  Hypergraphs are widely employed to represent complex higher-order
relationships in real-world applications. Most hypergraph learning research
focuses on node- or edge-level tasks. A practically relevant but more
challenging task, edge-dependent node classification (ENC), is only recently
proposed. In ENC, a node can have different labels across different hyperedges,
which requires the modeling of node-hyperedge pairs instead of single nodes or
hyperedges. Existing solutions for this task are based on message passing and
model within-edge and within-node interactions as multi-input single-output
functions. This brings three limitations: (1) non-adaptive representation size,
(2) node/edge agnostic messages, and (3) insufficient interactions among nodes
or hyperedges. To tackle these limitations, we develop CoNHD, a new solution
based on hypergraph diffusion. Specifically, we first extend hypergraph
diffusion using node-hyperedge co-representations. This extension explicitly
models both within-edge and within-node interactions as multi-input
multi-output functions using two equivariant diffusion operators. To avoid
handcrafted regularization functions, we propose a neural implementation for
the co-representation hypergraph diffusion process. Extensive experiments
demonstrate the effectiveness and efficiency of the proposed CoNHD model.
