HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

While Graph Neural Networks (GNNs) have proven highly effective at modeling relational data, pairwise connections cannot fully capture multi-way relationships naturally present in complex real-world systems. In response to this, Topological Deep Learning (TDL) leverages more general combinatorial representations -- such as simplicial or cellular complexes -- to accommodate higher-order interactions. Existing TDL methods often extend GNNs through Higher-Order Message Passing (HOMP), but face critical scalability challenges due to (i) a combinatorial explosion of message-passing routes, and (ii) significant complexity overhead from the propagation mechanism. To overcome these limitations, we propose HOPSE (Higher-Order Positional and Structural Encoder) -- a message passing-free framework that uses Hasse graph decompositions to derive efficient and expressive encodings over arbitrary higher-order domains. Notably, HOPSE scales linearly with dataset size while preserving expressive power and permutation equivariance. Experiments on molecular, expressivity and topological benchmarks show that HOPSE matches or surpasses state-of-the-art performance while achieving up to 7 times speedups over HOMP-based models, opening a new path for scalable TDL.