# A Study of OWA Operators Learned in Convolutional Neural Networks
## 6 Results and Discussion
### Initialization and Convergence  
The top 5 accuracy results of this first experiment are shown in Table 2, where we can observe the final top 5 accuracy of the network when we varied the initialization parameter \(\alpha\). Although there is a statistically significant improvement in the top 5 accuracy over the reference of about 2%, there is little difference between the different initialization parameters accuracy-wise.  
In Figure 2, we focus on the orness and dispersion of the learned features as we vary \(\alpha\). To create this figure, we first sorted the operators in each matrix, from lowest to highest orness, and then averaged the 40 weighting matrices of the repetitions of each experiment to denoise them. From this average matrix, we can then computed the orness and dispersion of each operator.  
As expected, we can observe how larger values of \(\alpha\) (a smaller variance of the initial weights) allows the network to generate a larger diversity of learned features. With an initialization with \(\alpha=128\), we already observe an orness range of 0.1 to 0.9, covering OWA operators from close to \(min\) to close to \(max\). There is a clear tendency to \(max\)-like operators, with 54 operators covering the range of ornessness 0.6 to 0.9, and only 10 operators in the range 0.1 to 0.6 (\(min\)-like). We can also observe how lower values of \(\alpha\) (\(\alpha<128\)) have a similar shape in the orness profile, but reduced range, and how higher values (\(\alpha>128\)) do not increase the range or change the shape of the profile anymore.  
About the dispersion, we can observe a maximal value of dispersion (around 4) for the features with an orness close to the average, dropping to lower values of dispersion for the features with minimum and maximum orness. The fact that operators with similar values of orness (operators 10 to 20 on all \(\alpha\) configurations show an orness of 0.4 to 0.6) show also a similar value of dispersion suggests a high stability in the shape of the operators across initialization configurations.  
Even though these matrices are averaged over the experiments, there is a notable decrease in the stability of the dispersion values for larger values of \(\alpha\). This suggests that increasing the value of \(\alpha\) not only does not increase the diversity of the operators anymore (as seen on the orness figure) but also leads to noisier and less stable operators.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Init \(\alpha\)** & **Top 5 acc.** & \(p\)**-Value** \\ \hline Reference & \(52.17\pm 1.5\) & \\
1 & \(54.28\pm 1.4\) & \(<\)0.001 * \\
2 & \(54.47\pm 1.4\) & \(<\)0.001 * \\
4 & \(53.95\pm 1.5\) & \(<\)0.001 * \\
8 & \(54.50\pm 1.6\) & \(<\)0.001 * \\
16 & \(54.38\pm 1.4\) & \(<\)0.001 * \\
32 & \(54.13\pm 1.7\) & \(<\)0.001 * \\
64 & \(54.61\pm 1.4\) & \(<\)0.001 * \\
128 & \(54.26\pm 1.4\) & \(<\)0.001 * \\
256 & \(54.43\pm 1.3\) & \(<\)0.001 * \\
512 & \(54.49\pm 1.3\) & \(<\)0.001 * \\
1024 & \(54.18\pm 1.7\) & \(<\)0.001 * \\ \hline \hline \end{tabular}
\end{table}
Table 2: Accuracy results for initialization variants.  
Figure 2: Orness and dispersion profiles of learned features for the initialization variants (\(\alpha\)).