Learning low-dimensional data representations is the central issue in modern machine learning. In many models the data are represented by vectors in Euclidean spaces. The corresponding architectures build upon these representations by leveraging apparatus of linear algebra. However, it has been recently recognized that some ubiquitous types of data are more faithfully represented in negatively curved manifolds. We briefly overview recent trends in this direction and explain the motivation. For further advances in hyperbolic deep learning it is necessary to enhance the corresponding mathematical framework. We introduce the notion of conformal barycenter in hyperbolic balls. Furthermore, we derive the gradient descent algorithm for computing this point. For statistical modeling, we introduce novel families of probability distributions in hyperbolic balls and explain the sampling procedure and maximum likelihood estimation. We emphasize significance of these mathematical results for future advances in geometric deep learning.  

Keywords: data representations, barycenter, Poincaré and Bergman balls, conformal invariance