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.

本研究提出一种新的查询嵌入模型，称为锥体嵌入（ConE），通过将实体和查询表示为二维锥形的笛卡尔积，处理与所有的一阶逻辑运算，包括合取，析取和否定，并在查询中引入几何补充算子，处理否定运算，从而在基准数据集上显著优于现有其他模型。