# Unraveling Convolution Neural Networks: A Topological Exploration of Kernel Evolution
## 3 Experiments
### Betti Curves on Diverse Datasets
#### 3.2.2 Study on Color Images (Category B)  
In the color image category (B), experiments were conducted on datasets such as CIFAR-10, BALL-10, and BUTTERFLY-100. Similar to the grayscale studies, the focus was on the evolution of convolutional kernels in the later layers of the networks. The analysis involved tracking the Betti curves throughout the course of the training to understand the topological changes.  
Figure 4 showcases the Betti curves for Experiments B-I and B-II. These curves represent two distinct sets for each experiment, covering the CIFAR-10, BALL-10, and BUTTERFLY-100 datasets. Across both B-series experiments, the Betti curves exhibited consistent behaviors. The Betti-0 curve displayed a rightward shift with advancing iterations,  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Network** & **Hyper-Parameters** & **Layer 1** & **Layer 2** & **Layer 3** & **Layer 4** & **Fully Connected Layer 1** & **Fully Connected Layer 2** \\ \hline \multirow{6}{*}{Network I} & Convolution Kernel Size & 3 \(\times\) 3 & 3 \(\times\) 3 & 3 \(\times\) 3 & N/A & N/A & N/A \\  & Dropout (\(p\)) & N/A & N/A & N/A & N/A & 02 & 0.2 \\  & Quantity & 16 & 32 & 64 & N/A & 64 \(\times\)\(H\times\)\(W\) & 500 \\  & Maxpooling & (2,2) & (2,2) & (2,2) & N/A & N/A & N/A \\  & Layer Identifier & II & I2 & I3 & N/A & FC1 & FC2 \\ \hline \multirow{6}{*}{Network II} & Convolution Kernel Size & 3 \(\times\) 3 & 3 \(\times\) 3 & 3 \(\times\) 3 & N/A & N/A \\  & Dropout (\(p\)) & N/A & N/A & N/A & N/A & 0.2 & 0.2 \\ \cline{1-1}  & Quantity & 16 & 32 & 64 & 32 & 32 \(\times\)\(H\times\)\(W\) & 500 \\ \cline{1-1}  & Maxpooling & (2,2) & (2,2) & (2,2) & N/A & N/A & N/A \\ \cline{1-1}  & Layer Identifier & II & II2 & II3 & II4 & FC1 & FC2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The detailed architecture and hyper-parameter selection of the neural networks. Here, \(H\) and \(W\) represent the height and width of the image after passing through the convolutional layer and Maxpooling.  
while the Betti-1 curve exhibited an initial increase, subsequently followed by a decrease. There was a reduction in the peak values and a shift of the curve toward the right.  
From the results of both experiments, a shared pattern emerges in the Betti curves, consistently observed across different network architectures, classification targets, and the convolutional layers in focus. This pattern is clear in the analysis of Betti curves from both grayscale (Figure 3) and color (Figure 4) image datasets. Specifically, in both sets of experiments, the Betti-0 curve consistently shifts rightward as training iterations increase. Concurrently, the Betti-1 curve demonstrates an initial rise followed by a decline, accompanied by a reduction in peak values and a corresponding rightward shift. This trend becomes more pronounced when early iterations are compared to later ones, indicating an initial rapid evolution in the topological structure that stabilizes over time.  
Figure 3: A comparative analysis of Betti curves, derived from convolutional kernels of two different CNN structures trained with grayscale images, showcasing the consistency and variations in their topological features. (**a**) Betti curves from convolutional kernels in Experiment A-I. (**b**) Betti curves from convolutional kernels in Experiment A-II.