Empool: Collapse via Discrete Morse Theory for Representation Learning

Despite their success on non-Euclidean data, Message Passing Neural Networks (MPNNs) face fundamental depth and expressivity limitations rooted in topological bottlenecks, notably over-smoothing and over-squashing. To address these structural constraints, we propose Empool, an end-to-end differentiable graph pooling framework grounded in discrete Morse theory. By modeling graphs as Lefschetz complexes, Empool learns a discrete Morse function through a parameterized multi-layer perceptron that governs a Forman reduction, progressively collapsing the graph topology into a compact set of critical cells while rigorously preserving its homology. The resulting sparse representation mitigates the topological bottlenecks responsible for information degradation in deep networks. Extensive experiments on established node- and graph-prediction benchmarks demonstrate competitive and robust performance relative to state-of-the-art architectures.