# Unraveling Convolution Neural Networks: A Topological Exploration of Kernel Evolution
## 2 Theoretical Background
### Representations of Persistent Homology  
Persistent homology results, illustrated through the evolution of a simplicial complex with an increasing radius in Figure 1, can be visually represented through persistence diagrams, barcode plots, or Betti curves [28], as shown in Figure 2. In the persistence diagram (left panel of Figure 2), each point represents a topological feature, with the vertical position indicating when the feature appears and the horizontal position indicating when it becomes insignificant. Red points in this diagram correspond to the most basic connectivity structures, while blue points represent primary cycles or loops within the space. Similarly, the barcode plot (middle panel of Figure 2) uses red bars to denote the persistence of the basic connectivity structures and blue bars for the primary cycles, with the length of each bar representing the lifespan of the corresponding feature within a specific filtration range.  
Figure 1: Evolution of a simplicial complex with increasing radius.  
The Betti curve (right panel of Figure 2) is another visualization tool in persistent homology. Unlike persistence diagrams or barcodes that summarize feature persistence across filtration levels, Betti curves provide a detailed view of the evolution of individual Betti numbers. Betti numbers denote the count of \(k\)-dimension holes in a topological space, such as the number of connected components (0 th Betti number), 1-dimensional loops (1 st Betti number), and so on.  
A Betti curve plots Betti numbers against the filtration level, illustrating how the count of topological features changes with the filtration. This curve offers insights into the development and disappearance of topological features at varying scales. The Betti curve provides a more granular analysis of individual Betti numbers, complementing the overall view provided by persistence diagrams or barcodes. It enables a deeper understanding of the topological structure of the data, revealing intricate patterns and transitions in the formation of holes across different dimensions.  
While persistent homology provides a robust framework for analyzing topological structures, the exploration of CNNs' topological complexity is not limited to this method alone. Specifically, multidimensional persistent homology broadens the scope of traditional persistent homology, enabling the examination of data's topological features through a multi-faceted lens [29]. Furthermore, the Mapper algorithm reveals the shape structure of data by creating a simplified representation. It is particularly suited for exploring the geometric and topological properties of high-dimensional data [30]. The application of the Mapper algorithm in the context of CNN kernels may uncover new insights into their high-dimensional learning processes, highlighting geometric and topological features that influence learning effectiveness and efficiency. In pursuit of clarity and to lay the groundwork for our analysis, we initially embrace Betti curves derived from persistent homology to map out and scrutinize the topological features of CNN kernel spaces. This approach simplifies our initial investigations and provides a comprehensive platform for delving into the evolutionary patterns of these features throughout the training process.