Algebraic Frameworks and Computational Experiments for Enhanced Machine Learning

In this thesis, we explore both the theory and practical considerations that come with tasks in machine learning. First, we apply a deep model to segment and highlight cancerous tumors in 3D medical scans. The data we use is uniquely produced by technology that was pioneered by a medical imaging company called Clarix Imaging, and the model we produce is unseen in existing literature. Second, we introduce two methods for reducing computational requirements for making predictions with neural networks, including by algebraically representing linear neural networks as a single layer, and by removing skip connections throughout training. We demonstrate these methods with computational examples on Residual Networks (ResNet) architecture. Third, we use tropical geometry to derive an algorithm that finds all linear regions of nonlinear neural networks, which are regions where the network behaves as a linear map. Through experiments, we find that neural networks with skip connections have more linear regions than regular networks, even before training begins, and that unseen data lies in unseen linear regions more often, providing an understanding of the efficacy of skip connections. We also visualize linear regions and observe the underlying patterns of their distribution. Lastly, we explore how an explainable artificial intelligence model called a GA^2M can be trained to fit computer-generated data with known underlying structure, providing insights into its interpretability and reliability.