Learning with Geometry: Advancing Representation and Federated Learning

This thesis investigates how geometric principles can enhance the performance and efficiency of modern learning systems. We first explore non-Euclidean representation learning, introducing novel techniques that demystify the application of curved geometries in deep learning. By moving beyond traditional Euclidean constraints, our methods achieve state-of-the-art results in prototype learning for image classification. Building on these foundational insights, we extend the application of geometry to Federated Learning (FL), examining how margin-based information and prototype algorithms can be utilized to stabilize and improve decentralized systems. Given that communication overhead remains the primary barrier to FL scalability, this work places a vertical emphasis on reducing transmission costs. To this end, we propose a comprehensive scalability benchmark for federated frameworks, along with novel methods that integrate traditional machine learning algorithms to achieve superior communication efficiency compared to standard approaches. Collectively, this research demonstrates that incorporating geometric priors results in learning systems that are not only more accurate but also more scalable and resource-efficient.