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### Hyperbolic Disk Embeddings for Directed Acyclic Graphs

Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of nodes are exponentially increasing is difficult. Tackling in this problem, we develop Disk Embeddings, which is a framework for embedding DAGs into quasi-metric spaces. Existing state-of-the-art methods, Order Embeddings and Hyperbolic Entailment Cones, are instances of Disk Embedding in Euclidean space and spheres respectively. Furthermore, we propose a novel method Hyperbolic Disk Embeddings to handle exponential growth of relations. The results of our experiments show that our Disk Embedding models outperform existing methods especially in complex DAGs other than trees.

### Large-Margin Classification in Hyperbolic Space

Representing data in hyperbolic space can effectively capture latent hierarchical relationships. With the goal of enabling accurate classification of points in hyperbolic space while respecting their hyperbolic geometry, we introduce hyperbolic SVM, a hyperbolic formulation of support vector machine classifiers, and elucidate through new theoretical work its connection to the Euclidean counterpart. We demonstrate the performance improvement of hyperbolic SVM for multi-class prediction tasks on real-world complex networks as well as simulated datasets. Our work allows analytic pipelines that take the inherent hyperbolic geometry of the data into account in an end-to-end fashion without resorting to ill-fitting tools developed for Euclidean space.

### Numerically Accurate Hyperbolic Embeddings Using Tiling-Based Models

Hyperbolic embeddings achieve excellent performance when embedding hierarchical data structures like synonym or type hierarchies, but they can be limited by numerical error when ordinary floating-point numbers are used to represent points in hyperbolic space. Standard models such as the Poincar{\'e} disk and the Lorentz model have unbounded numerical error as points get far from the origin. To address this, we propose a new model which uses an integer-based tiling to represent \emph{any} point in hyperbolic space with provably bounded numerical error. This allows us to learn high-precision embeddings without using BigFloats, and enables us to store the resulting embeddings with fewer bits. We evaluate our tiling-based model empirically, and show that it can both compress hyperbolic embeddings (down to $2\%$ of a Poincar{\'e} embedding on WordNet Nouns) and learn more accurate embeddings on real-world datasets.

### Hyperbolic Graph Convolutional Neural Networks

Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs.