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Modeling directed graphs with differentiable representations is a fundamental requirement for performing machine learning on graph-structured data. Geometric embedding models (e.g. hyperbolic, cone, and box embeddings) excel at this task, exhibiting useful inductive biases for directed graphs. However, modeling directed graphs that both contain cycles and some element of transitivity, two properties common in real-world settings, is challenging. Box embeddings, which can be thought of as representing the graph as an intersection over some learned super-graphs, have a natural inductive bias toward modeling transitivity, but (as we prove) cannot model cycles. To this end, we propose binary code box embeddings, where a learned binary code selects a subset of graphs for intersection. We explore several variants, including global binary codes (amounting to a union over intersections) and per-vertex binary codes (allowing greater flexibility) as well as methods of regularization. Theoretical and empirical results show that the proposed models not only preserve a useful inductive bias of transitivity but also have sufficient representational capacity to model arbitrary graphs, including graphs with cycles.more » « less
Clustering algorithms are often evaluated using metrics which compare with ground-truth cluster assignments, such as Rand index and NMI. Algorithm performance may vary widely for different hyperparameters, however, and thus model selection based on optimal performance for these metrics is discordant with how these algorithms are applied in practice, where labels are unavailable and tuning is often more art than science. It is therefore desirable to compare clustering algorithms not only on their optimally tuned performance, but also some notion of how realistic it would be to obtain this performance in practice. We propose an evaluation of clustering methods capturing this ease-of-tuning by modeling the expected best clustering score under a given computation budget. To encourage the adoption of the proposed metric alongside classic clustering evaluations, we provide an extensible benchmarking framework. We perform an extensive empirical evaluation of our proposed metric on popular clustering algorithms over a large collection of datasets from different domains, and observe that our new metric leads to several noteworthy observations.more » « less
Learning representations of words in a continuous space is perhaps the most fundamental task in NLP, however words interact in ways much richer than vector dot product similarity can provide. Many relationships between words can be expressed set-theoretically, for example, adjective-noun compounds (eg. “red cars”⊆“cars”) and homographs (eg. “tongue”∩“body” should be similar to “mouth”, while “tongue”∩“language” should be similar to “dialect”) have natural set-theoretic interpretations. Box embeddings are a novel region-based representation which provide the capability to perform these set-theoretic operations. In this work, we provide a fuzzy-set interpretation of box embeddings, and learn box representations of words using a set-theoretic training objective. We demonstrate improved performance on various word similarity tasks, particularly on less common words, and perform a quantitative and qualitative analysis exploring the additional unique expressivity provided by Word2Box.more » « less
Multi-label classification is a challenging structured prediction task in which a set of output class labels are predicted for each input. Real-world datasets often have natural or latent taxonomic relationships between labels, making it desirable for models to employ label representations capable of capturing such taxonomies. Most existing multi-label classification methods do not do so, resulting in label predictions that are inconsistent with the taxonomic constraints, thus failing to accurately represent the fundamentals of problem setting. In this work, we introduce the multi-label box model (MBM), a multi-label classification method that combines the encoding power of neural networks with the inductive bias and probabilistic semantics of box embeddings (Vilnis, et al 2018). Box embeddings can be understood as trainable Venn-diagrams based on hyper-rectangles. Representing labels by boxes rather than vectors, MBM is able to capture taxonomic relations among labels. Furthermore, since box embeddings allow these relations to be learned by stochastic gradient descent from data, and to be read as calibrated conditional probabilities, our model is endowed with a high degree of interpretability. This interpretability also facilitates the injection of partial information about label-label relationships into model training, to further improve its consistency. We provide theoretical grounding for our method and show experimentally the model's ability to learn the true latent taxonomic structure from data. Through extensive empirical evaluations on both small and large-scale multi-label classification datasets, we show that BBM can significantly improve taxonomic consistency while preserving or surpassing the state-of-the-art predictive performance.more » « less
A wide variety of machine learning tasks such as knowledge base completion, ontology alignment, and multi-label classification can benefit from incorporating into learning differentiable representations of graphs or taxonomies. While vectors in Euclidean space can theoretically represent any graph, much recent work shows that alternatives such as complex, hyperbolic, order, or box embeddings have geometric properties better suited to modeling real-world graphs. Experimentally these gains are seen only in lower dimensions, however, with performance benefits diminishing in higher dimensions. In this work, we introduce a novel variant of box embeddings that uses a learned smoothing parameter to achieve better representational capacity than vector models in low dimensions, while also avoiding performance saturation common to other geometric models in high dimensions. Further, we present theoretical results that prove box embeddings can represent any DAG. We perform rigorous empirical evaluations of vector, hyperbolic, and region-based geometric representations on several families of synthetic and real-world directed graphs. Analysis of these results exposes correlations between different families of graphs, graph characteristics, model size, and embedding geometry, providing useful insights into the inductive biases of various differentiable graph representations.more » « less