# Research on Generalized Hybrid Probability Convolutional Neural Network
## 2 Research on the Generalization Ability of the CNN Feature Mapper (CFM) to Extract Image Features
\begin{table}
\begin{tabular}{l l} \hline \(S^{*}\) & The source dataset for training the CNN \\ \hline \(s\) & The number of classes in the source dataset \(S^{*}\) \\ \hline \(T^{*}\) & The target dataset for classification \\ \hline \(t\) & The number of classes in the target dataset \(T^{*}\) \\ \hline \(P_{s}\) & The original training accuracy of VGG-16 in the source dataset with \(s\) class samples \\ \hline \(P_{t}^{s}\) & The test accuracy of target sets with \(t\) class samples \\ \hline \(\widehat{P}_{t}^{s}\) & The average accuracy of multiple \(P_{t}^{s}\) \\ \hline **CFM** & Convolutional feature mapper \\ \hline \(\textbf{CFM}_{s}\) & Convolutional feature mapper trained by the source dataset with \(s\) class samples \\ \hline \(T^{*}\subset S^{*}\) & The samples in \(T^{*}\) are selected from \(S^{*}\) \\ \hline \(S^{*}\bigcap T^{*}=\emptyset\) & The number of common samples between \(T^{*}\) and \(S^{*}\) is zero \\ \hline \(r_{t}^{s}\) & The increased amplitude of the \(t\) class target dataset compared with the \(s\) class source dataset \\ \hline \(tr(\textbf{A})\) & The trace of matrix **A** \\ \hline \(det(\textbf{A})\) & The determinant of matrix **A** \\ \hline \end{tabular}
\end{table} 
Table 1: Important notations and variables used in this paper. isons between the magnitudes of \(\widetilde{P}_{s}^{t}\) and \(P_{s}\) show the generalization ability of the \(\mathbf{CFM_{s}}\) in the t-class target dataset. \[r_{s}^{t}=\frac{\widetilde{P}_{s}^{t}-P_{s}}{P_{s}} \tag{1}\]  
The size of \(r_{s}^{t}\) can measure the strength of the \(\mathbf{CFM_{s}}\)'s generalization ability. That is, the greater the value of \(r_{s}^{t}\) is, the stronger the generalization ability of the \(\mathbf{CFM_{s}}\). When \(r_{s}^{t}>0\), the \(\mathbf{CFM_{s}}\) can extract features with better separability for the t-class targets and can obtain a higher accuracy than \(P_{s}\) under subsequent classifiers. That is, a \(\mathbf{CFM}\) trained by the s-class source dataset \(S^{*}\) has a better generalization ability for a t-class target dataset \(T^{*}\). When \(r_{s}^{t}<0\), the \(\mathbf{CFM_{s}}\) fails to extract features with better separability in the t-class target dataset, which results in a lower accuracy than the original \(P_{s}\) under subsequent classifiers. The corresponding classification accuracy can be obtained by changing the membership relationship between \(S^{*}\) and \(T^{*}\), and the generalization ability of the \(\mathbf{CFM_{s}}\) under different conditions can be further reflected by increasing or decreasing the magnitude of the classification accuracy compared with the original accuracy. Analysis of the Generalization Ability of the Feature Mapper to the Target Set after Category Reduction (\(T^{*}\subset S^{*}\))  
When \(T^{*}\subset S^{*}\), the trained \(\mathbf{CFM_{s}}\) on multiclass source dataset \(S^{*}\) extracts the features of target dataset \(T^{*}\) and trains the classification by the MLP classifier. In the CIFAR-100 dataset, we selected the source dataset with the number of categories for \(s\) (\(s=15,20,25,\cdots,100\)) to be fully trained with the VGG-16 algorithm, we obtained different original precisions \(P_{s}\) and the corresponding 15 \(\mathbf{CFM_{s}}\)s, and then randomly extracted the target dataset \(T^{*}\) with the number of categories \(t\) (\(t=5,10,\cdots,30\)) from the source dataset \(S^{*}\). The features of these target dataset \(T^{*}\) were extracted by the \(\mathbf{CFM_{s}}\), and then the training classification was performed by the MLP classifier. To make the results more universal, the target dataset of each category are randomly extracted eight times. The classification accuracy \(P_{s}^{t}\) was counted each time, and the final result was the average value of the multiple classification results, where \(t\) represented the number of categories in the target dataset \(T^{*}\), and \(s\) represented the number of categories in the source dataset \(S^{*}\).