This paper introduces Poincaré embeddings for learning hierarchical representations of symbolic data. Unlike traditional Euclidean embeddings, which are limited in their ability to capture hierarchical structures, Poincaré embeddings are learned in hyperbolic space, specifically the Poincaré ball model. This allows for more efficient and accurate representation of hierarchical data by capturing both hierarchy and similarity simultaneously. The paper presents an efficient algorithm based on Riemannian optimization to compute these embeddings, which are shown to outperform Euclidean embeddings in terms of representation capacity and generalization ability on data with latent hierarchies. The paper discusses the limitations of Euclidean embeddings in modeling complex hierarchical patterns and introduces Poincaré embeddings as a solution. It explains how hyperbolic geometry naturally supports hierarchical structures and how the Poincaré ball model is well-suited for gradient-based optimization. The paper evaluates Poincaré embeddings on tasks such as taxonomy embedding, link prediction in networks, and lexical entailment. Results show that Poincaré embeddings provide high-quality representations of large taxonomies, outperform Euclidean embeddings in link prediction, and achieve state-of-the-art performance in lexical entailment tasks. The paper also discusses the structural bias of Poincaré embeddings, which can lead to reduced overfitting on hierarchical data. It highlights the advantages of using Poincaré embeddings for modeling complex symbolic data, including their ability to capture hierarchical relationships in a continuous space. The paper concludes that Poincaré embeddings offer important advantages over Euclidean embeddings for hierarchical data, particularly in terms of representation capacity and generalization performance. Future work includes expanding the applications of Poincaré embeddings to multi-relational data and deriving models tailored to specific applications.This paper introduces Poincaré embeddings for learning hierarchical representations of symbolic data. Unlike traditional Euclidean embeddings, which are limited in their ability to capture hierarchical structures, Poincaré embeddings are learned in hyperbolic space, specifically the Poincaré ball model. This allows for more efficient and accurate representation of hierarchical data by capturing both hierarchy and similarity simultaneously. The paper presents an efficient algorithm based on Riemannian optimization to compute these embeddings, which are shown to outperform Euclidean embeddings in terms of representation capacity and generalization ability on data with latent hierarchies. The paper discusses the limitations of Euclidean embeddings in modeling complex hierarchical patterns and introduces Poincaré embeddings as a solution. It explains how hyperbolic geometry naturally supports hierarchical structures and how the Poincaré ball model is well-suited for gradient-based optimization. The paper evaluates Poincaré embeddings on tasks such as taxonomy embedding, link prediction in networks, and lexical entailment. Results show that Poincaré embeddings provide high-quality representations of large taxonomies, outperform Euclidean embeddings in link prediction, and achieve state-of-the-art performance in lexical entailment tasks. The paper also discusses the structural bias of Poincaré embeddings, which can lead to reduced overfitting on hierarchical data. It highlights the advantages of using Poincaré embeddings for modeling complex symbolic data, including their ability to capture hierarchical relationships in a continuous space. The paper concludes that Poincaré embeddings offer important advantages over Euclidean embeddings for hierarchical data, particularly in terms of representation capacity and generalization performance. Future work includes expanding the applications of Poincaré embeddings to multi-relational data and deriving models tailored to specific applications.