Leonardo Cotta is currently a fifth year PhD candidate at Purdue University under Professor Bruno Ribeiro. His doctoral work spans higher-order graph representations with applications in recommender systems and the life sciences. He held a Qatar Computing Research Institute fellowship in 2017 and 2018 and was a research intern at Intel Labs in 2021. He will join the Vector Institute in August as a postdoctoral fellow to work with Professor Chris J. Maddison.
Equivariant poset representations
Partially-ordered data is pervasive across a wide range of domains: from online user forums, to natural language understanding, to computer programs and, to bioinformatics. To develop successful applications in these domains, we need to learn representations mapping a hierarchy to a meaningful vector space. Partially ordered sets (posets) are combinatorial objects encoding such hierarchies. Despite the recent plethora of works on equivariant representations for other combinatorial structures, such as graphs and sets, posets have been consistently neglected in this context. In this project we will first discuss learning equivariant representations over combinatorial structures in general and then posets in particular. As a first model we will consider representing the poset as a Directed Acyclic Graph (DAG) and applying a Graph Neural Network (GNN) over it. We will then show how the DAG view does not explicitly encode all known aspects of a poset. Instead, we will consider applying a GNN over the poset zeta matrix (the analogous of the adjacency matrix for a poset). Finally, we will explore non-GNN-based equivariant architectures for representing posets. We will take inspiration from related notions of convolution over powersets to motivate the development of such a machinery.