The paper "Poincaré Embeddings for Learning Hierarchical Representations" by Maximilian Nickel introduces a novel approach to learning hierarchical representations of symbolic data, such as text and graphs, by embedding them into hyperbolic space, specifically an $n$-dimensional Poincaré ball. This method leverages the hyperbolic geometry to capture both hierarchy and similarity, leading to more parsimonious and effective representations. The authors propose an efficient algorithm based on Riemannian optimization to compute these embeddings, which scales well to large datasets and outperforms Euclidean embeddings in terms of representation capacity and generalization ability, especially on data with latent hierarchies. Experimental results on various tasks, including taxonomy embedding, link prediction in networks, and lexical entailment, demonstrate the superior performance of Poincaré embeddings. The paper also discusses the theoretical foundations of hyperbolic geometry and the optimization methods used to compute the embeddings.The paper "Poincaré Embeddings for Learning Hierarchical Representations" by Maximilian Nickel introduces a novel approach to learning hierarchical representations of symbolic data, such as text and graphs, by embedding them into hyperbolic space, specifically an $n$-dimensional Poincaré ball. This method leverages the hyperbolic geometry to capture both hierarchy and similarity, leading to more parsimonious and effective representations. The authors propose an efficient algorithm based on Riemannian optimization to compute these embeddings, which scales well to large datasets and outperforms Euclidean embeddings in terms of representation capacity and generalization ability, especially on data with latent hierarchies. Experimental results on various tasks, including taxonomy embedding, link prediction in networks, and lexical entailment, demonstrate the superior performance of Poincaré embeddings. The paper also discusses the theoretical foundations of hyperbolic geometry and the optimization methods used to compute the embeddings.