Robust multi-view locality preserving regression embedding

GE

The single-view feature extraction methods based on sample structures can be unified under the GE framework. In GE, a similarity matrix $W^S\in {\mathbb{R}}^{n\times n}$ is constructed using distinct structural information of orignal samples, where ${W^S}_{ij}$ represents the similarity of samples $x_i$ and $x_j$. This framework entails learning the projection matrix $P$ by embedding the graph structures into a regression model. The GE framework can be summarized as follows:

(1) $$\begin{array}{r}{\mathrm{m}\mathrm{i}\mathrm{n}}_{P}G(Y)=\sum _{i,j=1}^n\parallel y_i-y_j{\parallel }_2^2{W}_{ij}^s\\ \mathrm{s}.\mathrm{t}.YH{Y}^T=Ior{P}^{T}P=I,\end{array}$$where I is an identity matrix. $H$ represents the constraint matrix designed to prevent a trivial solution for the objective function. Therefore, the specific models in the GE framework are determined by the specific constraint and the matrices $W^S$ and $H$. The above objective function can be further transformed into:

(2) $$G(Y)=Y^{T}LY,$$where $L$ is the Laplacian matrix of $W^S$.

PCA

PCA is an unsupervised GE model that aims to identify a set of orthogonal axes representing directions with the highest variance in the original data. It then projects the data onto the first $m$ of these axes, selectively retaining the feature dimensions that contain the majority of the variance while disregarding dimensions with nearly zero variance, resulting in the reduction of data feature dimensionality. The constraint of PCA is given by:

(3) $$P^{T}P=I.$$

The similarity matrix $W^S$ is defined as follows:

(4) $${W^S}_{ij}=\{\begin{array}{cc}\frac{1}{n},& \mathrm{i}\mathrm{f}i\ne j\hfill \\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\hfill \end{array}$$

Thus, the Laplacian matrix of $W^S$ is computed as follows:

(5) $$L=I-\frac{1}{n}11^T,$$where $1$ is an n-dimensional column vector of all 1.

LDA

LDA is a supervised GE model that aims to ensure that during the projection process, the data in the new subspace has the maximum inter-class distance and the minimum intra-class distance. This is done to enhance the separability of the data after projection, aiming to improve classification or clustering performance. The constraint of LDA is given by:

(6) $$YH{Y}^T=I,$$where

(7) $$\begin{array}{l}H=I-\frac{1}{n}11^T.\end{array}$$

The similarity matrix $W^S$ is defined as follows:

(8) $$W_{ij}^S=\{\begin{array}{cc}\frac{1}{n_i},\hfill & \mathrm{i}\mathrm{f}{\mathrm{c}}_{\mathrm{i}}={\mathrm{c}}_{\mathrm{j}}\hfill \\ 0,\hfill & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\hfill \end{array}$$

Thus, the Laplacian matrix of $W^S$ is computed as follows:

(9) $$L=I-{\sum }_{c=1}^{n_c}\frac{1}{n_c}{1}^c{1^c}^T,$$

$n_c$ is the number of samples in $c$ th class. $1^c$ is an n-dimensional column vector with $1^c(i)=1$ if $c=c_i$; 0 otherwise.

MRE

To comprehensively account for the consistency and complementarity information among views while extending the single-view GE framework to multi-view feature extraction, we apply GE by incorporating the structural information from distinct views into shared embedding. Additionally, we introduce adaptive weights to the regression terms and the GEs in distinct views. The optimization problem is as follows:

(10) $$\begin{array}{ll}& \underset{\alpha ,Y,P_{\left(i\right)}}{min}\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{\left(i\right)})+\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2],\\ & \text{ s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1,\\ & YH{Y}^T=I\text{or}{P^T}_{\left(i\right)}{P}_{\left(i\right)}=I,i=1,...,v,\end{array}$$where

(11) $$G(Y,X^{(i)})=Y{L}^{(i)}{Y}^T,i=1,\dots ,v,$$

The parameter $r$ is a positive value, and $\alpha =[{\alpha }_1,\dots ,{\alpha }_v]$. $L^{(i)}$ represents the Laplacian matrix of the similarity matrix in $i$ th view. In this context, the second term of the objective function employs regression techniques to fit linear projections of different views to the shared embedding $Y$. This fitting process yields fitting residuals $Y_r^{(i)}$, which has the following representation:

(12) $$Y=P_{(i)}^T{X}^{(i)}+Y_r^{(i)},i=1,2,\dots ,v,$$

This implies that the shared embedding $Y$ is a nonlinear embedding due to this regression-based transformation. Therefore, this approach effectively prevents the loss of crucial information associated with linear projection methods.

To avoid over-fitting and prevent degenerate trivial solutions, we introduced a regularization term for the norm of the projection matrices $\Vert P_{(i)}^T{\Vert }_F^2$. Additionally, we introduced a structured regularization term $\Vert Y_i-\overline{Y}{\Vert }_F^2$ to maximize the overall divergence of the samples after projection, thereby retaining more complete information from each view. Here, $\overline{Y}$ is calculated as:

(13) $$\overline{Y}=\frac{1}{v}Y\cdot 1\cdot 1^T.$$

The optimization problem is as follows:

(14) $$\begin{array}{ll}& \underset{{\alpha }_i^r,Y,P_{\left(i\right)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{\left(i\right)})+\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{\left(i\right)}^T{\Vert }_F^2]}{\Vert Y_i-\bar{Y}{\Vert }_F^2},\\ & \text{ s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

In Eq. (14), the constraint $YH{Y}^T=I$ or ${P^T}_{(i)}{P}_{(i)}=I$, $i=1,\dots ,v$ has been replaced by the regularization term $||P_{(i)}^T|{|}_F^2$ in the objective function. Therefore, it can be omitted from the constraints.

MLPRE

To further explore the consistency information, we leverage the shared $k$-nearest neighbor structure across all views to constrain the shared embedding. The optimization problem can be stated as follows:

(15) $$\begin{array}{rl}& \underset{{\alpha }_i,Y,P_{\left(i\right)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{\left(i\right)})+\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{\left(i\right)}^T{\Vert }_F^2+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP\left(i\right)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\bar{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1,\end{array}$$where

(16) $$W_{jk}^{LP(i)}=\{\begin{array}{cc}exp\left(\frac{-\Vert x_j^{(i)}-x_k^{(i)}{\Vert }_2^2}{t}\right),\hfill & x_i\text{ and }{x}_j\text{ are}k\text{-neighbors in all views}.\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$

Note that GE uses the graph structure under each view to constrain the shared embedding $Y$ separately, whereas here the graph structure common to all views is used to constrain the shared embedding $Y$. Thus it further facilitates the exploration of the consistency information, and thus also weakens the detrimental effect of the inaccuracy of the graph structure in a particular view due to the noise and redundant features.

RMLPRE

To enhance robustness against noisy data, we employ distinct norms for the projection matrices within the proposed framework. The resulting model is expressed as follows:

(17) $$\begin{array}{rl}& \underset{{\alpha }_i,Y,P_{\left(i\right)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{\left(i\right)})+\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{\left(i\right)}^T{\Vert }_{\beta }+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP\left(i\right)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\bar{Y}{\Vert }_F^2},\\ & \text{ s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

In Eq. (17), the symbol $\Vert \cdot {\Vert }_{\beta }$ denotes a particular matrix norm, for example L1-norm or L2,1-norm. Notably, research based on L1-norm and L2,1-norm has shown that these norms exhibit improved performance in the presence of outliers compared to F-norm-based methods. Moreover, L2,1-norm-based techniques are often easier to solve than their L1-norm counterparts.

Framework application

In this section, we extend the single-view PCA and LDA into six multi-view feature extraction models by the three frameworks proposed above, respectively, namely pcMRE, pcMLPRE, pcRMLPRE, daMRE, daMLPRE, and daRMLPRE.

To derive the optimization problems for pcMRE, pcMLPRE, and pcRMLPRE, the Laplacian matrix for each view is calculated as follows:

(18) $$L^{(i)}=I-\frac{1}{n}1\cdot 1^T,i=1,\dots ,v,$$with the L2,1-norm regularization applied to objective function of pcRMLPRE. The optimization problems can be formulated as follows:

1) pcMRE:

(19) $$\begin{array}{ll}& \underset{{\alpha }_i,Y,P_{(i)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[tr(Y(I-\frac{1}{n}1\cdot 1^T)Y^T)+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_F^2]}{\Vert Y-\bar{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

2) pcMLPRE:

(20) $$\begin{array}{rl}& \underset{{\alpha }_i,Y,P_{(i)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[tr(Y(I-\frac{1}{n}1\cdot 1^T)Y^T)+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_F^2+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP(i)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\bar{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

3) pcRMLPRE:

(21) $$\begin{array}{ll}& \underset{{\alpha }_i,Y,P_{(i)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[tr(Y(I-\frac{1}{n}1\cdot 1^T)Y^T)+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_{2,1}+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP(i)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\overline{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

To derive the optimization problems for daMRE, daMLPRE and daRMLPRE, the Laplacian matrix for each view is calculated as follows:

(22) $$L^{(i)}=I-\sum _{c=1}^{n_c}\frac{1}{n_c}{1}^c\cdot {1^c}^T,i=1,\dots ,v,$$with the L2,1-norm regularization applied to objective function of daRMLPRE. The optimization problems can be formulated as follows:

1) daMRE:

(23) $$\begin{array}{rl}& \underset{{\alpha }_i,Y,P_{(i)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[tr(Y(I-\sum _{c=1}^{n_c}\frac{1}{n_c}{1}^c\cdot {1^c}^T)Y^T)+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_F^2]}{\Vert Y-\bar{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

2) daMLPRE:

(24) $$\underset{{\alpha }_i,Y,P_{(i)}}{min}$$

(25) $$\begin{array}{ll}& \frac{\sum _{i=1}^v{\alpha }_i^r[tr(Y(I-\sum _{c=1}^{n_c}\frac{1}{n_c}{1}^c\cdot {1^c}^T)Y^T)+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_F^2+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP(i)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\overline{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

3) daRMLPRE:

(26) $$\underset{{\alpha }_i,Y,P_{(i)}}{min}$$

(27) $$\begin{array}{ll}& \frac{\sum _{i=1}^v{\alpha }_i^r[tr(Y(I-\sum _{c=1}^{n_c}\frac{1}{n_c}{1}^c\cdot {1^c}^T)Y^T)+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_{2,1}+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP(i)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\overline{Y}{\Vert }_F^2},\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1.\end{array}$$

In summary, the integration of the Laplacian matrices $L^{(i)}$, $i=1,2,\dots ,v$ from PCA and LDA into our proposed frameworks allows for the development of tailored methods. Moreover, the Laplacian matrices from other single-view graph embedding methods can also be seamlessly incorporated into these frameworks. As a result, our frameworks effectively bridge the gap, expanding single-view graph embedding techniques into the domain of multi-view feature extraction and, in turn, propelling the progress of multi-view learning.

Optimization strategy

We designed a simplified iterative algorithm to replace the traditional alternating iterative algorithm to reduce the cost of each iteration.

Optimization of MRE and MLPRE

Let Eq. (15) be $f(Y,P_{(i)},{\alpha }_i)$:

(28) $$\frac{\mathrm{\partial }f}{\mathrm{\partial }{P}_{(i)}^T}=g(P_{(i)}^T{X}^{(i)}{X}^{(i)T}+\gamma P_{(i)}^T),$$where $g$ is independent of $P_{(i)}^T$. Let the partial derivative be $0$:

(29) $$P_{(i)}^T=Y{X}^{(i)T}(X^{(i)}{X}^{(i)T}+\gamma I{)}^{-1}.$$

Substituting Eq. (29) into Eq. (15), we have

(30) $$\begin{array}{l}\underset{{\alpha }_i,Y,P_{(i)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{(i)})+\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_F^2+\frac{\lambda }{2}\sum _{j,k=1}^n{W}_{jk}^{LP(i)}\Vert Y_j-Y_k{\Vert }_F^2]}{\Vert Y-\overline{Y}{\Vert }_F^2}\\ =>\underset{{\alpha }_i,Y}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{(i)})+tr[Y(M^{(i)})Y^T]]}{tr[YN{Y}^T]},\end{array}$$where

(31) $$M^{(i)}=I-X^{(i)T}(X^{(i)}{X}^{(i)T}+\gamma I{)}^{-1}{X}^{(i)}+\frac{\lambda }{{\alpha }_i^r}(D-W)$$and

(32) $$N=I-\frac{1}{n}1\cdot 1^T.$$

If $G(Y,X^{(i)})$ can be written as $tr[[Y(L^{(i)})Y^T]$, we have

(33) $$\begin{array}{ll}& \underset{{\alpha }_i,Y}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{(i)})+tr[Y(M^{(i)})Y^T]]}{tr[YN{Y}^T]}\\ =>& \underset{{\alpha }_i,Y}{min}\frac{tr\left[Y\left(\sum _{i=1}^v{\alpha }_i^r(L^{(i)}+M^{(i)})\right)Y^T\right]}{tr\left[YN{Y}^T\right]}\\ =>& \underset{{\alpha }_i,Y}{min}tr\left[Y\left(\sum _{i=1}^v{\alpha }_i^r(L^{(i)}+M^{(i)})\right)Y^T\right],s.t.YN{Y}^T=I.\end{array}$$

Step 1: Updating $Y$ while fixing ${\alpha }_i$. By constructing the multivariate Lagrange function and set the partial derivative of $Y^T$ to $0$, the optimization problem Eq. (33) can be solved by the following generalized eigenvalue problem

(34) $$\left(\sum _{i=1}^v{\alpha }_i^r(L^{(i)}+M^{(i)})\right)Y^T=\mu N{Y}^T.$$

Step 2: Updating ${\alpha }_i$ while fixing $Y$. The optimal solution of problem Eq. (33) can be calculated as:

(35) $$\begin{array}{r}{\alpha }_i=\frac{[1/tr(Y(L^{(i)}+M^{(i)}))Y^T){]}^{(1/(r-1))}}{\sum _{i=1}^v[1/tr(Y(L^{(i)}+M^{(i)}))Y^T){]}^{(1/(r-1))}}.\end{array}$$

We alternatively update $Y$ and ${\alpha }_i$ until convergence. $P_{(i)}^T$ can be solved by Formula (29). The complete procedures are described in Algorithm 1.

Optimization of RMLPRE

Substituting Eq. (29) into Eq. (17), Eq. (17) can be further written as follows:

If $G(Y,X^{(i)})$ can be written as $tr[Y(L^{(i)})Y^T]$, we have

(36) $$\begin{array}{ll}& \underset{{\alpha }_i,Y,P_{\left(i\right)}}{min}\frac{\sum _{i=1}^v{\alpha }_i^r[G(Y,X^{\left(i\right)})+\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{\left(i\right)}^T{\Vert }_{2,1}+\lambda \sum _{i,j=1}^n\frac{1}{2}{W}_{ij}^{LP}\Vert Y_i-Y_j{\Vert }_F^2]}{\Vert Y-\overline{Y}{\Vert }_F^2}\\ =>& \underset{{\alpha }_i,Y,P_{\left(i\right)}}{min}\sum _{i=1}^v{\alpha }_i^r[\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{\left(i\right)}^T{\Vert }_{2,1}+tr\left[Y[L^{\left(i\right)}+\frac{\lambda }{{\alpha }_i}(D^{\left(i\right)}-W^{LP\left(i\right)})]Y^T\right]],\\ & \text{s.t. }{\alpha }_i\ge 0,\sum _{i=1}^v{\alpha }_i=1,YN{Y}^T=I,\end{array}$$where $N=I-\frac{1}{n}1\cdot 1^T$.

Step 1: Updating $P_{(i)}$ while fixing ${\alpha }_i$ and $Y$. Problem Eq. (36) can be derived by minimizing the following function

(37) $$\begin{array}{l}f(P_{(i)})={\alpha }_i^r[\frac{1}{2}\Vert P_{(i)}^T{X}^{(i)}-Y{\Vert }_F^2+\frac{\gamma }{2}\Vert P_{(i)}^T{\Vert }_{2,1}].\end{array}$$

Take the derivative of $f(P_{(i)})$ and set it to zero

(38) $$P_{(i)}^T=Y{X}^{(i)T}(X^{(i)}{X}^{(i)T}+\frac{1}{2}\gamma W^{(i)}{)}^{-1},$$where

(39) $$W^{(i)}=\left[\begin{array}{ccc}1\Vert P_{(i)1}^T{\Vert }_2\hfill & & \\ & \ddots & \\ & & 1\Vert P_{(i)d_i}^T{\Vert }_2\hfill \end{array}\right],$$and $d_i$ is the dimension of the $i$ th view.

Step 2: Updating $Y$ while fixing ${\alpha }_i$ and $P_{(i)}$. Problem Eq. (36) is equivalent to

(40) $$\begin{array}{ll}& \underset{{\alpha }_i,Y,P_{\left(i\right)}}{min}\sum _{i=1}^v{\alpha }_i^r[\frac{1}{2}\Vert P_{\left(i\right)}^T{X}^{\left(i\right)}-Y{\Vert }_F^2+tr\left[Y[L^{\left(i\right)}+\frac{\lambda }{{\alpha }_i^r}(D^{\left(i\right)}-W^{LP\left(i\right)})]Y^T\right]],\\ & \text{s.t.}YN{Y}^T=I.\end{array}$$

Substituting Eq. (38) into Eq. (40), the optimization equation can be solved by the following generalized eigenvalue problem

(41) $$[{\sum }_{i=1}^v{\alpha }_i^r{G}^{(i)}+\lambda (D-W^{LP})]Y^T=\mu N{Y}^T,$$where $G^{(i)}=X^{(i)T}[(X^{(i)}{X}^{(i)T}+\frac{1}{2}\gamma W^{(i)}{)}^{-1}{X}^{(i)}{X}^{(i)T}(X^{(i)}{X}^{(i)T}+\frac{1}{2}\gamma W^{(i)}{)}^{-T}-(X^{(i)}{X}^{(i)T}+\frac{1}{2}\gamma W^{(i)}{)}^{-1}-(X^{(i)}{X}^{(i)T}+\frac{1}{2}\gamma W^{(i)}{)}^{-T}]X^{(i)}+I+L^{(i)}$.

Step 3: Updating ${\alpha }_i$ while fixing $P_{(i)}$ and $Y$. The optimal solution of problem Eq. (36) can be calculated as:

(42) $$\begin{array}{r}{\alpha }_i=\frac{{\left[1/tr(Y{G}^{\left(i\right)}{Y}^T+P_{\left(i\right)}^T{W}^{\left(i\right)}{P}_{\left(i\right)})\right]}^{\left(1/(r-1)\right)}}{\sum _{i=1}^v{\left[1/tr(Y{G}^{\left(i\right)}{Y}^T+P_{\left(i\right)}^T{W}^{\left(i\right)}{P}_{\left(i\right)})\right]}^{\left(1/(r-1)\right)}}.\end{array}$$

We alternatively update $P_{(i)}^T$, $Y$ and ${\alpha }_i$ until convergence. The complete procedures are described in Algorithm 2.

Time complexity analysis

The time complexity of Algorithm 1 is mainly determined by the computational costs of its key equations. Assuming

(43) $$d_{max}=max\{d_{(i)}|i=1,2,\dots ,v\}.$$

For Algorithm 1, the time complexity of Eqs. (31), (34), (35) and (29) are $O(n\ast d_{max}^2+d_{max}^3+n^2\ast d_{max})$, $O(n^3)$, $O(n^2+m\ast n^2+n\ast m^2)$, $O(n\ast d_{max}^2+d_{max}^3+m\ast n\ast d_{max})$, respectively. Given that $m\ll d_{(i)}$, the main time complexity of Algorithm 1 is $O(d_{max}^3+n\ast d_{max}^2+n^2\ast d_{max}+n^3)$. Similarly, for Algorithm 2, the operations and their associated costs are comparable, resulting in the main time complexity of $O(d_{max}^3+n\ast d_{max}^2+n^2\ast d_{max}+n^3)$.

Datasets description

Coil (https://cave.cs.columbia.edu/repository/COIL-20): The Coil Dataset (Nene, Nayar & Murase, 1996), originating from Columbia University, contains a diverse collection of 1,400 images, featuring 20 different objects. Each object is represented by a substantial set of 72 images, offering rich variability for analysis.

Orl (http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html): Hailing from the Olivetti Laboratory in Cambridge, England, the ORL Dataset (Samaria & Harter, 1994) comprises 400 images showcasing the faces of 40 distinct individuals. These images were captured under a range of conditions, including differing lighting, positions, and expressions.

Yale (https://vision.ucsd.edu/datasets/yale-face-database): The Yale Face Dataset contains 165 images of 15 people’s faces, with variations in lighting, expression, and posture.

Our multi-view graph embedding method is designed to process data in vector form. For example, for the image data, we first preprocess it to extract numerical vector features of gray-scale intensity (GSI), local binary patterns (LBP), and histogram of oriented gradients (HOG), respectively. Then, the proposed method, as well as comparison methods, is applied to extract the features of these numerical vector representations. For some details, please refer to Table 1. As shown in Fig. 2, the image features extracted using the three described techniques display significant differences, representing three distinct perspectives of the image data. Furthermore, we evaluated the robustness of our proposed methods by introducing salt-and-pepper noise with densities of 0.1 and 0.3 to each dataset.

Experiments setup

In our experiments, we employed a random split of the data, allocating 80% of the samples for training and reserving the remaining 20% for testing. The number of training samples for each dataset is provided in Table 1. For multi-view datasets $\{X_1,\dots ,X_v\}$, once we obtained the projection matrices $\{P_1,\dots ,P_v\}$, we extracted embedding features for each view as follows:

(44) $$Y^{(i)}=P_{(i)}^T{X}^{(i)},i=1,\dots ,v.$$

We applied a 1NN classifier for classification, and these experiments were repeated five times. The evaluation criteria were based on the average classification accuracies of the embedding representations. Optimal parameters were determined through grid search with $\gamma ,\lambda \in \{2^{-5},2^{-3},2^{-1},2,2^3,2^5\}$. The parameters of the algorithms employed for comparison were rigorously adhered to as delineated within the original publication.

Experiment results

On the real-world datasets

In our results, we commence by presenting the classification accuracy results across feature dimensions ranging from 10 to 90 for each dataset. The summarized results for all methods can be found in Tables 2–4, with the best performance highlighted in bold. Furthermore, we provide a visual representation of the classification accuracy of all methods under various reduced dimensions for each dataset in Fig. 3. Based on our experimental findings, several key observations are noted.

Table 2:

1NN classification accuracy (%) for Coil Dataset.

Bold entries indicate the best results.

	10	20	30	40	50	60	70	80	90
PCA (Oh & Kwak, 2016)	9.03	9.14	9.18	9.20	9.17	9.19	9.20	9.20	9.21
LDA (Belhumeur, Hespanha & Kriegman, 1997)	48.53	49.42	49.52	49.63	49.66	49.67	49.68	49.69	49.69
ALPCCA (Wang & Zhang, 2013)	18.77	32.69	44.30	53.13	58.39	62.90	65.26	66.40	66.45
LPCCA (Sun & Chen, 2007)	41.81	44.13	45.82	46.04	46.47	46.82	47.18	47.45	47.38
MCCA (Rupnik & Shawe-Taylor, 2010)	46.04	59.38	65.39	69.65	72.03	73.78	75.21	76.21	77.04
DMCCA (Gao et al., 2018)	67.11	72.46	74.73	75.84	76.31	77.21	70.12	74.69	78.29
MvDA (Shu et al., 2019)	53.80	50.76	45.02	39.03	40.06	41.51	42.60	43.91	44.73
MvPLS (Cao et al., 2018)	44.06	44.89	45.13	45.08	44.98	44.80	44.65	44.64	44.56
EWNMF (Wei et al., 2023)	69.23	71.57	72.78	73.95	74.23	73.38	74.61	76.93	77.27
MUNPE (Jayashree, Shiva Prakash & Venugopal, 2024)	68.83	70.86	67.75	67.11	66.67	64.86	65.42	63.61	61.81
pcMRE	80.11	82.58	81.93	80.71	78.97	76.52	74.79	73.17	71.74
pcMLPRE	80.05	82.52	81.91	80.72	78.91	76.51	74.69	73.03	71.59
pcRMLPRE	78.83	80.94	80.40	80.03	79.34	78.64	78.13	78.08	78.44
daMRE	85.44	86.83	87.30	86.09	84.67	82.01	79.50	76.93	74.89
daMLPRE	85.94	86.84	87.21	86.41	85.37	84.08	83.65	83.41	83.23
daRMLPRE	84.46	86.53	84.12	81.73	80.32	79.65	78.96	78.44	78.45

DOI: 10.7717/peerj-cs.2619/table-2

Table 3:

1NN classification accuracy (%) for ORL dataset.

Bold entries indicate the best results.

	10	20	30	40	50	60	70	80	90
PCA (Oh & Kwak, 2016)	8.80	9.38	9.43	9.43	9.50	9.45	9.50	9.50	9.53
LDA (Belhumeur, Hespanha & Kriegman, 1997)	28.38	29.13	29.38	29.63	29.63	29.63	29.63	29.75	29.75
ALPCCA (Wang & Zhang, 2013)	10.42	18.33	26.04	32.10	38.29	44.58	49.79	52.56	55.92
LPCCA (Sun & Chen, 2007)	23.38	35.94	41.42	43.08	45.40	48.15	50.67	53.65	56.25
MCCA (Rupnik & Shawe-Taylor, 2010)	15.63	27.50	38.75	45.50	50.83	57.67	61.92	66.17	68.92
DMCCA (Gao et al., 2018)	55.50	68.25	74.92	78.67	80.04	81.29	83.33	84.29	85.63
MvDA (Shu et al., 2019)	56.96	56.33	60.17	53.75	45.08	45.17	44.79	44.67	45.21
MvPLS (Cao et al., 2018)	41.58	44.54	45.71	46.08	46.21	46.33	46.04	46.08	46.13
EWNMF (Wei et al., 2023)	65.50	73.16	81.86	85.20	86.47	87.19	87.41	87.85	88.26
MUNPE (Jayashree, Shiva Prakash & Venugopal, 2024)	49.58	55.00	55.83	58.33	60.00	55.42	60.00	59.58	54.17
pcMRE	68.15	81.83	86.08	88.50	88.08	89.42	88.58	88.67	87.67
pcMLPRE	68.17	82.00	86.08	88.42	87.92	89.17	88.67	88.83	87.75
pcRMLPRE	72.58	83.25	85.08	86.83	87.67	89.33	89.42	88.58	87.75
daMRE	80.79	86.17	86.97	90.91	92.42	89.17	89.92	90.83	90.00
daMLPRE	80.83	86.25	86.92	90.83	91.75	88.58	88.33	89.75	89.42
daRMLPRE	80.17	87.42	89.33	94.17	92.08	90.42	87.75	86.25	84.92

DOI: 10.7717/peerj-cs.2619/table-3

Table 4:

1NN classification accuracy (%) for Yale Face Database.

Bold entries indicate the best results.

	10	20	30	40	50	60	70	80	90
PCA (Oh & Kwak, 2016)	7.20	7.96	8.13	8.27	8.31	8.27	8.44	8.49	8.49
LDA (Belhumeur, Hespanha & Kriegman, 1997)	21.56	22.22	22.22	22.22	22.22	22.22	22.22	22.22	22.22
ALPCCA (Wang & Zhang, 2013)	31.52	50.78	60.33	65.59	69.11	71.37	72.26	73.85	73.70
LPCCA (Sun & Chen, 2007)	24.56	30.48	35.74	38.30	40.74	42.41	43.56	44.52	45.37
MCCA (Rupnik & Shawe-Taylor, 2010)	43.33	56.74	63.04	66.44	69.41	70.22	71.48	72.74	72.89
DMCCA (Gao et al., 2018)	80.30	87.63	92.74	94.67	89.56	80.30	73.85	75.04	76.37
MvDA (Shu et al., 2019)	36.30	38.81	39.04	40.30	40.30	40.44	40.59	40.59	40.74
MvPLS (Cao et al., 2018)	36.15	39.33	40.00	40.07	40.22	40.07	40.44	40.74	41.26
EWNMF (Wei et al., 2023)	63.33	74.64	79.47	82.64	83.75	83.96	84.10	84.61	84.97
MUNPE (Jayashree, Shiva Prakash & Venugopal, 2024)	35.56	38.52	39.26	41.48	37.04	39.26	33.33	35.46	37.78
pcMRE	75.46	82.87	89.93	93.63	92.00	89.48	88.44	86.96	84.30
pcMLPRE	75.85	82.96	89.78	93.48	92.59	89.93	89.48	87.85	85.78
pcRMLPRE	85.78	87.70	90.67	92.44	91.41	90.22	90.07	89.93	88.89
daMRE	94.81	96.15	92.59	95.70	98.52	98.07	97.48	96.74	96.44
daMLPRE	94.96	98.37	96.30	96.30	98.48	98.93	98.07	97.19	95.70
daRMLPRE	94.67	95.70	93.78	91.11	89.48	88.85	87.19	86.56	85.48

DOI: 10.7717/peerj-cs.2619/table-4

The tables and figure unequivocally illustrate that, in nearly all instances, our proposed methods achieve superior classification accuracy compared to the comparison method. Specifically, in the realm of supervised methods, the MLPRE-based methods typically attain the highest accuracy, often peaking around 50 dimensions. However, it is notable that the classification accuracy curve does not always maintain a high level as dimensionality reduction dimension increases. This behavior is primarily due to the F-norm regularization term’s inclination to retain more feature information, and with higher dimensions, redundant information becomes more prominent, adversely affecting model accuracy. In sum, when using the F-norm regularization term, it is advisable to aim for a dimension reduction of around 50.

In the unsupervised methods, the proposed RMLPRE-based methods usually achieve the highest classification accuracy and tends to be more stable compared to the MLPRE-based methods. This stability is attributed to the L2,1-norm regularization, which tends to select a small number of features close to zero, rendering it more robust. In addition, based on the classification results, methods that incorporate shared structural information items generally outperform those that lack these items. This superiority is because the MLPRE-based methods generally outperform the MRE-based methods in terms of classification accuracy.

Considering the classification results, it is evident that different norm regularization terms have distinct applicable ranges. The L2,1-norm regularizer is more suitable for unsupervised methods, while the F-norm regularizer is better suited for supervised methods.