We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincaré-ball model. We show that the proposed approach allows us to learn high-quality embeddings of large taxonomies which yield improvements over Poincaré embeddings, especially in low dimensions. Lastly, we apply our model to discover hierarchies in two real-world datasets: we show that an embedding in hyperbolic space can reveal important aspects of a company's organizational structure as well as reveal historical relationships between language families.

Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry. Second, we view hierarchical relations as partial orders defined using a family of nested geodesically convex cones. We prove that these entailment cones admit an optimal shape with a closed form expression both in the Euclidean and hyperbolic spaces, and they canonically define the embedding learning process. Experiments show significant improvements of our method over strong recent baselines both in terms of representational capacity and generalization.

Metric learning has the aim to improve classification accuracy by learning a distance measure which brings data points from the same class closer together and pushes data points from different classes further apart. Recent research has demonstrated that metric learning approaches can also be applied to trees, such as molecular structures, abstract syntax trees of computer programs, or syntax trees of natural language, by learning the cost function of an edit distance, i.e. the costs of replacing, deleting, or inserting nodes in a tree.However, learning such costs directly may yield an edit distance which violates metric axioms, is challenging to interpret, and may not generalize well.In this contribution, we propose a novel metric learning approach for trees which learns an edit distance indirectly by embedding the tree nodes as vectors, such that the Euclidean distance between those vectors supports class discrimination. We learn such embeddings by reducing the distance to prototypical trees from the same class and increasing the distance to prototypical trees from different classes.In our experiments, we show that our proposed metric learning approach improves upon the state-of-the-art in metric learning for trees on six benchmark data sets, ranging from computer science over biomedical data to a natural-language processing data set containing over 300,000 nodes.

Conventional embedding methods directly associate each symbol with a continuous embedding vector, which is equivalent to applying a linear transformation based on a ``one-hot'' encoding of the discrete symbols. Despite its simplicity, such approach yields the number of parameters that grows linearly with the vocabulary size and can lead to overfitting. In this work, we propose a much more compact K-way D-dimensional discrete encoding scheme to replace the ``one-hot" encoding. In the proposed ``KD encoding'', each symbol is represented by a $D$-dimensional code with a cardinality of $K$, and the final symbol embedding vector is generated by composing the code embedding vectors. To end-to-end learn semantically meaningful codes, we derive a relaxed discrete optimization approach based on stochastic gradient descent, which can be generally applied to any differentiable computational graph with an embedding layer. In our experiments with various applications from natural language processing to graph convolutional networks, the total size of the embedding layer can be reduced up to 98% while achieving similar or better performance.

Word embedding is a useful approach to capture co-occurrence structures in large text corpora. However, in addition to the text data itself, we often have additional covariates associated with individual corpus documents---e.g. the demographic of the author, time and venue of publication---and we would like the embedding to naturally capture this information. We propose CoVeR, a new tensor decomposition model for vector embeddings with covariates. CoVeR jointly learns a \emph{base} embedding for all the words as well as a weighted diagonal matrix to model how each covariate affects the base embedding. To obtain author or venue-specific embedding, for example, we can then simply multiply the base embedding by the associated transformation matrix. The main advantages of our approach are data efficiency and interpretability of the covariate transformation. Our experiments demonstrate that our joint model learns substantially better covariate-specific embeddings compared to the standard approach of learning a separate embedding for each covariate using only the relevant subset of data, as well as other related methods. Furthermore, CoVeR encourages the embeddings to be ``topic-aligned'' in that the dimensions have specific independent meanings. This allows our covariate-specific embeddings to be compared by topic, enabling downstream differential analysis. We empirically evaluate the benefits of our algorithm on datasets, and demonstrate how it can be used to address many natural questions about covariate effects.