Unraveling Convolution Neural Networks: A Topological Exploration of Kernel Evolution

Abstract

Convolutional Neural Networks (CNNs) have become essential in deep learning applications, especially in computer vision, yet their complex internal mechanisms pose significant challenges to interpretability, crucial for ethical applications. Addressing this, our paper explores CNNs by examining their topological changes throughout the learning process, specifically employing persistent homology, a core method within Topological Data Analysis (TDA), to observe the dynamic evolution of their structure. This approach allows us to identify consistent patterns in the topological features of CNN kernels, particularly through shifts in Betti curves, which is a key concept in TDA. Our analysis of these Betti curves, initially focusing on the zeroth and first Betti numbers (respectively referred to as Betti-0 and Betti-1, which denote the number of connected components and loops), reveals insights into the learning dynamics of CNNs and potentially indicates the effectiveness of the learning process. We also discover notable differences in topological structures when CNNs are trained on grayscale versus color datasets, indicating the need for more extensive parameter space adjustments in color image processing. This study not only enhances the understanding of the intricate workings of CNNs but also contributes to bridging the gap between their complex operations and practical, interpretable applications.