Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds.

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

It imposes lower and upper bound constraints on the number of facilities opened for each label, ensuring fair representation of all demographic groups by the selected facilities.

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).

Recent research indicates that the performance of machine learning models can be improved by aligning the geometry of the latent space with the underlying data structure.

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural networklayers.

From a theoretical computer science point of view, many aspects of distance preservation of emed-dings are well understood.