Dual Graph Laplacian RPCA Method for Face Recognition Based on Anchor Points

1. Preliminaries

1.1. Quaternion and Quaternion Matrix

Introduced by mathematician Hamilton in 1843 [27], quaternion generalizes complex numbers by incorporating one real part and three imaginary parts. A quaternion can be expressed as

$$a=a_0+a_{1}i+a_{2}j+a_{3}k,$$

(5)

where $a_0,a_1,a_2,a_3\in \mathbb{R}$ , and $i,j,k$ are three imaginary units that follow the multiplication rules

$$i^2=j^2=k^2=-1,ij=-ji=k,jk=-kj=i,ki=-ik=j.$$

(6)

The conjugate and modulus of quaternion a are defined as follows

$$\overline{\mathrm{a}}={\mathrm{a}}_0-{\mathrm{a}}_1\mathrm{i}-{\mathrm{a}}_2\mathrm{j}-{\mathrm{a}}_3\mathrm{k},$$

(7)

$$\left|a\right|=\sqrt{{\left(a_0\right)}^2+{\left(a_1\right)}^2+{\left(a_2\right)}^2+{\left(a_3\right)}^2}.$$

(8)

A quaternion matrix $A\in {\mathbb{Q}}^{m\times n}$ takes the form

$$A=A_0+A_{1}i+A_{2}j+A_{3}k,$$

(9)

with $A_0,...,A_3\in {\mathbb{R}}^{m\times n}$, and its conjugate transposed matrix is given by $A^{*}=A_0^T-A_1^{T}i-A_2^{T}j-A_3^{T}k$. A pure quaternion matrix provides a representation for color images, with its three imaginary components $A_1,A_2,A_3$ corresponding to the red, green, and blue channels, respectively. Furthermore, the real representation method is a widely used technique for transforming quaternion matrices into the corresponding real matrices, the real representation of the matrix A is shown as

$${\gamma }_{\mathrm{A}}=\left(\begin{array}{cccc}{\mathrm{A}}_0& -{\mathrm{A}}_2& -{\mathrm{A}}_1& -{\mathrm{A}}_3\\ {\mathrm{A}}_2& {\mathrm{A}}_0& -{\mathrm{A}}_3& {\mathrm{A}}_1\\ {\mathrm{A}}_1& {\mathrm{A}}_3& {\mathrm{A}}_0& -{\mathrm{A}}_2\\ {\mathrm{A}}_3& -{\mathrm{A}}_1& {\mathrm{A}}_2& {\mathrm{A}}_0\end{array}\right),$$

(10)

and the first block column of ${\gamma }_A$ is denoted by

$${\gamma }_{A,c}=\left[\begin{array}{c}{A}_0\\ A_2\\ A_1\\ A_3\end{array}\right].$$

(11)

Properties 1.1.

For $A,B\in {\mathbb{Q}}^{m\times n},C\in {\mathbb{Q}}^{n\times s}$, the following properties hold [28].

• ${\gamma }_{A+B}={\gamma }_A+{\gamma }_B,{\gamma }_{AC}={\gamma }_A{\gamma }_C$.
• ${\gamma }_{A^{*}}={\gamma }_A^T$, where ${(·)}^T$ denotes transpose operation.
• A is a column unitary matrix if and only if ${\gamma }_A$ is a column orthogonal matrix.

1.2. Graph

Let $G=(V,E)$ be an undirected weighted graph. The weight value between $v_i$ and $v_j$ is denoted by $w_{ij}$. If there is no edge between $v_i$ and $v_j$, i.e., $(v_i,v_j)\notin E$, then $w_{ij}=0$. The matrix $W=\left(w_{ij}\right)(i,j=1,...,n)$ is called the adjacency matrix of G. We define the degree matrix D, which is a diagonal matrix whose diagonal entry $D_{ii}$ equals to the sum of weights of all edges incident to i, i.e., $d_i={\sum }_{j=1}^n{w}_{ij}$. Subsequently, the graph Laplacian matrix is defined as follows

$$L=D-W.$$

(12)

In some applications, the normalized graph Laplacian is used, defined as

$$L_{norm}=I-D^{-{\displaystyle \frac{1}{2}}}W{D}^{-{\displaystyle \frac{1}{2}}}.$$

(13)

Properties 1.2.

Let L denote a graph Laplacian matrix. Then the following properties hold [29].

• For each vector ${(f_1,...,f_n)}^{\prime }=f\in {\mathbb{R}}^n$, we have

  $$f^{\prime }Lf={\displaystyle \frac{1}{2}}\sum _{i=1}^n{w}_{ij}{\left(f_i-f_j\right)}^2.$$

  (14)
• L is symmetric and positive semi-definite.
• The smallest eigenvalue of L is 0, and corresponding eigenvector is 1 whose elements are all ones.
• L has n non-negative real eigenvalues $0={\lambda }_1\le {\lambda }_2\le ...\le {\lambda }_n$.

1.3. Graph Laplacian Embedding

Graph Laplacian embedding has emerged as a widely-used technique in nonlinear manifold learning, aiming to preserve the local geometric structure of data. The underlying assumption is that points which are close in the original data space should maintain their proximity in the embedded space. To achieve this, a nearest neighbor graph is constructed to capture and model the local relationships among data points.

Given the data matrix $X=(x_1,...,x_n)\in {\mathbb{R}}^{m\times n}$, where $x_i$$(i=1,...,n)$ denotes a data sample or one vertex in the graph. For each data point $x_i$, we connect each $x_i$ to its k nearest neighbors. Here, we adopt Euclidean distance $w_{ij}=\sqrt{{\sum }_{d=1}^m{(x_{i,d}-x_{j,d})}^2}$ to measure the similarity between the data samples.

Let $Z=(z_1,...,z_n)\in {\mathbb{R}}^{k\times n}$ represent embedding coordinates of data samples $X=(x_1,...,x_n)\in {\mathbb{R}}^{m\times n}$. The dissimilarity of the two data points in the lower-dimensional space can be measured by the Euclidean distance. Define the dissimilarity of the two data points in the lower-dimensional space as $d(z_i,z_j)=\left|\right|z_i-z_j{\left|\right|}^2$, combining with the weight matrix W, the smoothness of the low-dimensional representation can be measured by minimizing the following equation [30]

$$\begin{array}{cc}\hfill \mathrm{S}& ={\displaystyle \frac{1}{2}}\sum _{i,j=1}^n{\parallel z_i-z_j\parallel }^2{w}_{ij}\hfill \\ & =\sum _{i=1}^n{z}_i^T{z}_i{D}_{ii}-\sum _{i,j=1}^n{z}_i^T{z}_j{w}_{ij}\hfill \\ & =Tr\left(V^{T}DV\right)-Tr\left(V^{T}WV\right)=Tr\left(V^{T}LV\right),\hfill \end{array}$$

(15)

where $D=diag(d_1,...,d_n)$ is a diagonal matrix, and $L=D-W$ is the graph Laplacian matrix.

1.4. Anchors

Anchor point techniques play a critical role in data representation and learning, effectively reducing computational costs while capturing the underlying structure of the data. In machine learning, anchor points are a set of representative samples or features selected from a dataset. Given a dataset $X=(x_1,x_2,...,x_n)\in {\mathbb{R}}^{m\times n}$, anchor points $A=(a_1,a_2,...,a_k)\in {\mathbb{R}}^{m\times k}$ with $k\ll n$, act as reference points in the data space, which can be strategically selected based on prior knowledge (e.g., class centers) or learned through methods like K-means clustering. Anchor points facilitate the construction of similarity matrices or lower-dimensional embeddings, thereby reducing computational complexity and enhancing model generalization, particularly for large-scale data. Following this introduction, we will explore two methodologies for deriving these anchor points, which are crucial for subsequent steps in constructing similarity matrices.

• K-means method for anchor points
  We adopt the K-means clustering algorithm to derive a set of anchor points, which serve as representative prototypes for the underlying data distribution. Considering a dataset $X=(x_1,x_2,...,x_n)\in {\mathbb{R}}^{m\times n}$, the K-means algorithm aims to partition X into k disjoint clusters $C=\{C_1,C_2,...,C_k\}$ by minimizing the within-cluster sum of squares, formulated as [31]

  $$f\left(x\right)=\sum _{i=1}^k\sum _{x\in C_i}{\parallel x-u_i\parallel }^2,$$

  (16)

  where $C_i$ denotes the i-th cluster and $u_i$ is its centroid, namely the anchor points.
• BKHK method for anchor points
  The Balanced and Hierarchical K-means (BKHK) algorithm is a hierarchical anchor point selection method that combines K-means and hierarchical clustering to recursively construct evenly distributed anchor points, thereby improving representation ability. In contrast to conventional K-means algorithms that execute a single-step partitioning of the dataset into a predetermined number of clusters, BKHK employs a hierarchical partitioning strategy. It recursively divides the dataset X into m sub-clusters, performing binary K-means at each step. This process continues until the desired number of clusters is achieved. The objective function of Balanced Binary K-means is defined as follows [32]

  $$\underset{H\in {\mathbb{B}}^{n\times 2},1_n^{T}H=[\lfloor {\displaystyle \frac{n}{2}}\rfloor ,n-\lfloor {\displaystyle \frac{n}{2}}\rfloor ]}{\arg \min }{\parallel X-C{H}^T\parallel }_F^2,$$

  (17)

  where $C\in {\mathbb{R}}^{d\times 2}$ represents the two class centers, and $H\in {\mathbb{B}}^{n\times 2}$ is the class indicator matrix.

The construction of the similarity matrix based on representative anchors is a well-explored problem. In our approach, we choose to measure the distance using the Euclidean distance. To be specific, our objective is to learn a similarity matrix $Z\in {\mathbb{R}}^{n\times m}$, in a way that a smaller distance corresponds to a larger affinity value. This allows us to formulate the optimization problem as follows

$$\underset{z^i{1}_m=1,z^i>0}{\arg \min }\sum _{j=1}^m{\parallel x_i-u_j\parallel }_2^2+\gamma \sum _{j=1}^m{z}_{ij}^2.$$

(18)

In many cases, we prefer Z to be a sparse matrix, such that $z^i$ has k nonzero values corresponding to the k-nearest anchors. Consequently, the maximal $\gamma$ is determined by solving the following problem

$$\underset{\gamma ,\parallel z^i{\parallel }_0=k}{max}\gamma .$$

(19)

Suppose $e_{ij}={\parallel x_i-u_i\parallel }_2^2$, and $e_{ij}$ are arranged from small to large. The solution to (19) is $\gamma ={\displaystyle \frac{k}{2}}{e}_{i,k+1}-{\displaystyle \frac{1}{2}}{\sum }_{j=1}^k{e}_{ij}$. Therefore, the solution to (18) can be obtained as follows

$$z_{ij}=\left\{\begin{array}{cc}{\displaystyle \frac{e_{i,k+1}-e_{ij}}{k{e}_{i,k+1}-{\sum }_{h=1}^k{e}_{ih}}},\hfill & j\le k,\hfill \\ 0,\hfill & j>k.\hfill \end{array}\right.$$

(20)

After obtaining the matrix Z, we construct the similarity matrix A using the formula

$$A=Z{\Delta }^{-1}{Z}^T.$$

(21)

Here, $\Delta \in {\mathbb{R}}^{m\times m}$ is a diagonal matrix with the j-th diagonal element calculated as ${\sum }_{i=1}^n{z}_{ij}$. In practical applications, we can obtain A without explicitly computing it by utilizing $A=P{P}^T$, where $P=Z{\Delta }^{-{\displaystyle \frac{1}{2}}}$.

2. Methodology

The sample dataset comprises l color facial images, where n labeled images form the training set X, denoted as $X=\{X_1,...,X_n\}$, and $l-n$ unlabeled images constitute the test set Y, denoted as $Y=\{X_{n+1},...,X_l\}$. Each color facial image is initially represented as a quaternion matrix $Q=Q_0+Q_{1}i+Q_{2}j+Q_{3}k\in {\mathbb{Q}}^{p\times q}$, where $\mathbb{Q}$ denotes the quaternion space. To facilitate numerical computation, the quaternion matrix Q is transformed into its real representation matrix ${\gamma }_Q\in {\mathbb{R}}^{4p\times 4q}$. Due to the information redundancy of the real representation matrix, the first column block of ${\gamma }_Q$, ${[Q_0,Q_2,Q_1,Q_3]}^T$ is vectorized into $X_i\in {\mathbb{R}}^{m\times 1}$, where $m=4p\times q$. The data matrix X and the test matrix Y are constructed by column-wise concatenation of these vectors, resulting in $X=\{X_1,...,X_n\}\in {\mathbb{R}}^{m\times n}$ and $Y=\{X_{n+1},...,X_l\}\in {\mathbb{R}}^{m\times (l-n)}$.

In robust principal component analysis (RPCA), it is well-established that data signals are typically decomposed into a sum of low-rank terms, sparse terms, and noise terms that are statistically modeled. This decomposition can be mathematically expressed as

$$X=L+S+E.$$

(22)

The columns of matrix L span a low dimensional subspace, while matrix S contains only a sparse set of non-zero entries. Typically, it is assumed that the elements of matrix E are drawn from a Gaussian distribution with a mean of zero. The low-rank component L can be factorized as $L=U{V}^T$ , allowing equation (22) to be reformulated as

$$X=U{V}^T+S+E,$$

(23)

where $U\in {\mathbb{R}}^{m\times k},V\in {\mathbb{R}}^{n\times k}$ and $k\ll m,n$. It is important to note that, due to the inherent constraints imposed by this factorization, a low-rank structure is implicitly enforced on matrix L, even though k might still exceed the true rank of L. In equation (23), the columns of matrix U define the low-dimensional subspace in which the columns of matrix L reside, while matrix V represents the corresponding coefficients within this subspace.

To capture the geometric structures of the sample and feature manifolds in facial data, we use two graphs, namely a sample graph for columns and a feature graph for rows, each tailored to its respective dimension.

Initially, a k-nearest-neighbors sample graph is constructed, with its vertices corresponding to the data points $(x_1,...,x_n)$. We adopt a 0-1 weighting scheme to establish the k-nearest-neighbors data graph. Consequently, the sample weight matrix can be formally defined as

$${\left[W_V\right]}_{ij}=\left\{\begin{array}{cc}1,\hfill & if{x}_j\in N\left(x_i\right),\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.i,j=1,...,n,$$

(24)

where $N\left(x_i\right)$ is the k-nearest neighbor of $x_i$ and $L_V=D_V-W_V$ is the graph Laplacian matrix of the sample, where ${\left[D_V\right]}_{ii}={\sum }_j{\left[W_V\right]}_{ij}$ is a diagonal degree matrix.

Similarly to the methodology employed for constructing the sample weight matrix, we obtain the feature weight matrix as follows

$${\left[W_U\right]}_{ij}=\left\{\begin{array}{cc}1,\hfill & if{x}^{j,T}\in N\left(x^{i,T}\right),\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.i,j=1,...,m.$$

(25)

The feature graph Laplacian matrix is defined as $L_U=D_U-W_U$.

Via integrating graph regularization for both sample and feature manifolds with the RPCA model, we propose a novel dual graph regularized principal component analysis model. This model is designed to simultaneously capture the inherent geometric structures within the data across both manifolds. The corresponding objective function is formulated as follows

$$\begin{array}{cc}\hfill \underset{U,V}{min}{\parallel X-U{V}^T-S\parallel }_F^2& +\gamma Tr\left(U^T{L}_{U}U\right)+\gamma Tr\left(V^T{L}_{V}V\right)+\lambda {\parallel S\parallel }_1,\hfill \\ & s.t.U^{T}U=I,\hfill \end{array}$$

(26)

where the parameter $\gamma$ is used to balance the contribution of the graph Laplacian regularization term and the parameter $\lambda$ is used to balance the contribution of the sparse term.

We employ the Augmented Lagrangian Multiplier (ALM) method to optimize the objective function. This approach reformulates the constrained optimization problem into a sequence of unconstrained subproblems by incorporating Lagrange multipliers and penalty terms. Through this iterative process, the method progressively converges towards the optimal solution. In the context of utilizing the ALM method to derive the optimal solution, we substitute Z for $U{V}^T+S$, and (26) can be equivalently reformulated as follows

$$\begin{array}{cc}\hfill \underset{U,V}{min}{\parallel X-Z\parallel }_F^2& +\gamma Tr\left(U^T{L}_{U}U\right)+\gamma Tr\left(V^T{L}_{V}V\right)+\lambda {\parallel S\parallel }_1,\hfill \\ & s.t.U^{T}U=I,Z=U{V}^T+S.\hfill \end{array}$$

(27)

According to the ALM method, (27) can be equivalently minimized as

$$\begin{array}{cc}\hfill L_{\mu ,\nu ,Y,\Gamma }(S,Z,U,V)=& {\parallel X-Z\parallel }_F^2+\gamma Tr\left(U^T{L}_{U}U\right)+\gamma Tr\left(V^T{L}_{V}V\right)\hfill \\ & +{\lambda \parallel S\parallel }_1+Tr\left(Y^T(Z-U{V}^T-S)\right)+{\displaystyle \frac{\mu }{2}}{\parallel Z-U{V}^T-S\parallel }_F^2\hfill \\ & +Tr\left({\Gamma }^T(U^{T}U-I)\right)+{\displaystyle \frac{\nu }{2}}{\parallel U^{T}U-I\parallel }_F^2,\hfill \end{array}$$

(28)

where $\Gamma$ and $\Lambda$ are Lagrange multipliers, and $\mu$ and $\nu$ are the step size in the update rule.

Given that there are four variables requiring solution, the Alternating Direction Method (ADM) is employed to address this problem. It simplifies the solution process by allowing us to solve for a single variable while keeping the others fixed. By applying this method, the optimization problem represented by (28) can be naturally decomposed into four subproblems.

Problem 1: With variables $U,V,Z$ fixed, the variable S is solved by rewriting (28)

$$L_{\mu ,\nu ,Y,\Gamma }(S,Z,U,V)={\lambda \parallel S\parallel }_1+{\displaystyle \frac{\mu }{2}}{\parallel Z-U{V}^T-S+{\displaystyle \frac{Y}{\mu }}\parallel }_F^2.$$

(29)

(29) can be solved by the proximal shrink operator denoted as follows

$$\begin{array}{cc}\hfill S^{k+1}& =soft(Z^{k+1}-U^{k+1}{V}^{k+1,T}+{\displaystyle \frac{Y^k}{\mu }},{\displaystyle \frac{\lambda }{\mu }}),\hfill \\ & soft(a,\tau )=sign\left(a\right)max\left(\right|a|-\tau ,0).\hfill \end{array}$$

(30)

Problem 2: With variables $V,S,Z$ fixed, the variable U is solved by rewriting (28)

$$\begin{array}{cc}\hfill L_{\mu ,\nu ,Y,\Gamma }(S,Z,U,V)=& \gamma Tr\left(U^T{L}_{U}U\right)+Tr\left(Y^T(Z-U{V}^T-S)\right)\hfill \\ & +{\displaystyle \frac{\mu }{2}}\parallel Z-U{V}^T{-S\parallel }_F^2+Tr\left({\Gamma }^T(U^{T}U-I)\right)+{\displaystyle \frac{\nu }{2}}{\parallel U^{T}U-I\parallel }_F^2,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill L=& \gamma Tr\left(U^T{L}_{U}U\right)-Tr\left(V^T{Y}^{T}U\right)-\mu Tr\left(Z^{T}U{V}^T\right)+{\displaystyle \frac{\mu }{2}}Tr\left(V{U}^{T}U{V}^T\right)\hfill \\ & +\mu Tr\left(V^T{S}^{T}U\right)+Tr\left({\Gamma }^T{U}^{T}U\right)+{\displaystyle \frac{\nu }{2}}{\parallel U^{T}U-I\parallel }_F^2.\hfill \end{array}$$

(32)

By computing the partial derivative with respect to the variable U in (32), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial L}{\partial U}}& =2\gamma L_{U}U-YV-\mu ZV+\mu U\left(V^{T}V\right)\hfill \\ & +\mu SV+2U\Gamma +\nu (2U{U}^{T}U-U).\hfill \end{array}$$

(33)

By setting the equation (33) equal to 0 , the updated iteration formula for variable U can be addressed via the Sylvester equation, as showed below

$$(2\gamma L_U+\nu I)U+U(\mu V^{T}V+2\Gamma )=YV+\mu (Z-S)V.$$

(34)

Problem 3: With variables $U,S,Z$ fixed, the variable V is solved by rewriting (28)

$$\begin{array}{cc}\hfill L_{\mu ,\nu ,Y,\Gamma }(S,Z,U,V)=& \gamma Tr\left(V^T{L}_{V}V\right)+Tr\left(Y^T(Z-U{V}^T-S)\right)\hfill \\ & +{\displaystyle \frac{\mu }{2}}{\parallel Z-U{V}^T-S\parallel }_F^2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill L=& \gamma Tr\left(V^T{L}_{V}V\right)-Tr\left(V^T{Y}^{T}U\right)-\mu Tr\left(Z^{T}U{V}^T\right)\hfill \\ & +{\displaystyle \frac{\mu }{2}}Tr\left(V{U}^{T}U{V}^T\right)+\mu Tr\left(V^T{S}^{T}U\right).\hfill \end{array}$$

(36)

Similar to the procedure outlined in (33), we derive the partial derivative of the variable U in equation (36)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial L}{\partial V}}& =2\gamma L_{V}V-Y^{T}U-\mu Z^{T}U+\mu V\left(U^{T}U\right)+\mu S^{T}U.\hfill \end{array}$$

(37)

The updated iteration formula for variable V also involves solving the Sylvester equation, as detailed below

$$2\gamma L_{V}V+V\left(\mu U^{T}U\right)=Y^{T}U+\mu (Z^T-S^T)U.$$

(38)

Problem 4: With variables $U,V,S$ fixed, the variable Z is solved by rewriting (28)

$$\begin{array}{c}\hfill L_{\mu ,\nu ,Y,\Gamma }(S,Z,U,V)={\parallel X-Z\parallel }_F^2+{\displaystyle \frac{\mu }{2}}{\parallel Z-U{V}^T-S+{\displaystyle \frac{Y}{\mu }}\parallel }_F^2\end{array}.$$

(39)

The problem described by equation (39) is convex, which facilitates a straightforward update of the variable Z as follows

$$Z^{k+1}={\displaystyle \frac{2X+\mu (U{V}^T+S-\frac{Y}{\mu })}{2+\mu }}.$$

(40)

Algorithm 1: The solution to optimize (26)

Require:  Facial image data matrix: $X_{m\times n}$, Parameters: $k,r,\lambda ,\gamma ,\mu ,\nu$ Ensure:  $U_{m\times k},V_{n\times k}$ 1: Initialize $Z,Y,S,U,V$ 2: repeat 3:    Update S by (30). 4:    Update U by (34). 5:    Update V by (38). 6:    Update Z by (40). 7:    Update Y by $Y^{r+1}=Y^r+{\mu }^r(Z^T-U{V}^T-S)$. 8:    Update $\mu$ by ${\mu }^{r+1}=\rho {\mu }^r$. 9: until convergence

3. Experiments

3.1. Datasets

The proposed model is rigorously evaluated on the Georgia Tech face dataset [33], a well-established benchmark in face recognition research comprising facial images from 50 distinct subjects, with each subject contributing 15 unique samples. These samples exhibit significant variations in lighting conditions, facial expressions (e.g., neutral, smiling, surprised), and head poses (e.g., frontal, side views), rendering the dataset highly effective for assessing the robustness and generalization capabilities of face recognition models. This diversity facilitates robust assessment of model generalization under challenging real-world conditions. All images in the GTdatabase are preprocessed by manual cropping and resizing to a uniform resolution of $44\times 33$ pixels, with representative samples visualized in Fig.Figure 1.

3.2. Parameter Selection

Our model involves six critical parameters: $\gamma$, $\lambda$, $\mu$, $\nu$, k and r, which must be carefully tuned in the formulation (28). Based on empirical evidence from prior studies and our experimental validation, we set $\gamma =0.005,\mu =0.005$ and $\nu =0.001$. For the regularization parameter $\gamma$, we adopt $\lambda ={\displaystyle \frac{1}{\sqrt{max(m,n)}}}$ as suggested in [5]. Additionally, we select $k=15$ as the number of nearest neighbors for graph construction, ensuring a balanced representation of local data structures. Furthermore, we respectively set $r=200$ and $r=100$ to compare the final results. These parameter choices are carefully calibrated to optimize the model’s performance across diverse scenarios.

3.3. Experiment Results

Using the learned projection matrix U, we project the test samples into the low-dimensional space and compute their similarity to the training samples in this reduced space using Euclidean distance. This enables nearest neighbor classification, where recognition results are determined by identifying the closest matches based on similarity. To evaluate the model’s performance, we employ two metrics: the standard accuracy (ACC) and the convergence time. ACC measures the face matching accuracy, defined as the ratio of correctly classified samples to the total number of test samples, providing a quantitative assessment of the model’s recognition capability. Specifically, ACC is calculated as

$$ACC={\displaystyle \frac{{\sum }_{i=1}^n\sigma (p_i,map\left(q_i\right))}{n}}\times 100\%,$$

(41)

where $p_i$ is the predicted label of a test sample and $map\left(q_i\right)$ is the real label of the test sample. $\sigma (x,y)$ equals 1 if $x=y$ and 0 otherwise.

For the $50\times 15$ samples in the GTdatabase, we partitioned the dataset into training and test sets with a ratio of 0.7. To ensure statistical reliability and robustness, the results reported are the average values obtained from 10 independent trials, providing a more precise and representative assessment of the model’s performance.

For the $k=15$-nearest neighbors, we set the dimensionality reduction parameter to $r=200$. The corresponding accuracy and runtime of the model, evaluated using the original graph Laplacian matrix, the graph Laplacian matrix constructed via K-means, and the graph Laplacian matrix generated by the BKHK method, are summarized in Table 2.

Similarly, for the $k=15$-nearest neighbors, we conducted an additional experiment with the dimensionality reduction parameter set to $r=100$. The accuracy and runtime of the model with original graph Laplacian matrix, graph Laplacian matrix constructed via K-means, and graph Laplacian matrix generated by the BKHK method are summarized in Table 3.

From the experimental results summarized in Table 2 and Table 3, we can draw the following conclusions. First, all three methods—standard graph Laplacian, K-means-based graph Laplacian, and BKHK-based graph Laplacian—demonstrate high recognition accuracy, with the standard method exhibiting slightly superior performance due to its ability to capture the underlying data structure more comprehensive. Second, the anchor-based methods, particularly the BKHK approach, exhibit significantly lower computational complexity, especially in lower-dimensional settings (e.g., r = 100). This efficiency advantage positions the BKHK-based method as a highly scalable solution for large-scale datasets, where computational efficiency is paramount. In conclusion, while all methods deliver comparable accuracy, the anchor-based approaches, particularly BKHK, provide substantial computational savings, making them more suitable for real-world applications requiring both performance and scalability. 

4. Conclusions

In this study, we present an advanced RPCA framework incorporating bidirectional graph Laplacian constraints and an anchor point strategy, achieving enhanced computational efficiency while preserving recognition accuracy in facial analysis. The dual-graph architecture captures intrinsic geometric relationships from both sample and feature perspectives, maintaining structural and discriminative characteristics of high-dimensional data. Anchor points effectively reduce computational complexity without sacrificing robustness. Extensive experiments on the GTdatabase demonstrate superior processing speed and sustained accuracy under challenging conditions including illumination variations, expressions, and pose changes. Our work advances dimensionality reduction techniques while providing a practical solution for face recognition. Future research will extend this framework to larger datasets and explore broader applications in computer vision.