MorseFormer: Scalable Topology-Aware Graph Transformers via Discrete Morse Theory

Topological Deep Learning extends graph neural networks to higher-order domains, but its dominant message-passing schemes are bounded by cellular WL, and attention-based extensions operate on the full cell complex, where sequence length grows with the lifting and global attention quickly becomes infeasible. Independently, persistent homology and Hodge Laplacian descriptors add expressivity to graph learners but are expensive and thus rarely used inside the training loop. We introduce **MorseFormer**, a topology-aware graph transformer built on Discrete Morse Theory. After a (possibly learnable) lifting, an acyclic matching collapses the cell complex onto its Morse complex — a CW complex on the critical cells, homotopy-equivalent to the original. The Morse retraction transports cell attributes basin-wise to the critical cells, and the resulting stream becomes the token sequence of a dual-branch transformer: a graph branch over the original graph and a Morse branch over critical cells with cross-rank global attention. Because the Morse complex is smaller, persistent homology and Hodge eigendecompositions become tractable as a fixed-size topological readout inside training. On graph classification and expressivity benchmarks MorseFormer is competitive with or improves on state-of-the-art topological baselines. Under static liftings, the entire topological machinery is one-shot preprocessing; training-time cost reduces to a transformer over the Morse complex.