Hyperbolic Optimization

This work explores optimization methods on hyperbolic manifolds. Building on Riemannian optimization principles, we extend the Hyperbolic Stochastic Gradient Descent (a specialization of Riemannian SGD) to a Hyperbolic Adam optimizer. While these methods are particularly relevant for learning on the Poincar\'e ball, they may also provide benefits in Euclidean and other non-Euclidean settings, as the chosen optimization encourages the learning of Poincar\'e embeddings. This representation, in turn, accelerates convergence in the early stages of training, when parameters are far from the optimum. As a case study, we train diffusion models using the hyperbolic optimization methods with hyperbolic time-discretization of the Langevin dynamics, and show that they achieve faster convergence on certain datasets without sacrificing generative quality.