# Robust Bilinear Probabilistic Principal Component Analysis
## 4 Numerical Examples
In this example, we also test the impact of the initializations on \(C^{\odot_{g}}\) and \(R^{\odot_{g}}\), and the sample size \(N\), respectively, based on the synthetic data having 10% outliers. Three different types of initializations of \(C^{\odot_{g}}\) and \(R^{\odot_{g}}\) are set as follows:  
(1) initialization 1: \(C^{\odot_{g}}=\mathrm{rand}(d_{c},q_{c})\) and \(R^{\odot_{g}}=\mathrm{rand}(d_{r},q_{r})\);  
(2) initialization 2: \(C^{\odot_{g}}=\mathrm{randn}(d_{c},q_{c})\) and \(R^{\odot_{g}}=\mathrm{randn}(d_{r},q_{r})\);(3) initialization 3:  
\[C^{\text{(t)}}=R^{\text{(t)}}= \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&0&0&0&1\\ \frac{9}{d_{c}}&\frac{9}{d_{c}}+9&\sin(9)&\cos(9)&\tan(9)&\cot(9)&\sec(9)& \csc(9)\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{d_{c}}{d_{c}}&\frac{d_{c}}{d_{c}}+d_{c}&\sin(d_{c})&\cos(d_{c})&\tan(d_{c })&\cot(d_{c})&\sec(d_{c})&\csc(d_{c})\end{bmatrix} ,\]  
and other parameters are fixed to be the same. The results associated with the different initializations are shown in Figure 2. Inspection of the plot illustrates that our RBPPCA method appears to be insensitive to different initializations, since the convergence behaviors of the RBPPCA method based on different initializations do not have a significant difference. Figure 3 presents the required CPU time in seconds and the quantities of arc length distances of the BPPCA and RBPPCA methods with the number of iterations being 25 with respect to the number of samples, where the sample size \(N\) varies from 200 to 5000 with a stride length of 200. Such graphs convey the fact that with the increase of the sample size \(N\), the BPPCA and RBPPCA methods both required more CPU time for 25 iterations, and the BPPCA method needs less time complexity than RBPPCA as we stated in the remarks of Algorithm 1. However, the bigger \(N\) does not lead to the improved arc length distances of the BPPCA method. Figure 1: Convergence behaviors of RBPPCA and BPPCA with the ratio of outliers being 0%, 10%, 20% and 30%, respectively. **Example 2** (**Experiments on face image databases)**. : _In this example, all of the experiments are based on two publicly available face image databases: the Yale face database ( Available from [[http://vision.ucsd.edu/content/yale-face-database](http://vision.ucsd.edu/content/yale-face-database)]([http://vision.ucsd.edu/content/yale-face-database](http://vision.ucsd.edu/content/yale-face-database))) (accessed on 22 September 2021), and the Yale B face database (Available from [[http://vision.ucsd.edu/~leek/ExtYaleDatabase/ExtYaleB.html](http://vision.ucsd.edu/~leek/ExtYaleDatabase/ExtYaleB.html)]([http://vision.ucsd.edu/~leek/ExtYaleDatabase/ExtYaleB.html](http://vision.ucsd.edu/~leek/ExtYaleDatabase/ExtYaleB.html))) (accessed on 22 September 2021).The Yale face database includes different face orientations, facial expression, and whether there exist glasses, and the Yale B face database is collected under different illumination conditions. We select 265 face images of \(15\) individuals for each database. Each person has \(11\) images, and these images are cropped to \(64\times 64\) pixels. For each individual, eight images are randomly selected as the training set, and the rest are the test set. Then, we randomly select two and four images from the eight images in the training set to be corrupted as outliers, respectively. Half of the corrupted images are generated by replacing part of the original image with a \(30\times 30\) rectangle of noise, and the other half of corrupted images use a \(50\times 50\) rectangle of noise to replace it. Within each rectangle, the pixel value comes from a uniform distribution on the interval \([0,255]\). Then, we rescale the images from the range of [0, 255] to the range [0, 1]. Some original and corrupted images of the Yale and Yale B databases are shown in Figures 4-7, respectively.