Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not capture. To address this, we propose a model that embeds multi-relational graph data in the Poincaré ball model of hyperbolic space. Our Multi-Relational Poincaré model (MuRP) learns relation-specific parameters to transform entity embeddings by Möbius matrix-vector multiplication and Möbius addition. Experiments on the hierarchical WN18RR knowledge graph show that our Poincaré embeddings outperform their Euclidean counterpart and existing embedding methods on the link prediction task, particularly at lower dimensionality.

Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models. To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion. We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding. We integrate curvature regularization into five popular proximity-preserving embedding methods, and empirical results in two applications show significant improvements on a wide range of open graph datasets.

Graph embedding, which represents real-world entities in a mathematical space, has enabled numerous applications such as analyzing natural languages, social networks, biochemical networks, and knowledge bases.It has been experimentally shown that graph embedding in hyperbolic space can represent hierarchical tree-like data more effectively than embedding in linear space, owing to hyperbolic space's exponential growth property. However, since the theoretical comparison has been limited to ideal noiseless settings, the potential for the hyperbolic space's property to worsen the generalization error for practical data has not been analyzed.In this paper, we provide a generalization error bound applicable for graph embedding both in linear and hyperbolic spaces under various negative sampling settings that appear in graph embedding. Our bound states that error is polynomial and exponential with respect to the embedding space's radius in linear and hyperbolic spaces, respectively, which implies that hyperbolic space's exponential growth property worsens the error.Using our bound, we clarify the data size condition on which graph embedding in hyperbolic space can represent a tree better than in Euclidean space by discussing the bias-variance trade-off.Our bound also shows that imbalanced data distribution, which often appears in graph embedding, can worsen the error.

In this paper, we propose MuRP, a theoretically inspired method to embed hierarchical multi-relational data in the Poincaré ball model of hyperbolic space. By considering the surface area of a hypersphere of increasing radius centered at a particular point, Euclidean space can be seen to "grow" polynomially,

Additional Feedback: * One philosophical question that comes to mind when reading the three observations in the Introduction and while going over the example in Figure 1 is the following: Could it be that the exact reason that the representation methods learn interesting patterns is the fact that they are allowed to twist and curve the space as required, in order to bring nodes that are far apart in terms of graph distance close together in terms of Euclidean distance? It is not obvious to me that constraining the optimization algorithm of this ability can only have positive outcome. There's a parallel to be drawn here with the kernel trick in Support Vector Machines, where we are allowed to embed the data in a higher dimension, where the classes become linearly separable. This way, the sum could be defined over (q', q'') \in \Gamma_{i, j} * The sectional curvature could have been more thoroughly introduced. Unfortunately, for a paper that is heavily based on geometric notions, there's a clear shortage of pictures.

Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models. To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion. We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding.

Geospatial Knowledge Graphs (GeoKGs) model geoentities (e.g., places and natural features) and spatial relationships in an interconnected manner, providing strong knowledge support for geographic applications, including data retrieval, question-answering, and spatial reasoning. However, existing methods for mining and reasoning from GeoKGs, such as popular knowledge graph embedding (KGE) techniques, lack geographic awareness. This study aims to enhance general-purpose KGE by developing new strategies and integrating geometric features of spatial relations, including topology, direction, and distance, to infuse the embedding process with geographic intuition. The new model is tested on downstream link prediction tasks, and the results show that the inclusion of geometric features, particularly topology and direction, improves prediction accuracy for both geoentities and spatial relations. Our research offers new perspectives for integrating spatial concepts and principles into the GeoKG mining process, providing customized GeoAI solutions for geospatial challenges.

Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not capture. To address this, we propose a model that embeds multi-relational graph data in the Poincaré ball model of hyperbolic space. Our Multi-Relational Poincaré model (MuRP) learns relation-specific parameters to transform entity embeddings by Möbius matrix-vector multiplication and Möbius addition. Experiments on the hierarchical WN18RR knowledge graph show that our Poincaré embeddings outperform their Euclidean counterpart and existing embedding methods on the link prediction task, particularly at lower dimensionality.

Graph embedding, which represents real-world entities in a mathematical space, has enabled numerous applications such as analyzing natural languages, social networks, biochemical networks, and knowledge bases.It has been experimentally shown that graph embedding in hyperbolic space can represent hierarchical tree-like data more effectively than embedding in linear space, owing to hyperbolic space's exponential growth property. However, since the theoretical comparison has been limited to ideal noiseless settings, the potential for the hyperbolic space's property to worsen the generalization error for practical data has not been analyzed.In this paper, we provide a generalization error bound applicable for graph embedding both in linear and hyperbolic spaces under various negative sampling settings that appear in graph embedding. Our bound states that error is polynomial and exponential with respect to the embedding space's radius in linear and hyperbolic spaces, respectively, which implies that hyperbolic space's exponential growth property worsens the error.Using our bound, we clarify the data size condition on which graph embedding in hyperbolic space can represent a tree better than in Euclidean space by discussing the bias-variance trade-off.Our bound also shows that imbalanced data distribution, which often appears in graph embedding, can worsen the error.

Knowledge graphs (KGs), which store an extensive number of relational Knowledge Graphs (KGs), e.g., the widely used YAGO [23], Freebase facts (h,,), serve various applications. While [3], DBpedia [2], WordNet [19], have been serving multiple many downstream tasks highly rely on the expressive modeling and downstream applications such as information retrieval [30], recommender predictive embedding of KGs, most of the current KG representation systems [36, 38], natural language processing [32, 34], learning methods, where each entity is embedded as a vector in the multimedia network analysis [31, 35], question answering [14, 16], Euclidean space and each relation is embedded as a transformation, fact checking [15, 17]. To utilize the extensive amount of knowledge follow an entity ranking protocol. On one hand, such an embedding in the KG, many works have studied Knowledge Graph Embedding design cannot capture many-to-many relations. On the other hand, (KGE), which learns low-dimensional representations of entities in many retrieval cases, the users wish to get an exact set of answers and relations of them [10, 21, 26, 27, 29]. Starting from TransE [4], without any ranking, especially when the results are expected to be a group of translation-based methods TransH [28], TransR [13], precise, e.g., which genes cause an illness. Such scenarios are commonly TransD [9], TorusE [6] model the relation as translations between referred to as "set retrieval". This work presents a pioneering entities in the embedding space. However, the translation-based study on the KG set retrieval problem.