A novel semi-supervised learning for face recognition

doi:10.1016/j.neucom.2014.11.018

Neurocomputing

Volume 152, 25 March 2015, Pages 69-76

https://doi.org/10.1016/j.neucom.2014.11.018 Get rights and content

Highlights

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We points out the shortcoming of LE in learning the local structure.
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Our approach characterizes both the diversity and similarity of data.
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Our approach helps encode the local discriminating information of data.

Abstract

Laplacian embedding (LE) has been widely used to learn the intrinsic structure of data. However, LE ignores the diversity and may impair the local topology of data, resulting in unstable and inexact intrinsic structure representation. In this article, we build an objective function to learn the intrinsic structure that well characterizes both the similarity and diversity of data, and then incorporate this structure representation into linear discriminant analysis to build a semi-supervised approach, called stable semi-supervised discriminant learning (SSDL). Experimental results on two databases demonstrate the effectiveness of our approach.

Introduction

Dimensionality reduction has been an active research topic in face recognition, human action recognition, image classification, and machine learning [1], [2], [3], [4]. The goal of dimensionality reduction is to reveal the meaningful structures and unexpected relationships embedded in multivariate data for different tasks such as data representation and classification. For classification, many previous researches have demonstrated that the performance can be significantly improved by discriminant approaches [5], [6], [7], [8]. Although their motivations are different, all these approaches can be unified in the graph embedding framework [9] or patch alignment framework [10], and the performance of these approaches will seriously deteriorate when the number of the labeled training samples is small or very small. In real-world applications, the labeled data are hard or expensive to obtain and meanwhile it is easy to acquire abundant unlabeled data at very low cost. Thus, how to efficiently exploit the large unlabeled data is becoming an interesting research topic for improving the performance of discriminant approaches [11], [12], [13], [14].

Unsupervised learning especially unsupervised manifold learning has demonstrated that some intrinsic geometric structures embedded in the unlabeled data are of great importance for data or image classification [15], [16], [17]. Motivated by this, many semi-supervised learning approaches were developed by combining supervised and unsupervised learning [11], [13], [14], [18], [19], [20]. These algorithms not only consider the label information, but also utilize a consistency assumption, namely, nearby points are likely to have the same label in classification tasks [12], [13], [18]. For semi-supervised dimensionality reduction, graph-based semi-supervised manifold learning techniques are successful and effective in many applications such as face recognition, action recognition, and image retrieval [19], [20], [21]. These semi-supervised approaches can be unified within the graph-based semi-supervised framework [22], [23].

Semi-supervised discriminant analysis (SDA) [18] and maximum margin projection (MMP) [24] are two of the most representative graph-based semi-supervised manifold learning approaches. SDA imposes a smoothness constraint into the objective function of linear discriminant analysis (LDA). This constraint based on graph Laplacian regularization [16] aims to enforce nearby points to have similar representations in the low-dimensional feature space. Different from SDA, MMP adds the same constraint into the local discriminant objective function. Motivated by SDA and MMP, some graph embedding based semi-supervised approaches were developed by different similarity metrics [25] or label propagation for the unlabeled data [26], [27], [28]. Although the motivations of the above- mentioned semi-supervised approaches are different, all these semi-supervised approaches employ Laplacian embedding (LE) to learn the intrinsic structure of data or the smoothness constraint.

It is generally considered that LE has the local topology preserving property. However, LE emphasizes the large distance pairs and may enforce that the larger the distance between two nearby points is, the closer they are embedded in the reduced space, resulting in the impairment of the local topology of data [29], [30], [31]. This impairs the intrinsic structure embedded in data. Moreover, LE maps nearby data in the observed data space to nearby points with the low- dimensional representation, and mainly characterizes the similarity of data. In the ideal case, all nearby points may be mapped to a single point in the reduced space. This leads to over- fitting and impairs the stableness of the algorithms.

Recently, some works have showed that the diversity of data, which can be obtained by maximizing the variation of data, also reflects the intrinsic structure of data and is of great importance for data representation and classification [32], [33], [34], [35]. Moreover, in many real-world applications, the intrinsic structure of data is unknown and complex, and testing data is usually different from the training data due to many factors. Thus, only similarity or diversity may not be sufficient to characterize the intrinsic structure of data [31]. Motivated by the analysis of great insight, we propose a novel semi-supervised dimensionality reduction algorithm called stable semi- supervised discriminant learning (SSDL), which explicitly takes into account both the diversity and similarity among nearby data points in this article. To be specific, we construct an adjacency graph, which characterizes the diversity of data, to overcome the shortcoming of LE, and combine it with LE to learn the intrinsic structure, and then incorporate the intrinsic structure into the objective function of LDA. Experiments on two databases demonstrate the effectiveness of our approach.

Section snippets

Semi-supervised discriminant analysis (SDA)

SDA respects the intrinsic geometrical structure inferred from both labeled and unlabeled data points and integrates the intrinsic geometrical structure of data into the objective function of LDA. Given a labeled set ${(x_{i}, τ_{i})} {i = 1}^{l}$ belonging to c classes and an unlabeled set ${x_{i}} {i = l + 1}^{n}$ , where $x_{i} \in R^{d}$ , $τ_{i}$ denotes the label of training data $x_{i}$ . Suppose that the $k$ -th class has l_k samples, then $\sum_{k = 1}^{c} l_{k} = l$ . The objective function of SDA is as follows [18]: $\underset{α^{T} α = 1}{arg} \max \frac{α^{T} S_{b} α}{α^{T} (S_{w} + β \sum_{i, j = 1}^{n} ((x_{i} - x_{j}) {(x_{i} - x_{j})}^{T} S_{i j}}$

Intrinsic structure representation

We construct two adjacency graphs $G_{g - s} = {X, B}$ and $G_{g - v} = {X, D}$ with a vertex set $X = {x_{1}, x_{2}, \dots, x_{n}}$ and two weight matrices $B$ and $D$ to model the intrinsic structure of data. Where matrix B characterizes the similarity relationship among nearby points, and matrix D characterizes the diversity relationship among nearby points. Motivated by [16], [30], [31], we define the elements $B_{i j}$ in B and $D_{i j}$ in D as follows: $B_{i j} = {\begin{array}{l} \exp (- d^{2} (x_{i}, x_{j}) / t^{2}), & i f x_{i} \in N_{k} (x_{j}) o r x_{j} \in N_{k} (x_{i}) \\ 0, & O t h e r w i s e \end{array}$ $D_{i j} = {\begin{array}{l} 1 - \exp (- d^{2} (x_{i}, x_{j}) / t^{2}), & i f x_{i} \in N_{k} (x_{j} \end{array}$

Experiments

We evaluated the proposed approach on two image databases (PIE and FERET), and compared its performance with Fisherface [5], MFA [8], MMP [24], and SDA [18]. In the experiments, we used Euclidean metric and nearest classifier for classification due to its simplicity. It is an open problem to select suitable values for the parameters $t$ and $k$ in MFA, MMP, SDA, and SSDL approaches, therefore, we selected $t$ within the interval $[0.1 \times d_{\min}^{2}, 10 \times d_{\max}^{2}]$ , where $d_{\min}$ and $d_{\max}$ denote the smallest and

Conclusion

In this article, we construct two adjacency graphs to learn the intrinsic structure that well characterizes both the local topology and geometrical properties of similarity and diversity of data, and then incorporate it into the objective function of LDA to build semi-supervised discriminant learning approach, namely stable semi-supervised discriminant learning (SSDL), for dimensionality reduction. Experimental results on PIE and FERET databases demonstrate the effectiveness of our approach.

Acknowledgments

We would like to thank the anonymous reviewers and AE for their constructive comments and suggestions. This work is supported by National Natural Science Foundation of China under Grant 61271296, Natural Science Basic Research Plan in Shaanxi Province of China under Grant 2012JM8002, China Postdoctoral Science Foundation under grant 2012M521747, the 111 Project of China (B08038), Fundamental Research Funds for the Central Universities of China under Grant BDY21, and the Open Project Program of

Quanxue Gao received the Ph.D. degree in control theory and control engineering from Northwestern Polytechnical University, China, in 2005. From 2006 to 2007, he was a research associate in the Department of Computing at The Hong Kong Polytechnic University. He is currently a Professor in School of Telecommunications Engineering, Xidian University. His research interests include pattern recognition, dimensionality reduction, semi-supervised learning, and sparse representation.

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Yunfang Huang received the B.Eng. degree from Hebei University, China, in 2012. He is currently working toward the M.S. degree in Electronic and Telecommunications engineering from Xidian University, China. His research interests include sparse representation, Semi-supervised learning, and face recognition.

Xinbo Gao (M׳02-SM׳07) received the B.Eng., M.Sc. and Ph.D. degrees in signal and information processing from Xidian University, China, in 1994, 1997 and 1999 respectively. From 1997 to 1998, he was a research fellow in the Department of Computer Science at Shizuoka University, Japan. From 2000 to 2001, he was a postdoctoral research fellow in the Department of Information Engineering at the Chinese University of Hong Kong. Since 2001, he joined the School of Electronic Engineering at Xidian University. Currently, he is a Professor of Pattern Recognition and Intelligent System, and Director of the VIPS Lab, Xidian University. His research interests are computational intelligence, machine learning, computer vision, pattern recognition and wireless communications. In these areas, he has published 5 books and around 150 technical articles in refereed journals and proceedings including IEEE TIP, TCSVT, TNN, TSMC, Pattern Recognition etc.

Weiguo Shen Received the B. Eng. degree in measurement and control technology from Xi׳an University of Posts and Telecommunications, P R China, in 2009 and the M.S. degree in traffic information engineering and control from Xidian University, China, in 2012. Her research interests include pattern recognition, and dimensionality reduction.

Hailin Zhang received the B.S. and M.S. degrees from Northwestern Polytechnical University, China, in 1985 and 1988 respectively, and the Ph.D. form Xidian University, China, in 1991, all in electronic information engineering. He is now a senior Professor with School of Telecommunications Engineering at Xidian University. He is currently the Dean of this school, the Director of Key Laboratory in Wireless Communications Sponsored by China Ministry of Information Technology, a key member of State Key Laboratory of Integrated Services Networks, a field leader in Telecommunications and Information Systems in Xidian University, and an associate Director for National 111 Project.

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