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Logical query answering over Knowledge Graphs (KGs) is a fundamental yetcomplex task. A promising approach to achieve this is to embed queries andentities jointly into the same embedding space. Research along this linesuggests that using multi-modal distribution to represent answer entities ismore suitable than uni-modal distribution, as a single query may containmultiple disjoint answer subsets due to the compositional nature of multi-hopqueries and the varying latent semantics of relations. However, existingmethods based on multi-modal distribution roughly represent each subset withoutcapturing its accurate cardinality, or even degenerate into uni-modaldistribution learning during the reasoning process due to the lack of aneffective similarity measure. To better model queries with diversified answers,we propose Query2GMM for answering logical queries over knowledge graphs. InQuery2GMM, we present the GMM embedding to represent each query using aunivariate Gaussian Mixture Model (GMM). Each subset of a query is encoded byits cardinality, semantic center and dispersion degree, allowing for preciserepresentation of multiple subsets. Then we design specific neural networks foreach operator to handle the inherent complexity that comes with multi-modaldistribution while alleviating the cascading errors. Last, we define a newsimilarity measure to assess the relationships between an entity and a query'smulti-answer subsets, enabling effective multi-modal distribution learning forreasoning. Comprehensive experimental results show that Query2GMM outperformsthe best competitor by an absolute average of $5.5\%$. The source code isavailable at \url{https://anonymous.4open.science/r/Query2GMM-C42F}.

Query2GMM: Learning Representation with Gaussian Mixture Model for Reasoning over Knowledge Graphs