We consider a structured multi-label prediction problem where the labels are organized under implication and mutual exclusion constraints. A major concern is to produce predictions that are logically consistent with these constraints. To do so, we formulate this problem as an embedding inference problem where the constraints are imposed onto the embeddings of labels by geometric construction. Particularly, we consider a hyperbolic Poincaré ball model in which we encode labels as Poincaré hyperplanes that work as linear decision boundaries. The hyperplanes are interpreted as convex regions such that the logical relationships (implication and exclusion) are geometrically encoded using the insideness and disjointedness of these regions, respectively. We show theoretical groundings of the method for preserving logical relationships in the embedding space. Extensive experiments on 12 datasets show 1) significant improvements in mean average precision; 2) lower number of constraint violations; 3) an order of magnitude fewer dimensions than baselines.

We consider a structured multi-label prediction problem where the labels are organized under implication and mutual exclusion constraints. A major concern is to produce predictions that are logically consistent with these constraints. To do so, we formulate this problem as an embedding inference problem where the constraints are imposed onto the embeddings of labels by geometric construction. Particularly, we consider a hyperbolic Poincaré ball model in which we encode labels as Poincaré hyperplanes that work as linear decision boundaries. The hyperplanes are interpreted as convex regions such that the logical relationships (implication and exclusion) are geometrically encoded using the insideness and disjointedness of these regions, respectively.

Relational representation learning transforms relational data into continuous and low-dimensional vector representations. However, vector-based representations fall short in capturing crucial properties of relational data that are complex and symbolic. We propose geometric relational embeddings, a paradigm of relational embeddings that respect the underlying symbolic structures. Specifically, this dissertation introduces various geometric relational embedding models capable of capturing: 1) complex structured patterns like hierarchies and cycles in networks and knowledge graphs; 2) logical structures in ontologies and logical constraints applicable for constraining machine learning model outputs; and 3) high-order structures between entities and relations. Our results obtained from benchmark and real-world datasets demonstrate the efficacy of geometric relational embeddings in adeptly capturing these discrete, symbolic, and structured properties inherent in relational data.