Locality Regularization Embedding for face verification
Introduction
Face verification has been researched over the past few decades. However, how to design a reliable dimensionality reduction technique is remained an open problem. There are several issues from pattern recognition point of views. For instance, the facial data often resides on high dimension space despite limited training samples are available. Poor performance is expected if it is used directly due to curse of dimensionality [1]. Furthermore, a high dimensional data always contains redundant information and noises. Hence, a dimensionality reduction technique is required to break the curse while extracting useful features.
In general, there are two important concerns in designing a dimensionality reduction technique for face verification: (1) How to effectively exploit the limited available training samples? (2) How to seek the most discriminative facial feature representation? Conventionally, the most popular techniques include Principal Component Analysis (PCA) [2] and Linear Discriminant Analysis (LDA) [3]. PCA projects face samples onto linear directions with maximal variances. Unlike PCA which is an unsupervised method, LDA is a supervised method that utilizes available class specific information. LDA seeks a linear projection that is optimal in data discrimination. Both linear subspace techniques have demonstrated a fairly good performance under strictly controlled environment.
Graph Embedding Framework (GEF) was proposed as a method to unify several well-known dimensionality reduction techniques and it provides an insight to design new methods for dimensionality reduction [4]. The instances are Neighbourhood Preserving Embedding (NPE) [5], Locality Preserving Projection (LPP) [6], Marginal Fisher Analysis (MFA) [4] etc. In general, GEF seeks an embedded low-dimensional manifold based on data similarities on an affinity graph [4].
In this section, a brief account of GEF is given. In GEF, each data is represented as a vertex of a graph. Conforming manifold preserving criterion, graph embedding transforms the vertex to a low dimensional representation that best preserves the similarities between the vertex pairs [7]. The similarity is quantified by a similarity matrix of a locality graph that depicts certain geometrical properties of the data set.
Let with be a set of numbers of -dimensional data, be a weighted graph with , and weight matrix . Each element in signifies the similarity of vertex pairs [4]. can be formulated based on different similarity criteria, such as prior class information in supervised learning algorithms [4], local neighborhood coefficients [5] and Gaussian similarity [6]. In brief, different definitions of graph correspond to different graph embedding algorithms.
For simplicity, one-dimensional case is considered and the low dimensional representations of vertices are represented as a vector . is the low dimensional representation of vertex . The target of the mapping is to make the vertices stay as close as possible to each other via a locality preserving criterion, given aswhere is a constant and is a constraint matrix for avoiding trivial solution.
With some simple algebraic manipulations, we obtainwhere is the Laplacian matrix and is a diagonal matrix defined as
Hence, the above minimization problem can be reformulated toIn general, is a diagonal matrix for scale normalization. is set to 1 to eliminate an arbitrary scaling factor. Since , the above optimization problem is equal to
Graph embedding excessively concerns about truthful data representation. It is no doubt that a reliable data representation is important. However, discriminative features play more crucial role in pattern recognition. The discriminating capability of these embedding can be explicitly boosted through discriminant criteria such as Fisher criterion [3] and Maximum Margin criterion [8]. This enhanced treatment is known as discriminant graph embedding.
As mentioned previously, higher locality preserving leads to better class discrimination. Hence, various regularization methods [9], [10], [11], [12], [13] have been proposed to address this requirement.
Summaries of some well-known graph embedding techniques, including criterion function, characteristics and limitations, are tabulated in Table 1, Table 2, Table 3 which correspond to unsupervised, supervised and regularized techniques respectively.
Unsupervised techniques learn a good representation from unlabeled training samples. Since there is no proper guidance, the performance of unsupervised techniques is not as pleasing as that in supervised and regularized counterparts. On the other hand, supervised techniques leverage foreknowledge of class label for leaning. Hence, these techniques always appear to be superior as far as recognition task is concerned. As another alternative, regularized techniques are meant to relieve the ill-posed problems due to numerical instability or improper model fitting and parameter estimation [9], [12], [13], [15], [16], [17].
Graph Embedding Framework (GEF) attempts to maximally preserve data locality after embedding, so that the embedded samples are remained in proximity. GEF could achieve this ultimate goal provided that population information is available, which is unrealistic. Practically, data locality is estimated based on finite yet noisy training samples. The estimation could be severely biased if the training samples poorly reflect the population information. As a result, this could negatively affect a projection function and leads to recognition performance deterioration.
Jiang [16], [17] performed a thorough analysis on how to reliably restore the population statistics based on regularization technique. Besides that, the author has also proposed a few solutions, such as by regulating eigenvalues of the covariance matrix with a piece-wise weighting function, a probabilistic subspace learning approach and eigenfeatures regularization and fitting with a model. Works in [10], [11], [12], [13], [15] are the instances of these solutions.
Inspired from [17], a graph embedding regularization technique is proposed in this paper. The regularization technique, dubbed as Locality Regularization Embedding (LRE), adopts a local Laplacian matrix to restore data locality. Even though both LRE and RLPDE [15] are using local Laplacian matrix for data locality regularization, LRE is a more general method that considers various local Laplacian matrices, whereas RLPDE is one specific instantiation of the LRE that applied a specific local Laplacian matrix.
The main contributions of this work include (1) efficient feature extraction methods that employ regularization for better locality preserving, leading to better discrimination, and (2) a theoretical analysis on the effectiveness of LRE on data discrimination.
The robustness of the proposed techniques is examined thoroughly with five public available face databases: CMU Pose, Illumination, and Expression (CMU PIE) [18], Facial Recognition Technology (FERET) [19], ORL Database of Faces (ORL) [20], Yale Face Database B (YaleB) [21] and Face Recognition Grand Challenge (FRGC) [22]. Armed with Nemenyi post-hoc statistical of significant test, the effectiveness of the proposed techniques in face verification is attested.
Section snippets
The locality preserving in graph embedding
In graph embedding, data are distributed on an underlying manifold, . Suppose that there is a map . The gradient of from the manifold space to is denoted as . For small [7]From Eq. (6), it is noticed that data points near x is mapped to data points near if is small. Belkin and Niyogi [23] defined a function as a measure metric of locality preserving on average of ,The above equation can be
Databases
The experiments are conducted by using five publicly available face datasets, namely CMU Pose, Illumination, and Expression (CMU PIE), Facial Recognition Technology (FERET), ORL Database of Faces (ORL), Yale Face Database B (YaleB) and Face Recognition Grand Challenge (FRGC).
CMU PIE database was acquired by using 13 synchronized cameras and an array of flashes as light sources. These camera flashes were placed in specific positions relative to the subject. They were triggered with a very short
Conclusion
Graph embedding techniques attempt to produce a high data locality projection for better recognition performance. However, under a scenario of limited training samples, the estimation of population/true data locality could be severely biased. The biased estimation triggers overfitting problem resulting poor generalization. A locality regularization of graph embedding is studied. Manipulation of a local Laplacian matrix is performed to approach true data locality for better data manifold
Conflict of interest
None declared.
Acknowledgment
This research was supported by UM-MMU Collaboration and Fundamental Research Grant Scheme – FRGS (#MMUE/140020) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013006574).
Pang Ying Han received her B.E. degree in Electronic Engineering in 2002, M.E. degree in 2005 and Ph.D. degree in 2013 from the Multimedia University, Malaysia. Her research interests include face recognition, manifold learning, dimensionality reduction, image processing and pattern recognition.
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Pang Ying Han received her B.E. degree in Electronic Engineering in 2002, M.E. degree in 2005 and Ph.D. degree in 2013 from the Multimedia University, Malaysia. Her research interests include face recognition, manifold learning, dimensionality reduction, image processing and pattern recognition.
Andrew Beng Jin Teoh obtained his BEng (Electronic) in 1999 and Ph.D degree in 2003 from National University of Malaysia. He is currently an associate professor in Electrical and Electronic Engineering Department, College Engineering of Yonsei University, South Korea. His research interests are Pattern Recognition, Machine Learning and Information Security. He has published more than 220 international refereed journals, conference articles, and several book chapters. He has been a reviewer for more than 30 journals and conferences. He has served international conference committees worldwide.
Hiew Fu San received his B.E. degree in Computer Engineering in 2002 and M.E. degree in 2008 from the Multimedia University, Malaysia. His research interests include pattern recognition and remote sensing.