3.1.

Covariance Discriminative Learning

CDL uses a natural methodology to characterize image sets by the covariance descriptor. Let $\mathbf{X}=[x_1,x_2,\dots ,x_n]$ denote the data matrix of an image set with $n$ image vectors, where $x_i\in {\mathbb{R}}^D$ in the $D$-dimensional vector space. The covariance descriptor can be expressed as

The kernel function $k_{\log }$ is shown to be an SPD kernel^10,13 that obeys Mercer’s theorem.⁴³ Therefore, the manifold structure can be preserved by the LED Riemannian kernel.

The explicit kernel feature mapping allows application of any standard vector space learning algorithms. The discriminative learning of CDL is conducted by the kernel LDA¹⁸ with the kernel trick. The mapping of Riemannian manifold to an Euclidean space is defined by the function $\varphi (·)$. Therefore, if $L$ points of the specified Riemannian manifold are spanned by the ${\mathrm{sym}}_D^{+}$ matrices $\{{\mathbf{B}}_1,{\mathbf{B}}_2,\dots ,{\mathbf{B}}_L\}$, the mapped feature points on Euclidean space can be denoted as $\{f({\mathbf{B}}_1),f({\mathbf{B}}_2),\dots ,f({\mathbf{B}}_L)\}$. With the inner product $⟨\varphi ({\mathbf{B}}_i),\varphi ({\mathbf{B}}_j)⟩=k_{\log }({\mathbf{B}}_i,{\mathbf{B}}_j)$, CDL seeks to solve the following optimization:¹⁰

3.2.

Eigenspectrum Regularization Technique

Eigenspectrum regularization^24,26,29 was originally proposed to address the conventional problems (problems caused singularity of ${\mathbf{S}}_W$ and the numerical instability of its inverse) of LDA on the linear Euclidean space. In this section, we introduce LRE²⁹ as an instance since it is the prototype of the proposed method in this paper.

Consider $n$ samples of training data $\mathbf{X}=[x_1,x_2,\dots ,x_n]$ with $\{x_i\in {\mathbb{R}}^D|i=\mathrm{1,2},\dots ,n\}$. In LRE, intrinsic data structure is modeled to regularize the directions of data locality $\mathbf{X}{\mathbf{L}}_{\mathrm{loc}}{\mathbf{X}}^T$.²⁹ The eigenspectrum and directions can be obtained by decomposing

LRE decomposes the entire eigenspace $\mathbf{V}$ into two subspaces: (1) the disparity subspace ${\mathbf{V}}_{\text{disparity}}=[v_1,v_2,\dots ,v_q]$, which corresponds to lower locality preservation, and (2) the principal subspace ${\mathbf{V}}_{\text{principal}}=[v_{q+1},v_{q+2},\dots ,v_D]$ for higher locality preservation. LRE indicates that the first few eigenvectors of the eigenspace correspond to large eigenvalues that provide lower locality preserving capability, whereas the eigenvectors that correspond to smaller eigenvalues provide higher locality preserving capability. Hence, larger weights are imposed on the subspace with higher locality preservation, whereas smaller weights are assigned to the subspace with lower locality preservation. A method is devised by determining “fences” to separate the disparity subspace and principal subspace, and then regularize these two subspace according to an adaptive eigenspectrum regularization model. The fences is defined by a split point on eigenspectrum ${\lambda }_{\text{disparity}}=\gamma (Q3+1.5\times IQR)$, where $\mathbf{Q}3$ is the third quartile (cutting off the highest 75% or lowest 25% of the sum of $\mathbf{\lambda }$), and $\gamma$ is a parameter for adaptively scaling the separating value. The definition of $IRQ$ is $IRQ=Q3-Q1$, where $\mathbf{Q}1$ is the first quartile.

This adaptive eigenspectrum regularization model finds the $q$’th split eigenvalue that satisfies ${\lambda }_q=\max \{\forall {\lambda }_i|{\lambda }_i\le {\lambda }_{\text{disparity}}\}$. The piecewise regularization function of LRE is defined as

The regularization function is imposed on the corresponding eigenvectors to form a full-dimensional transformation matrix

Then, LRE can obtain a more localized feature by transforming the original training data

We indicate that there is no dimensional reduction has occurred in this transforming. The information of the original training data is preserved as much as possible.

In the subsequent step of LRE, feature extraction and dimensional reduction from the regularized and more compact data are performed. To further preserve the within-locality and between-locality power, a similarity weight matrix ${\mathbf{G}}^{\mathrm{LRE}}$ is utilized. The element of ${\mathbf{G}}^{\mathrm{LRE}}$ is defined as

This problem can be easily solved by converting it to a generalized eigenvalue problem $\tilde{\mathbf{X}}{\mathbf{G}}^{\mathrm{LRE}}{\tilde{\mathbf{X}}}^T{u}_i={\phi }_i{u}_i$. By retaining $d$ eigenvectors $\mathbf{U}=[u_1,u_2,\dots ,u_d]$ ($d\le D$), which correspond to the largest $d$ eigenvalues, the projection matrix ${\mathbf{Z}}^{\mathrm{LRE}}=\tilde{\mathbf{V}}\mathbf{U}$ is used for the final lower-dimensional eigenfeature extraction.

4.1.

Representation of SPD Manifold

At first, according to Harandi et al.¹² and Tan and Gao,¹⁶ the computational cost of the Riemannian kernel with a high-dimensional SPD matrix is quite high. Several strategies are available to lower the dimensionality of the SPD matrix and reduce the computational cost of constructing the Riemannian kernel matrix.¹² Here, we combine all the training data in different training sets to collaboratively produce the dimensional reduction projection matrix by PCA.

Consider $L$ training image sets $\mathbf{\chi }={\{{\mathbf{X}}_i\}}_{i=1}^L$, each set contains $L_i$ images ${\mathbf{X}}_i=[x_1,x_2,\dots ,x_{L_i}]$. We combine all images of all sets to build a sample data collection

The dimensional reduction projection matrix can be obtained by decomposing the following sample covariance matrix:

This simple PCA that is applied to all training sets not only alleviates the problem of the high computational complexity of constructing the SPD kernel matrix but also better preserves the main variations in the set data to build the covariance matrices, which form the SPD manifold. This operation of refining the high-dimensional SPD matrices can also be viewed as a transformation from the high-dimensional manifold to a low-dimensional manifold.¹⁶

4.2.

Eigenspectrum Regularization with SPD Manifold

The $i$’th dimensional reduced set can be represented as ${\mathbf{Y}}_i=[y_1^i,y_2^i,\dots ,y_{L_i}^i]$. ${\mathbf{Y}}_i$ can be modeled by the covariance descriptor [Eq. (1)] and represented as ${\mathbf{B}}_i\in {\mathbb{R}}^{d_1\times d_1}$. To ensure that ${\mathbf{B}}_i$ is nonsingular to form the SPD manifold, a small perturbation is added to the covariance matrix ${\mathbf{B}}_i+\eta \mathbf{I}$. Hence, the perturbed ${\mathbf{B}}_i$ is an SPD matrix ${\mathrm{sym}}_{d_1}^{+}$.

For $L$ image sets of $C$ classes, they can be denoted as a collection of ${\mathrm{sym}}_{d_1}^{+}$ matrices $\mathbf{B}=\{{\mathbf{B}}_1,{\mathbf{B}}_2,\dots ,{\mathbf{B}}_L\}$ that form an SPD manifold. By defining the Riemannian mapping $\varphi :\mathcal{M}\to \mathcal{H}$, we can obtain the samples $\mathbf{\Phi }(\mathbf{B})=[\varphi ({\mathbf{B}}_1),\varphi ({\mathbf{B}}_2),\dots ,\varphi ({\mathbf{B}}_L)]$ on the RKHS $\mathcal{H}$, which is homeomorphic to Euclidean space.

To further preserve the local structure, we incorporate the graph-embedded framework into our proposed method. The local Laplacian matrix ${\mathbf{L}}_{\mathrm{loc}}$ is utilized to preserve the locality information, whereas the global Laplacian matrix is adjacent regardless of the class membership of all vertices.²⁹ In this step, we aim to obtain the eigenspectrum and directions of the local structure in the SPD Riemannian kernel space. They can be implemented by decomposing the locality preserving matrix $\mathbf{\Phi }(\mathbf{B}){\mathbf{L}}_{\mathrm{loc}}\mathbf{\Phi }{(\mathbf{B})}^T$ on the mapped space; we denote $\mathbf{\Phi }(\mathbf{B})$ as $\mathbf{\Phi }$ for simplicity. Then, we have

Unlike the edge weights in ${\mathbf{L}}_{\text{bin}}$ and ${\mathbf{L}}_{\text{class}}$, which are fixed in values, the edge weights in ${\mathbf{L}}_{\mathrm{adj}}$ are variables that are based on different similarity definitions, such as the heat kernel in locality preserving projections⁴⁴ and neighborhood reconstruction coefficients in neighborhood preserving embedding.⁴⁵ ${\mathbf{L}}_{\mathrm{adj}}$ is a Laplacian matrix that is computed by

As known by linear algebra, the projection direction of $\mathbf{V}$ in Eq. (17) can be represented as a linear combination of the eigenspace on the mapped space

By substituting Eq. (21) into Eq. (17), we obtain $\mathbf{\lambda }={\mathbf{\alpha }}^T{\mathbf{\Phi }}^T\mathbf{\Phi }{\mathbf{L}}_{\mathrm{loc}}{\mathbf{\Phi }}^T\mathbf{\Phi }\mathbf{\alpha }$. We use the Riemannian kernel function [e.g., Eq. (4)] to build the kernel Gram matrix $\mathbf{K}={\mathbf{\Phi }}^T\mathbf{\Phi }$, Eq. (17) can be rewritten as

Equation (22) can be solved by the eigendecomposition of $\mathbf{K}{\mathbf{L}}_{\mathrm{loc}}\mathbf{K}$ subject to ${\mathbf{\alpha }}^T\mathbf{\alpha }=1$.

In the theory of eigenspectrum regularization, we need to regularize the whole feature space $\mathbf{V}$ [see Eq. (21)] on the mapped space. Assume that, the regularization can be generalized to the weighting function, which is defined as

According to Eq. (21), we have $v_k=\mathbf{\Phi }{\mathbf{\alpha }}_k$. By defining

The regularized eigenspace is known as a transformation matrix²⁴ that can transform the original feature data to an intermediate feature vector space. It is worth noting that the transformation matrix $\tilde{\mathbf{V}}$ is a full-dimensional matrix with size $L\times L$. Hence, the mapped data $\mathbf{\Phi }(\mathbf{B})$ from SPD manifold can be transformed to the new feature vector space $\tilde{\mathbf{\Phi }}(\mathbf{B})$ with no dimensional reduction, which can preserve information as much as possible. We denote $\tilde{\mathbf{\Phi }}(\mathbf{B})$ as $\tilde{\mathbf{\Phi }}$ for simplicity. The transformation is depicted as

Although $\mathbf{\Phi }$ is implicitly defined, the transformed feature $\tilde{\mathbf{\Phi }}$ can be explicitly expressed by the kernel trick. According to Eqs. (26) and (27), we can denote the transformed $\tilde{\mathbf{\Phi }}$ by $\tilde{\mathbf{\Phi }}={\mathit{\rho }}^T{\mathbf{\Phi }}^T\mathbf{\Phi }$. As $\mathbf{K}={\mathbf{\Phi }}^T\mathbf{\Phi }$, Eq. (27) can be rewritten as

In this aspect, according to the previous mathematical deduction, by defining the important regularized eigenspace $\mathit{\rho }$ [see Eq. (25)], the regularization of eigenspace $\mathbf{V}$ is turned into the regularization of the eigenspace $\mathbf{\alpha }$. In other words, the effectiveness of eigenspectrum regularization model on the eigenspace $\mathbf{\alpha }$ is equivalent to the eigenspace $\mathbf{V}$.

The selection of a suitable eigenspectrum regularization model is a critical aspect of the proposed method. The proper eigenspectrum regularization model ensures that the regularized data can be very close to the real population variances.⁴⁶ The eigenspectrum regularization of LRE is an adaptive model that estimates the optimal parameter $\gamma$ using training data. However, this process is usually time-consuming, and the performance decays quickly when the training data are insufficient. In this paper, we employ the data-independent eigenspectrum regularization models of ERE and CDEFE to regularize the eigenspace $\mathit{\alpha }$, which are more general and robust. The first model is the eigenspectrum regularization model of ERE.²⁴ The heuristic theory of ERE for designing the eigenspectrum regularization model is the median operation. The weighting function applied to the eigenspace $\mathit{\alpha }$ is defined as

The second regularization model is taken from CDEFE.²⁶ The eigenspectrum regularization model of CDEFE regularizes the eigenspace in a Gaussian kernel space, which may have a special effect on the proposed RGCDL in the Riemannian kernel space.

The second regularization model aims to find the minimum eigenratio from the eigenspectrum of $\mathbf{K}{\mathbf{L}}_{\mathrm{loc}}\mathbf{K}$, which is formed by the eigenvalues [see Eq. (17)] in descending order. Let ${\delta }_k$ denote the ratio of two adjacent eigenvalues ${\lambda }_k$ and ${\lambda }_{k+1}$ in the eigenspectrum, we have

The minimum eigenratio can be formulated as

4.3.

Feature Extraction and Dimensional Reduction

The new feature vectors $\tilde{\mathbf{\Phi }}$ are compact and full dimensional, the eigenfeature of $\tilde{\mathbf{\Phi }}$ should be decorrelated and dimension reduced for classification. According to Jiang et al.,²⁴ PCA is exploited to extract the final discriminative eigenfeatures since it is less sensitive to different training databases. However, the class affinity is not considered in Ref. 24, which may cause the missing discriminative information. In this work, we employ a graph-embedded framework to extract the final discriminative features, which incorporates a similarity weight matrix to form the final scatter matrix. Although LRE had extended the graph-embedded framework to eigenfeature extraction, our method is designed to address the problems in a Riemannian kernel space.

According to Eq. (13), the eigenfeature extraction and dimensional reduction of RGCDL can be achieved by solving the following eigendecomposition problem on the mapped space:

Obviously, $\mathbf{Z}$ does not have an explicit expression since $\tilde{\mathbf{V}}=\mathbf{\Phi }\mathit{\rho }$, and $\mathbf{\Phi }$ is the vector space mapped by the Riemannian mapping, which is implicitly defined.

However, an explicit expression can be provided by the kernel trick when calculated with the test samples. For a given test nonsingular covariance matrix ${\mathbf{B}}^{te}$, which is an element of the SPD manifold, we use ${\varphi }^{te}$ to denote the test feature vector that is mapped by the Riemannian mapping. Subsequently, we can extract the discriminative feature $\mathbf{F}$ by the transformation

4.4.

Complete RGCDL Algorithm

The steps of RGCDL algorithm are given in Algorithm 1.

Algorithm 1

RGCDL algorithm.

5.1.

Dataset and Parameter Settings

We employed the Extended Yale face database B (ExtYaleB)⁴⁷ for face recognition task and the RGB-D object database⁴⁸ for object categorization task. The ExtYaleB database is the extension of the Yale face database B; it contains 16,128 images of 28 human subjects with 64 illumination conditions and 9 poses for each subject. According to the 9 poses of each subject, we built 9 image sets ($\sim 60$ images per set), which correspond to 9 poses for each subject. We utilized a cascaded face detector⁴⁹ to collect faces from each image frame. The captured faces were then converted to grayscale and resized to $20\times 20\text{pixels}$. Some example images are shown in Fig. 2. We selected 2 to 5 image sets of 9 poses for discriminative training (103 sets) and employed the remaining sets for testing (149 sets). The experiments were repeated 10 times by randomly choosing the reference sets for training and the test sets for probe.

Fig. 2

Captured face examples from the ExtYaleB dataset.

The RGB-D object database is a large-scale dataset of 300 common household objects that are organized into 51 categories (classes). Each category has 3 to 14 objects that belong to the same category. For each object, 3 video sequences are recorded with a camera that is mounted at different heights so that the object is viewed from different angles with the horizon. The video sequences were captured by placing each object on a turntable for a whole rotation using a Kinect style three-dimensional camera. More than 100 images were extracted for each object’s video sequences; they involve RGB color channels and a depth channel. We removed the depth images in this study to ensure fair comparisons. We built image sets according to each object, forming a total of 300 image sets (102 sets for training and 198 sets for testing). Grayscale and resized images of $20\times 20\text{pixels}$ were adopted for RGB-D dataset. Some example objects are shown in Fig. 3. To obtain more general results, we also conducted 10 cross-validation experiments by randomly choosing different combinations of training sets and test sets. The NN classifier was applied for all evaluations to ensure fair comparisons.

Fig. 3

Some example objects from the RGB-D dataset.

5.2.

Stability of Extracted Features

In this section, we evaluated the stability of the extracted features from RGCDL. We show that by applying eigenspectrum regularization, the features extracted by RGCDL are more stable than those extracted by the original CDL method.

As described in Sec. 4, the proposed RGCDL aims to extract features from the whole regularized eigenfeature space. As the number of final extracted features increases [controlled by varying $d_2$ dimensions of $\mathbf{U}$ in Eq. (37)], a higher performance can be achieved by our RGCDL, whereas the original CDL algorithm cannot retain this characteristic. To confirm this assumption and provide evidence, we employed real data of face and object datasets to conduct these experiments.

The collaborative dimensional reduction of each image set is set to 100 dimensions, that is, the dimension-reduced covariance matrix of each image set is $100\times 100$. Attributed to the covariance descriptor, the computational cost of constructing the Riemannian kernel matrices is not associated with the number of images within a set. The Riemannian kernel induced by the LED in Eq. (4) was employed for the RGCDL and CDL methods. We vary the final feature dimensions of RGCDL and CDL to perform a comprehensive comparison. The comparison results are shown in Figs. 4 and 5. Each figure consists of the recognition rates against the number of final extracted features. The recognition rates are the average results of 10 cross-validation experiments. Two eigenspectrum regularization models of ERE and CDEFE were evaluated for the proposed RGCDL.

Fig. 4

Error rates using different numbers of features on the ExtYaleB dataset.

Fig. 5

Error rates using different numbers of features on the RGB-D dataset.

As shown in Figs. 4 and 5, with the increasing dimensions of the final extracted features, the proposed RGCDL methods with two regularization models generally produce low error rates, whereas the original CDL degrades rapidly from the dimension $C-1$. The dimension of $C-1$ represents the optimal performance number of features in LDA.¹⁷ The degradation of CDL is caused by the incorrectly scaled null kernel space of the within-class scatter matrix,²⁴ which causes overfitting and poor generalization. These results reveal that, with the eigenspectrum regularization models, the conventional problems (e.g., the singularity of within-class scatter matrix) caused by limited training samples in CDL can be alleviated. Since the new feature space is properly scaled, the estimated eigenvalues obey the true variances of the population;²⁴ hence, better generalization can be achieved. The final extracted features are learned from the regularized full-dimensional transformation matrix $\tilde{\mathbf{V}}$ [Eq. (26)], which can preserve discriminative information as much as possible. As a result, using an increasing number of features, the recognition rates of RGCDL preserve the stable performance.

5.3.

Performance Evaluation against RGDA

RGDA is the preliminary work of this paper, however, RGDA was proposed to solve the overfitting and poor generalization problems of Grassmann discriminative learning, whereas our RGCDL solves these problems against CDL on SPD manifold. Moreover, we further employed different local Laplacian graphs to analyze the locality preserving ability and improve the performance; the locality preserving is evaluated in Sec. 5.4. We discovered that the performance of the SPD manifold with the eigenspectrum regularization techniques is better than that of the Grassmann manifold. We conducted two experiments to evaluate the advantages of the proposed method. First, we compared the classification ability of RGDA and RGCDL when different dimensions of the extracted features were applied. As shown in Figs. 6 and 7, for both eigenspectrum regularization models in two datasets, the error rates of RGCDL with different number of features always lower than those of the RGDA. Moreover, the error rate curves of RGCDL are smoother and steadier for a different number of features, and RGCDL can achieve a lower error rate even with a low number of features, particularly for the ExtYaleB dataset. The RGDA method usually cannot achieve high performance using lower dimensions of features. This finding demonstrates that the SPD manifold formed by the covariance matrices has better discriminative information preservation ability than the Grassmann manifold formed by subspaces.

Fig. 6

Performance comparisons of RGCDL and RGDA on the ExtYaleB dataset.

Fig. 7

Performance comparisons of RGCDL and RGDA on the RGB-D dataset.

Subsequently, we conducted noisy set data to evaluate the robustness of the proposed method. Image sets may contain noisy data in real-world applications, for example, outliers of other categories or subjects within sets exist, which may degrade the performance of classifiers. Here, we show that the SPD manifold-based RGCDL method is more robust than the Grassmann manifold-based RGDA method. We conducted experiments by systematically corrupting the training (gallery) sets or test (probe) sets. The corruption is implemented by adding images from other classes. The data with no noise are denoted as “clean,” the data with noise in the gallery sets are denoted as “N_G,” and the data with noise in the probe sets are denoted as “N_P.” Experiments were evaluated using both face recognition and object categorization.

The average classification rates of several cross-validations with different noise-corrupted datasets are shown in Figs. 8 and 9. The classification rates of our RGCDL always outperform RGDA in clean and different corrupted data. Especially in the gallery corrupted data N_G, RGCDL-ERE and RGCDL-CDEFE exhibit great advantages than RGDA-ERE and RGDA-CDEFE. Once again, these results demonstrate that the SPD manifold formed by the second-order statistic covariance matrices can be able to account for the noisy set data better than the Grassmann manifold formed by subspaces; it reveals the robustness of RGCDL when dealing with noisy set data.

Fig. 8

Evaluated performances on the noisy ExtYaleB dataset.

Fig. 9

Evaluated performances on the noisy RGB-D dataset.

5.4.

Performance Comparison to Other Set-Based Classification Methods

We further evaluated the proposed RGCDL compared with other set-based classification methods. Multiple image set-based classification methods were evaluated for comprehensive comparison. The compared methods include the subspace-based methods DCC⁷ and ECCA;³⁶ the Grassmann manifold methods of GDA,⁸ KGDA,³⁸ GGDA,⁹ MMD,³ GNP,¹¹ and RGDA;² the SPD Riemannian manifold methods of CDL,¹⁰ PPCDL,¹⁶ and DARG.⁴¹

The parameter settings of different methods are depicted as follows. The final feature dimension of GDA, KGDA, GGDA, and RGDA was established as the recommendation of Ref. 2, and only the projection kernel⁸ was employed for Grassmannian mapping. The parameters (such as the nonlinearity score and number of NNs of data points) of MMD were tuned to be optimal with the code provided by the authors in our datasets. For CDL and PPCDL, the final feature dimension was set as the recommended value $C-1$. The dimension of the input covariance matrices of DARG and PPCDL was set to $100\times 100$, which is the same as the proposed RGCDL for fairness. The Riemannian kernel induced by LED was applied for CDL, PPCDL, and our RGCDL. We chose kernel-based DARG and the good performance of MD + LED⁴¹ distance matrix for evaluation. We fixed the dimension to 150 for DCC by preapplying PCA to the data.⁷ The number of canonical correlations of DCC and ECCA was set to 20, which is the same as the Grassmannian dimension of the GDA, KGDA, GGDA, and RGDA methods. For RGDA and our RGCDL, the eigenspectrum regularization models of ERE and CDEFE were applied for comparison. Three local Laplacian matrices of $L_{\text{bin}}$, $L_{\text{class}}$, and $L_{\mathrm{adj}}$ were evaluated in the proposed RGCDL.

Experiments were evaluated on the ExtYaleB and RGB-D datasets. The experimental results are formed by the average classification rates and standard deviations over 10-fold cross-validations. As shown in Table 1, the proposed RGCDL with regularization models of ERE and CDEFE achieves the best classification results among all methods. The SPD manifold with covariance matrices of CDL, PPCDL, DARG, and our RGCDL approaches usually achieves better performance than other methods in ExtYaleB, which has shown the better accommodative ability of the second-order statistic of covariance matrix on handling the illumination-varying face recognition. The inferior performances of PPCDL and DARG on RGB-D dataset may cause by the conventional problems of discriminative learning and the improperly estimated GMM. Benefitting from the eigenspectrum regularization with the graph-embedded framework, the proposed RGCDL with different models outperforms all other methods. For the evaluation of different local Laplacian graphs, the $L_{\text{class}}$ achieves the best results with the ERE regularization model. However, the performance of adjustable Laplacian matrix $L_{\mathrm{adj}}$ is also outstanding in ERE model, and it achieves the best results with the CDEFE model. The adjustable Laplacian matrix $L_{\mathrm{adj}}$ performs stable in different regularization models, it has revealed the good locality preserving ability. Obviously, $L_{\mathrm{adj}}$ is not the best local Laplacian matrix for locality preserving, better affinity matrix can be designed according to suitable theories.

Table 1

Average classification rates and standard deviations (%) on ExtYaleB and RGB-D datasets.