The choice of geometric space for knowledge graph (KG) embeddings can have significant effects on the performance of KG completion tasks.The hyperbolic geometry has been shown to capture the hierarchical patterns due to its tree-like metrics, which addressed the limitations of the Euclidean embedding models.Recent explorations of the complex hyperbolic geometry further improved the hyperbolic embeddings for capturing a variety of hierarchical structures.However, the performance of the hyperbolic KG embedding models for nontransitive relations is still unpromising, while the complex hyperbolic embeddings do not deal with multi-relations.This paper aims to utilize the representation capacity of the complex hyperbolic geometry in multi-relational KG embeddings.To apply the geometric transformations which account for different relations and the attention mechanism in the complex hyperbolic space, we propose to use the fast Fourier transform (FFT) as the conversion between the real and complex hyperbolic space.Constructing the attention-based transformations in the complex space is very challenging, while the proposed Fourier transform-based complex hyperbolic approaches provide a simple and effective solution.Experimental results show that our methods outperform the baselines, including the Euclidean and the real hyperbolic embedding models.

Hierarchical text classification has many real-world applications. However, labeling a large number of documents is costly. In practice, we can use semi-supervised learning or weakly supervised learning (e.g., dataless classification) to reduce the labeling cost. In this paper, we propose a path cost-sensitive learning algorithm to utilize the structural information and further make use of unlabeled and weakly-labeled data. We use a generative model to leverage the large amount of unlabeled data and introduce path constraints into the learning algorithm to incorporate the structural information of the class hierarchy. The posterior probabilities of both unlabeled and weakly labeled data can be incorporated with path-dependent scores. Since we put a structure-sensitive cost to the learning algorithm to constrain the classification consistent with the class hierarchy and do not need to reconstruct the feature vectors for different structures, we can significantly reduce the computational cost compared to structural output learning. Experimental results on two hierarchical text classification benchmarks show that our approach is not only effective but also efficient to handle the semi-supervised and weakly supervised hierarchical text classification.