Leveraging Manifold Geometry: Exploring Distance, Curvature, and Dimensions in Deep Learning Models

This thesis explores how the mathematical concept of manifolds—a structured way to represent high-dimensional data—can enhance machine learning. Neural networks, unlike traditional methods, transform data across layers using properties like distance, curvature, and dimension. We first propose a method to preserve distances in data, improving its structure and quality. Then, we explore curvature, developing techniques for efficient representation in hyperbolic spaces. Lastly, we introduce a method for low-dimensional mappings to personalize Text-to-Image models. These studies show how understanding manifold geometry improves neural networks' ability to learn, generalize, and create high-quality outputs efficiently.