As a significant direction in this line, hyperbolic representation learning has been shown to offer several advantages over conventional Euclidean geometry.

Our method overcomes this limitation by sampling in a single function evaluation. The key innovation is to optimize a neural network via maximum likelihood on the manifold, possible by adapting the free-form flow framework to Riemannian manifolds.

In Euclidean space, OCO boasts a robust theoretical foundation and numerous real-world applications, such as online load balancing (Molinaro, 2017), optimal control (Li et al., 2019), revenue maximization (Lin et al., 2019), and portfolio management (Jézéquel et al., 2022).

For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings andtensor decompositions, providing anearly-consistent noise rate estimator.