  Hyperbolic deep learning has become a growing research direction in computer
vision for the unique properties afforded by the alternate embedding space. The
negative curvature and exponentially growing distance metric provide a natural
framework for capturing hierarchical relationships between datapoints and
allowing for finer separability between their embeddings. However, these
methods are still computationally expensive and prone to instability,
especially when attempting to learn the negative curvature that best suits the
task and the data. Current Riemannian optimizers do not account for changes in
the manifold which greatly harms performance and forces lower learning rates to
minimize projection errors. Our paper focuses on curvature learning by
introducing an improved schema for popular learning algorithms and providing a
novel normalization approach to constrain embeddings within the variable
representative radius of the manifold. Additionally, we introduce a novel
formulation for Riemannian AdamW, and alternative hybrid encoder techniques and
foundational formulations for current convolutional hyperbolic operations,
greatly reducing the computational penalty of the hyperbolic embedding space.
Our approach demonstrates consistent performance improvements across both
direct classification and hierarchical metric learning tasks while allowing for
larger hyperbolic models.
