  Query embedding (QE) -- which aims to embed entities and first-order logical
(FOL) queries in low-dimensional spaces -- has shown great power in multi-hop
reasoning over knowledge graphs. Recently, embedding entities and queries with
geometric shapes becomes a promising direction, as geometric shapes can
naturally represent answer sets of queries and logical relationships among
them. However, existing geometry-based models have difficulty in modeling
queries with negation, which significantly limits their applicability. To
address this challenge, we propose a novel query embedding model, namely Cone
Embeddings (ConE), which is the first geometry-based QE model that can handle
all the FOL operations, including conjunction, disjunction, and negation.
Specifically, ConE represents entities and queries as Cartesian products of
two-dimensional cones, where the intersection and union of cones naturally
model the conjunction and disjunction operations. By further noticing that the
closure of complement of cones remains cones, we design geometric complement
operators in the embedding space for the negation operations. Experiments
demonstrate that ConE significantly outperforms existing state-of-the-art
methods on benchmark datasets.
