Hierarchical relations are prevalent and indispensable for organizing human knowledge captured by a knowledge graph (KG). The key property of hierarchical relations is that they induce a partial ordering over the entities, which needs to be modeled in order to allow for hierarchical reasoning.

Partial-label learning(PLL) is a multi-class classification problem, where each training example isassociated withasetofcandidate labels.

The last equation is derived as follows. Inaddition, we set the observation varianceσx to 0.25. Logistic(;µ,s) is the density function of a logistic distribution with the location parameterµand the scale parameters,andσ isthe logistic sigmoid function. Before each activation, we apply the layer normalization [Ba et al., 2016] to stabilize training. When the model has sufficiently high expressive power,b may diverge to infinity [Rezende and Viola, 2018], so we add a regularization term of(b+2ζ( b))/m to the loss function, wherem is the number of training examples.

Our main contributions are as follows.