# Hierarchical Prototypes Polynomial Softmax Loss Function for Visual Classification
## 4 Experimental Results and Discussion
### Ablation Study
We evaluated the impact of the two dimensionality reduction methods on the model, as shown in Table 5. It is shown that when the average pooling is used in each stage, the network's inference performance is better than the maximum value pooling performance. \begin{table}
\begin{tabular}{c c c c c} \hline
**Model** & **Alpha** & **Gamma** & **ImageNet100** & **Stanford Dogs** \\ \hline \multirow{4}{*}{Mobilenetv2 (\(\times\)1.0)} & & 0.5 & 79.39 & 59.10 \\  & 2 & 0.8 & 79.91 & 59.75 \\  & & 1 & 81.05 & 60.42 \\  & & 0.5 & 78.91 & 57.32 \\  & 3 & 0.8 & 79.61 & 59.51 \\  & & 1 & 79.97 & 59.37 \\ \hline \end{tabular}
\end{table} 
Table 4: The effect of different alpha and gamma values in the proposed polynomial loss function on network performance. The result is the top 1 of the test set, the unit is %. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**Add Prototypes Loss or Not**} & \multicolumn{2}{c}{**ImageNet100**} & \multicolumn{2}{c}{**Stanford Dogs**} \\ \cline{2-9}  & **FC** & **Stage 5** & **Stage 4** & **Stage 3** & **Stage 2** & **CE** & **PS** & **CE** & **PS** \\ \hline \multirow{4}{*}{Mobilenetv3-small (\(\times\)1.0)} & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 78.97 (Orig) & 48.43 (Orig) \\  & ✓ & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 78.07 & 78.68 & 53.99 & 53.05 \\  & ✓ & ✓ & \(\bigcirc\) & \(\bigcirc\) & 79.70 & 78.66 & 54.54 & 56.75 \\  & ✓ & ✓ & ✓ & \(\bigcirc\) & 79.85 & **81.05** & 57.61 & **60.42** \\  & ✓ & ✓ & ✓ & ✓ & \(\bigcirc\) & 79.23 & 80.54 & 58.86 & 59.73 \\  & ✓ & ✓ & ✓ & ✓ & ✓ & 79.30 & 78.91 & 59.42 & 58.76 \\  & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 70.00 (Orig) & 42.75 (Orig) \\  & ✓ & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 70.03 & 71.37 & 45.16 & 43.97 \\  & ✓ & ✓ & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 70.21 & 70.37 & 45.03 & 45.31 \\  & ✓ & ✓ & ✓ & \(\bigcirc\) & \(\bigcirc\) & 70.64 & **72.15** & 47.05 & 44.92 \\  & ✓ & ✓ & ✓ & ✓ & \(\bigcirc\) & 69.92 & 70.21 & 47.75 & **47.53** \\  & ✓ & ✓ & ✓ & ✓ & ✓ & 68.95 & 70.43 & 46.71 & 46.51 \\  & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 76.91 (Orig) & 47.01 (Orig) \\  & ✓ & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 76.45 & 76.89 & 50.61 & 52.51 \\  & ✓ & ✓ & \(\bigcirc\) & \(\bigcirc\) & \(\bigcirc\) & 76.22 & 76.71 & 51.13 & 51.90 \\  & ✓ & ✓ & ✓ & \(\bigcirc\) & \(\bigcirc\) & 78.14 & **78.56** & 56.76 & 56.40 \\  & ✓ & ✓ & ✓ & ✓ & \(\bigcirc\) & 77.42 & 77.73 & 56.40 & **57.31** \\  & ✓ & ✓ & ✓ & ✓ & \(\bigcirc\) & 76.93 & 77.83 & 55.68 & 57.14 \\ \hline \end{tabular}
\end{table} 
Table 3: Add our proposed polynomial loss function to different stages of Mobilenetv2 and Mobilenetv3. The result is the top 1 of the test set, the unit is %. To have a perceptual understanding of the network performance, we used the Class Activation Mapping (CAM) method [34] to visualize a feature map output at each stage. The CAM method can effectively highlight the area that the network pays attention to. It can be seen from Figure 5 that compared to cross-entropy, our proposed method can focus more on the dominant feature area of the object to be recognized at different stages. The loss function we proposed can effectively enhance the features extracted at each stage of the network, thereby improving the inference performance of the network.