# Robust Bilinear Probabilistic Principal Component Analysis
## 4 Numerical Examples
The iteration of BPPCA and RBPPCA is stopped when their corresponding relative changes of the log-likelihood defined in (29) are smaller than \(10^{-5}\)._the reconstructed performance of RBPPCA presents better images than BBPPCA because it tries to explain noise information. We also compare the average reconstruction errors \(\eta=\sum_{n=1}^{N}\|X_{n}-\widehat{X}_{n}\|_{\mathrm{F}}/N\) and the recognition accuracy rates for the RBPPCA and BPPCA algorithms in the cases where each person has two corrupted images and four corrupted images, respectively, where recognition accuracy rates are calculated based on the nearest neighbor classifier (1-NN) which is employed for the classification. The average reconstruction errors and recognition accuracy rates versus the order of \(Z\) are plotted in Figures 9 and 10, respectively. As expected, the average reconstruction errors decrease, while the recognition accuracy rates rise as the order of \(Z\) increases. In these cases, our proposed RBPPCA algorithm outperforms the BPPCA algorithm in reducing average reconstruction errors and enhancing the recognition accuracy. In addition, another advantage of robust probabilistic algorithms based on \(t\) distributions is outlier detection. By [14], we can compute \(\rho_{n}=[\mathrm{vec}(X_{n}-M)]^{\mathrm{T}}(\Sigma_{r}\otimes\Sigma_{c})^{ -1}\,\mathrm{vec}(X_{n}-M)\) as the standard of outlier detection. Figure 11 is the scatter chart for \(\rho_{n}\) of all the images in the training set of the Yale and Yale B databases with two corrupted images for one person, respectively. It is exhibited that the quantity of \(\rho_{n}\) can be divided into three parts. Notice that these three parts correspond to the images with no noise, a \(30\times 30\) rectangle of noise, and a \(50\times 50\) rectangle of noise, respectively. Hence, the comparison of \(\rho_{n}\) provides a method for judging the outliers. **Example 3** (**Experiments on the MNIST dataset**). : _In Example 2, it is shown that RBPPCA is superior to BPPCA when the data sets contain outliers. In this example, we compare the RBPPCA algorithm to the tPPCA [14] and L1-PPCA [15] algorithms based on handwritten digit images from the MNIST (Available from [[http://yann.lecun.com/exdb/mnist](http://yann.lecun.com/exdb/mnist)]([http://yann.lecun.com/exdb/mnist](http://yann.lecun.com/exdb/mnist))) (accessed on 22 September 2021) database in which each image has \(28\times 28\) pixels. We choose 59 images of the digit 4 as the training data set, and randomly select nine of them to be corrupted as outliers. The way of corrupting the images is to add noise from a uniform distribution on the interval \([0,510]\), and then normalize all images to the range \([0,1]\). The normalized corrupted images of the digit 4 are shown in Figure 12. The RBPPCA, tPPCA [14] and L1-PPCA [15] algorithms are implemented with 100 iterations for the original data set of the digit 4 and the corrupted data set, respectively._  
Figure 13 presents the average reconstruction errors of the RBBPCA, tPPCA and L1-PPCA algorithms where the feature number is the order of \(Z\) for the RBBPCA algorithm and is the dimension of the low-dimensional representation for the tPPCA and L1-PPCA algorithms. It is observed that the performance of our RBPPCA algorithm is superior to other algorithms. The reconstructed behaviors of different algorithms based on the original data set of the digit 4 and the corrupted data set with \(q_{c}=q_{r}=1\) are shown in Figure 14. In Figure 14, the first column is the original images, and the second, third and fourth columns are the reconstructed images by the RBBPCA, tPPCA and L1-PPCA algorithms, respectively. As shown in Figure 14, compared to the tPPCA and L1-PPCA algorithms, the RBPPCA algorithm performs better reconstruction outcomes in such a case. Figure 8: Original images (the first column), reconstructed images by RBPPCA (the second column), and reconstructed images by BPPCA (the third column) of the Yale (**left**) and Yale B (**right**) databases, respectively.