis of the relationship between the two structures, in this paper, we take both of them into account and propose two integrated approaches for dimensionality reduction. Concretely, we for achieving improvement directly integrate two available objectives, utilizing neighborhood structure and based on sparsity structure, to construct the combined method, briefly called CSNP. However, such rough strategy often results in its degradation in practice. Instead of the superficial combination, we exploit a hybrid structure by intergrading the two structures and then propose the Sparsity and Neighborhood Preserving Projections, dubbed SNPP, by preserving the hybrid structure into reduced subspace. The resulting optimization problems can be also interpreted as an instance of the general graph embedding framework and can reduce to the generalized eigenvalue decomposition problem. Finally, we conduct extensive experiments on publicly available data sets to verify the efficacy of our algorithms. From the experimental results, we roughly draw the conclusion that neighborhood structure is more important for low-dimensional data while sparsity structure is more useful for high-dimensional data.

Subclass Graph Embedding and a Marginal Fisher Analysis paradigm

Subspace learning techniques have been extensively used for dimensionality reduction (DR) in many pattern classification problem domains. Recently, methods like Subclass Discriminant Analysis (SDA) and Clustering-based Discriminant Analysis (CDA), which use subclass information for the discrimination between the data classes, have attracted much attention. In parallel, important work has been accomplished on Graph Embedding (GE), which is a general framework unifying several subspace learning techniques. In this paper, GE has been extended in order to integrate subclass discriminant information resulting to the novel Subclass Graph Embedding (SGE) framework, which is the main contribution of our work. It is proven that SGE encapsulates a diversity of both supervised and unsupervised unimodal methods like Locality Preserving Projections (LPP), Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA). The theoretical link of SDA and CDA methods with SGE is also established. Along these lines, it is shown that SGE comprises a generalization of the typical GE framework including subclass DR methods. Moreover, it allows for an easy utilization of kernels for confronting non-linearly separable data. Employing SGE, in this paper a novel DR algorithm, which uses subclass discriminant information, called Subclass Marginal Fisher Analysis (SMFA) has been proposed. Through a series of experiments on various real-world datasets, it is shown that SMFA outperforms in most of the cases the state-of-the-art demonstrating the efficacy and power of SGE as a platform to develop new methods.

Graph embedded nonparametric mutual information for supervised dimensionality reduction.

In this paper, we propose a novel algorithm for dimensionality reduction that uses as a criterion the mutual information (MI) between the transformed data and their corresponding class labels. The MI is a powerful criterion that can be used as a proxy to the Bayes error rate. Furthermore, recent quadratic nonparametric implementations of MI are computationally efficient and do not require any prior assumptions about the class densities. We show that the quadratic nonparametric MI can be formulated as a kernel objective in the graph embedding framework. Moreover, we propose its linear equivalent as a novel linear dimensionality reduction algorithm. The derived methods are compared against the state-of-the-art dimensionality reduction algorithms with various classifiers and on various benchmark and real-life datasets. The experimental results show that nonparametric MI as an optimization objective for dimensionality reduction gives comparable and in most of the cases better results compared with other dimensionality reduction methods.

Graph Embedded Extreme Learning Machine.

In this paper, we propose a novel extension of the extreme learning machine (ELM) algorithm for single-hidden layer feedforward neural network training that is able to incorporate subspace learning (SL) criteria on the optimization process followed for the calculation of the network's output weights. The proposed graph embedded ELM (GEELM) algorithm is able to naturally exploit both intrinsic and penalty SL criteria that have been (or will be) designed under the graph embedding framework. In addition, we extend the proposed GEELM algorithm in order to be able to exploit SL criteria in arbitrary (even infinite) dimensional ELM spaces. We evaluate the proposed approach on eight standard classification problems and nine publicly available datasets designed for three problems related to human behavior analysis, i.e., the recognition of human face, facial expression, and activity. Experimental results denote the effectiveness of the proposed approach, since it outperforms other ELM-based classification schemes in all the cases.

A framework of joint graph embedding and sparse regression for dimensionality reduction.

Over the past few decades, a large number of algorithms have been developed for dimensionality reduction. Despite the different motivations of these algorithms, they can be interpreted by a common framework known as graph embedding. In order to explore the significant features of data, some sparse regression algorithms have been proposed based on graph embedding. However, the problem is that these algorithms include two separate steps: (1) embedding learning and (2) sparse regression. Thus their performance is largely determined by the effectiveness of the constructed graph. In this paper, we present a framework by combining the objective functions of graph embedding and sparse regression so that embedding learning and sparse regression can be jointly implemented and optimized, instead of simply using the graph spectral for sparse regression. By the proposed framework, supervised, semisupervised, and unsupervised learning algorithms could be unified. Furthermore, we analyze two situations of the optimization problem for the proposed framework. By adopting an ℓ2,1-norm regularization for the proposed framework, it can perform feature selection and subspace learning simultaneously. Experiments on seven standard databases demonstrate that joint graph embedding and sparse regression method can significantly improve the recognition performance and consistently outperform the sparse regression method.

Local and Global Geometric Structure Preserving and Application to Hyperspectral Image Classification

Locality Preserving Projection (LPP) has shown great efficiency in feature extraction. LPP captures the locality by theK-nearest neighborhoods. However, recent progress has demonstrated the importance of global geometric structure in discriminant analysis. Thus, both the locality and global geometric structure are critical for dimension reduction. In this paper, a novel linear supervised dimensionality reduction algorithm, calledLocality and Global Geometric Structure Preserving(LGGSP) projection, is proposed for dimension reduction. LGGSP encodes not only the local structure information into the optimal objective functions, but also the global structure information. To be specific, two adjacent matrices, that is, similarity matrix and variance matrix, are constructed to detect the local intrinsic structure. Besides, a margin matrix is defined to capture the global structure of different classes. Finally, the three matrices are integrated into the framework of graph embedding for optimal solution. The proposed scheme is illustrated using both simulated data points and the well-known Indian Pines hyperspectral data set, and the experimental results are promising.

MiLDA: A graph embedding approach to multi-view face recognition

In a vast number of real-world face recognition applications, gallery and probe image sets are captured from different scenarios. For such multi-view data, face recognition systems often perform poorly. To tackle this problem, in this paper we propose a graph embedding framework, which can project the multi-view data into a common subspace of higher discriminability between classes. This framework can be readily utilized to extend classical dimensionality reduction methods to multi-view scenarios. Hence, by utilizing the framework for multi-view face recognition, we propose multi-view linear discriminant analysis (MiLDA). We also empirically demonstrate that, for several distinct multi-view face recognition scenarios, MiLDA has an excellent performance and outperforms many popular approaches.