# Unraveling Convolution Neural Networks: A Topological Exploration of Kernel Evolution
## 1 Introduction
Their research provides upper and lower bounds for the complexity of shallow and deep networks with the same number of hidden units, highlighting the higher complexity of deep structures. [PERSON] and [PERSON] [23] investigate the relationship between the topological complexity of datasets and the capacity of neural networks. By understanding the topological features of datasets, more efficient and smaller-scale network architectures can be designed, reducing computational resources and training time. Another line of research involves the visualization of neural networks using the Mapper algorithm. [PERSON] [24] applies the Mapper algorithm to visualize test datasets in deep neural network models. By clustering the activations of the test set on the neural network, they identify clusters representing misclassified samples, which provide insights into areas where the network's performance can be improved. [PERSON] and [PERSON] [25; 26] visualize the convolutional kernels of well-trained CNNs using the Mapper algorithm. They observe that the point cloud formed by the kernels exhibits a \"ring\" structure in the parameter space. This finding suggests that convolutional kernels tend to form a simple topological structure during the network learning process. Furthermore, they propose the notion of \"topological simplicity\" as a measure of network generalization performance, comparing the persistence lengths of one-dimensional homology classes. Other works also explore the parameter space analysis of neural networks. [PERSON] et al. [15] introduce a method to measure the complexity of neural network structures. By constructing a weighted graph, where neurons are considered vertices and connections are considered edges, they use persistent homology to quantify the 0-dimensional topological information of the graph. This complexity measure can monitor the impact of regularization techniques, such as dropout and batch normalization, during the network training process and serve as an indicator of overfitting. [PERSON] and [PERSON] [16] view the entire neural network as a directed weighted graph and apply the construction of Clique complexes to capture the topological features of the parameter space. This approach allows the introduction of higher-dimensional topology and provides insights into the neural network'sstructural organization and hierarchy. These studies demonstrate the diverse applications of topological analysis in interpreting neural networks, assessing their complexity, and guiding network design, optimization, and generalization. Our research specifically investigates the dynamic topological evolution of CNN kernels throughout the learning process. We analyze the shifts in the Betti-0 and Betti-1 curves of convolutional kernels, revealing patterns that highlight the network's learning dynamics and decision-making process. A key focus is the distinct topological adjustments observed when CNNs are trained on grayscale versus color datasets, emphasizing the need for significant parameter space modifications for color image learning. Our contributions are significant in two ways: (i) unveiling the dynamic topological patterns in CNN kernels during the iterative learning process and (ii) demonstrating how these topological changes can serve as indicators of effective learning within the network and influence the overall performance. This insight not only deepens our understanding of CNN internals but also opens new pathways for optimizing their design and application in complex scenarios.