Adaptive maximum margin analysis for image recognition
Introduction
In reality, the high-dimensional images may not be uniformly distributed in the whole ambient space and reside on a lower dimensional sub-manifold which is embedded in the high-dimensional ambient space [1], [2], [3], [4]. Thus, it is important to consider how to obtain a compact and effective low-dimensional representation, i.e., dimensionality reduction in the literature. Two of the most popular techniques are principal component analysis (PCA) [5], [6], which is an unsupervised method, and linear discriminant analysis (LDA) [6], [7] that is more suitable for image classification.
LDA aims to seek the projection vectors on which the data points of different classes are far from each other while requiring data points of the same class to be close to each other. Applied LDA to face recognition, Belhumeur et al. proposed Fisherfaces algorithm [7]. Motivated by Fisherfaces, many methods have been developed to solve the singularity of within-class scatter matrix caused by small sample size problem [8], [9], [10], [11], [12]. These methods mainly capture the global Euclidean structure and cannot well characterize the local intrinsic geometric structure of manifold on which images possibly reside [4], [13], [14].
Manifold learning is capable of well characterizing the local intrinsic structure. Two of the most popular linear approaches are locality preserving projection (LPP) [3] and neighborhood preserving embedding (NPE) [17], which are a linear approximation of Laplacian Eigenmap (LE) [4] and locally linear embedding (LLE) [13] respectively. LPP and NPE do not consider the label information of training data, which is useful for classification. To handle this problem, Yan et al. proposed margin Fisher analysis (MFA) [18] which is similar to local discriminant embedding (LDE) [19]. Zhang et al. proposed discriminant locality alignment (DLA) [20]. Motivated by MFA and DLA, many linear manifold learning approaches have been developed by imposing non-negative constraint [21], [22], orthogonal constraint [23], [24] and other constrains [25], [26], [27] with them. Although their motivations are different, these approaches can be unified within the graph-embedding framework (GEF) [18] and depend on the construction of adjacency graphs, which is constructed by hand in advance. This reduces the flexibility and generalization of algorithms.
Recently, sparse representation has shown its powerful performance for image classification and has been widely used for computer vision and pattern recognition [28], [29], [30], [31], [32], [33], [34]. For example, Wright et al. [28] proposed a sparse representation based classification (SRC) algorithm, which uses training data to sparsely represent a given test sample and employs residual errors to identify the test image. Based on this content, Chen et al. [29] and Wright et al. [30] respectively used the L1-graph to adaptively construct adjacency intrinsic graph and then performed manifold learning, which is similar to sparsity preserving projection (SPP) [32]. Gui et al. [33] used L1-graph to construct the adjacency graph of NPE and proposed discriminant sparse neighborhood preserving embedding by combining maximum margin criterion. However, in high-dimensional original space, sparse representation cannot well characterize the intrinsic structure of data [35], [36] due to the small sample size. Another limitation is that they do not best maximize the margin between different classes in the low-dimensional space, which is very important for classification [15], [16].
Margin is a critical important property in pattern recognition [16], [37], [38] and measures the confidence of a classifier with respect to its predictions. Many approaches have demonstrated that large margin in the low- dimensional space helps improve the generalization ability of classification [15], [16], [38], [39], [40]. Representative methods are MMP [39], MMPP [40], and MMDA [15]. MMPP, which is an extension of MMP, aims to find projection matrix such that SVM can obtain large margin in the low-dimensional space. MMDA aims to find the projection matrix by maximizing the minimum distance between different classes. Cheng and Zhou [41], [42] proposed a general minimax framework for high-dimensional data representation, which interprets the relationship between data and regularization terms. Although much progress has been made for learning the maximum margin, there is little attention to learn the margin in the low-dimensional space for discriminant manifold learning.
In this paper, we propose an adaptive maximum margin analysis (AMMA) for dimensionality reduction. Different from many existing discriminative manifold learning methods, our main results are (1) AMMA selects the nearby points in the low-dimensional space by using sparse representation and maximizes the margin which is determined by the selected nearby points in the low-dimensional space. Thus, AMMA can maximize the margin in the low-dimensional space. (2) AMMA adaptively constructs intrinsic graph and penalty graph by sparse representation. This improves the adaptability of the algorithm. (3) AMMA suits SRC in classification stage.
The rest of the paper is organized as follows. In Section 2, we provide a brief review for discriminant manifold learning. Adaptive maximum margin analysis (AMMA) is introduced in Section 3. Experimental results are described in Section 4. Conclusions are summarized in Section 5.
Section snippets
Discriminant manifold learning
Many existing manifold learning approaches can be unified within the graph embedding framework (GEF) [18]. Given a set of training samples (), the objective function of GEF iswhere denotes one-dimensional representation of training sample , is a projection vector, denotes the weight between the th node and th node in the intrinsic graph, is the weight between the th node and th node in the penalty
Motivation
Combing the aforementioned insight in GEF, our approach AMMA aims to seek the projection matrix such that the low-dimensional representations have the largest margin between different classes and simultaneously fit sparse representation. After it happens, sparse representation can well characterize the local geometric structure in the low-dimensional space. Thus, we can use L1-graph to adaptively construct intrinsic graph and penalty graph in the low-dimensional space. In the following
Experiments
In this section, we validate our proposed approach AMMA in image classification and compare it with several dimensionality methods including SPP [32], SRC-DP [35], MFA [18], DSNPE [33] and LDA [7]. In the following experiments, five image databases, including three face image databases (AR, Extended Yale B, and CMU-PIE) and two other image databases (USPS and COIL20) are used for image recognition. Similar to SRC-DP [35], we first employ PCA to reduce the dimensionality to be 200, and then use
Conclusions
In this paper, we present a supervised dimensionality reduction approach, namely AMMA which is formally similar to MFA. Different from MFA, which maximizes the margin in the high-dimensional original space, AMMA maximizes the margin among nearby data that is adaptively calculated in the low-dimensional space, and well encodes the local discriminant structure embedded in data. Moreover, AMMA adaptively constructs the weights of intrinsic and penalty graphs by L1-graph. Thus, AMMA is a free
Acknowledgments
The authors would like to thank the anonymous reviewers and AE for their constructive comments and suggestions, which improved the paper substantially. This work is supported by National Natural Science Foundation of China under Grant 61271296, Fundamental Research Funds for the Central Universities of China under Grant BDY21, China Postdoctoral Science Foundation under Grant 2012M521747, the 111 Project of China (B08038), and the the Basic Science Research of Shaanxi province under Grant
Qianqian Wang received the B.Eng. degree from Lanzhou University of Technology, Lanzhou, China, in 2014. Now, she is working toward the Ph.D. degree at Xidian University, Xi'an, China. Her research interests include dimensionality reduction, pattern recognition and deep learning.
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Qianqian Wang received the B.Eng. degree from Lanzhou University of Technology, Lanzhou, China, in 2014. Now, she is working toward the Ph.D. degree at Xidian University, Xi'an, China. Her research interests include dimensionality reduction, pattern recognition and deep learning.
Quanxue Gao received the B. Eng. degree from Xi'an Highway University, Xi'an, China, in 1998, the M.S. degree from the Gansu University of Technology, Lanzhou, China,in 2001, and the Ph.D. degree from Northwestern Polytechnical University, X'ian China, in 2005.He was a associate research with the Biometrics Center, The Hong Kong Polytechnic University, Hong Kong from 2006 to 2007. He is currently a professor with the School of Telecommunications Engineering, Xidian University, and also a key member of State Key Laboratory of Integrated Services Networks. His current research interests include pattern recognition and machine learning
Lan Ma received the B.Eng. degree from Lanzhou University of Technology, Lanzhou, China, in 2015. Now, she is working toward the Master. degree at Xidian University, Xi'an, China. Her research interests include dimensionality reduction and deep learning.
Yang Liu received the B.Eng. and M.S. degrees from Xidian University, Xi'an, China, in 2013, 2015, Now, he is working toward the Ph.D. degree at Xidian University, Xi'an, China. Her research interests include dimensionality reduction, pattern recognition, and machine learning.