# Unraveling Convolution Neural Networks: A Topological Exploration of Kernel Evolution
## 1 Introduction  
Convolutional Neural Networks (CNNs), a key part of deep learning methodologies, have become a part of many areas of daily life because of their significant advancements in computer vision. Introduced as early as the 1980s [1], the concept of CNNs gained significant attention following the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) in 2012 [2, 3], becoming a leading tool in computer vision tasks. CNNs employ weight sharing and pooling layers, techniques that preserve data features while simultaneously reducing computational complexity, thereby significantly reducing the need for computational power and enhancing the efficiency of the network structure. However, the lack of transparency and interpretability inherent in neural networks has slowed down the widespread acceptance and approval of CNNs by relevant institutions, despite their significant advancements and community applications. This lack of interpretability can pose significant ethical problems in the application of CNNs [4, 5, 6, 7]. Aiming to enhance the interpretability of neural networks, researchers have proposed a series of methods. These methods are broadly classified into local explanations and global explanations. Local explanations focus on the behavior of models on specific inputs or neurons, aiming to uncover the mechanisms through which models make predictions on specific samples. For instance, generalizable complex model interpretability methods, such as LIME [8] and SHAP [9], are quite popular. LIME creates an interpretable local linear model around input data to approximate the behavior of complex models in thatlocality, while SHAP assigns importance values to each sample feature based on game theory, thus explaining individual predictions. Additionally, visualizing the features and behaviors of neural network models also falls under local explanations. For example, in the literature, the authors of [10] visualize the outputs of hidden layer activation functions to observe the learned features, aiding in understanding how models recognize different types of input data. Local explanation methods focus on how specific samples or neurons operate in neural networks, not revealing more generalized patterns of neural networks. In contrast, global explanations aim to study the overall behavior and characteristics of models, uncovering the general working principles of the models. For example, through theoretical analysis, some studies have aimed to understand the capacity, stability, and generalization abilities of models [4; 11; 12]. Other works test the performance and robustness of models by constructing adversarial examples [13]. In recent years, there has been growing interest in utilizing topological tools, such as persistent homology, to interpret the behavior of neural networks [14; 15; 16; 17; 18; 19], which fits within the field of Topological Data Analysis (TDA). These methods offer valuable insights into the internal representations and dynamics of neural networks by analyzing their topological properties. Typically, this analysis involves extracting features from neural networks, such as neuron activations, weights, or other network attributes, which can be represented as high-dimensional point clouds or graphs. Topological techniques, including persistent homology and simplicial complexes, are then applied to extract and analyze the topological information encoded within these high-dimensional structures [20; 21]. Through this process, researchers gain a deeper understanding of the network's behavior and can leverage the obtained insights to improve network architecture, optimize hyper-parameters, enhance performance, reduce overfitting, or increase interpretability. One aspect of the work focuses on analyzing the feature space of neural networks. [PERSON] and [PERSON] [22] utilize topological concepts to measure the complexity of neural networks. They calculate the Betti numbers of decision boundaries in the data space to evaluate the complexity of neural networks.