The study of machine learning-based logical query-answering enables reasoning with large-scale and incomplete knowledge graphs. This paper further advances this line of research by considering the uncertainty in the knowledge. The uncertain nature of knowledge is widely observed in the real world, but does not align seamlessly with the first-order logic underpinning existing studies. To bridge this gap, we study the setting of soft queries on uncertain knowledge, which is motivated by the establishment of soft constraint programming. We further propose an ML-based approach with both forward inference and backward calibration to answer soft queries on large-scale, incomplete, and uncertain knowledge graphs. Theoretical discussions reveal that our method ensures there are no catastrophic cascading errors in our forward inference algorithm while maintaining the same complexity as state-of-the-art inference algorithms for first-order queries. Empirical results justify the superior performance of our approach against previous ML-based methods with number embedding extensions.

Site recommendation, which aims at predicting the optimal location for brands to open new branches, has demonstrated an important role in assisting decision-making in modern business. In contrast to traditional recommender systems that can benefit from extensive information, site recommendation starkly suffers from extremely limited information and thus leads to unsatisfactory performance. Therefore, existing site recommendation methods primarily focus on several specific name brands and heavily rely on fine-grained human-crafted features to avoid the data sparsity problem. However, such solutions are not able to fulfill the demand for rapid development in modern business. Therefore, we aim to alleviate the data sparsity problem by effectively utilizing data across multiple cities and thereby propose a novel Optimal Transport enhanced Cross-city (OTC) framework for site recommendation. Specifically, OTC leverages optimal transport (OT) on the learned embeddings of brands and regions separately to project the brands and regions from the source city to the target city. Then, the projected embeddings of brands and regions are utilized to obtain the inference recommendation in the target city. By integrating the original recommendation and the inference recommendations from multiple cities, OTC is able to achieve enhanced recommendation results. The experimental results on the real-world OpenSiteRec dataset, encompassing thousands of brands and regions across four metropolises, demonstrate the effectiveness of our proposed OTC in further improving the performance of site recommendation models.