# Robust Face Recognition Based on a New Supervised Kernel Subspace Learning Method
## 1 Introduction  
In the field of face recognition, many different dimensional recognition approaches have been developed in recent times [1; 2; 3]. Dimensionality reduction is a main problem in numerous recognition techniques, caused by the great amount of data with high dimensions in many real-world utilizations [4; 5; 6]. In fact, dimensionality reduction techniques have been recommended by researchers to avoid \"the cure of dimensionality,\" so as to amend the computational efficiency of image recognition [7; 8]. Generally, dimensionality reduction techniques can be classified into two main groups: i.e., linear and nonlinear. In linear methods, a significant low-dimensional subspace is intended to be discovered in the input data with high-dimensional space, where the embedded data in the input space has a linear structure [9; 10; 11; 12]. Principle component analysis (PCA) is one of the most famous linear methods [13; 14; 15]. PCA aims to retain global geometric information for data representation through enhancing the trace of the feature covariance matrix [13; 16; 17]. Linear discriminant analysis (LDA) is a linear technique that seeks to find out the discriminant information for data classification by enhancing the ratio between inter-class and intra-class scatters [16; 18]. Some of the limitations of both PCA and LDA are that they could suffer from the small sample size (SSS) [15] issue, and that they may fail to recognize many important data structures that are nonlinear [19; 20]. Scholars have developed abundant practical nonlinear dimensionality reduction strategies [21] to address these problems. They can be classified into two types: manifold-learning-based and kernel-based techniques [22; 23]. Manifold learning directly aims to discover principal nonlinear data with low-dimensional structures that are concealed in the input space. Isometric featuremapping (ISOMAP) [24; 25] and local linear embedding (LLE) [26; 27] are the most well-known manifold-learning-based techniques to find inherent low-dimensional embedding of data [28]. Based on some experiments carried out using these techniques, it is illustrated that these methods can discover meaningful embedded nonlinear data structures for face images adequately. However, manifold-learning-based techniques could suffer from two issues in terms of pattern recognition [15]. The first one is called \"overlearning of locality,\" [29] since manifold learning keeps locality data structures but there is no straight connection with classification. Out-of-sample is another issue that shows why most manifold-learning-based techniques are not appropriate for image recognition tasks [30; 31]. These techniques can yield an embedding directly from a training data set, but they are often not able to find the sample's image in the embedding space when it is implemented to a new point. These problems cannot be overcome by the currently proposed manifold learning methods. Although a few supervised forms have been proposed, they still suffer from these problems [30; 31; 32] because they are all founded based on \"locality\" characterization. The local quantity is sufficient for one manifold modeling, but it does not work well for classification tasks in multi-manifold modeling [17]. In contrast to manifold-learning-based techniques and in order to indirectly represent observed patterns into possibly much larger dimensional feature vectors, kernel-based techniques have been proposed by applying a kernelized nonlinear representation method. In this approach, the nonlinear data structure can be more separable in the observation space and become linear in the feature space. The representative strategies include kernel Fisher discriminant (KFD) [23; 33] and kernel principal component analysis (KPCA) [34; 35]. Both have shown that they can be practical in many real-world functions, such as face recognition, to preserve the nonlinear data structure [36].