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Answering complex queries on knowledge graphs is important but particularlychallenging because of the data incompleteness. Query embedding methods addressthis issue by learning-based models and simulating logical reasoning with setoperators. Previous works focus on specific forms of embeddings, but scoringfunctions between embeddings are underexplored. In contrast to existing scoringfunctions motivated by local comparison or global transport, this workinvestigates the local and global trade-off with unbalanced optimal transporttheory. Specifically, we embed sets as bounded measures in $\real$ endowed witha scoring function motivated by the Wasserstein-Fisher-Rao metric. Such adesign also facilitates closed-form set operators in the embedding space.Moreover, we introduce a convolution-based algorithm for linear timecomputation and a block-diagonal kernel to enforce the trade-off. Results showthat WFRE can outperform existing query embedding methods on standard datasets,evaluation sets with combinatorially complex queries, and hierarchicalknowledge graphs. Ablation study shows that finding a better local and globaltrade-off is essential for performance improvement.

Wasserstein-Fisher-Rao Embedding: Logical Query Embeddings with Local Comparison and Global Transport