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Personalized item recommendation typically suffers from data sparsity, which is most often addressed by learning vector representations of users and items via low-rank matrix factorization. While this effectively densifies the matrix by assuming users and movies can be represented by linearly dependent latent features, it does not capture more complicated interactions. For example, vector representations struggle with set-theoretic relationships, such as negation and intersection, e.g. recommending a movie that is “comedy and action, but not romance”. In this work, we formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent. To capture this set-theoretic dependence we represent each user and attribute by a hyper-rectangle or box (i.e. a Cartesian product of intervals). Box embeddings can intuitively be understood as trainable Venn diagrams, and thus not only inherently represent similarity (via the Jaccard index), but also naturally and faithfully support arbitrary set-theoretic relationships. Queries involving set-theoretic constraints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. We empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30% overall. 

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.