Hodge Laplacian Quasi-Harmonic Flows for Option Discovery

Prior Laplacian-based option-discovery methods build options from 0-form Laplacians defined on nodes, where the resulting eigenvectors induce routes toward isolated boundaries or extrema, thereby skewing visitation toward specific states. In contrast, kernel eigenvectors of the 1-form Hodge Laplacian represent interpretable, topology-aware edge flows and naturally capture cyclic behaviour known as harmonic flows. These kernel eigenvectors are particularly attractive because they define zero-divergence edge flows, offering a natural solution for non-isolated visitation to sources and sinks in regions in which they are active. Despite this appeal, harmonic flows have not previously been explored as a basis for option discovery in reinforcement learning. This is what we do here. We first introduce harmonic-flow skills and show that, on state graphs with tree-like “dead-end” appendages, purely harmonic flows can become nearly inactive in those regions, which can limit coverage. To address this limitation, we propose quasi-harmonic flow options: skill-inducing edge flows that preserve harmonic cycles where harmonic activity is strong, while injecting drift into appendages using the lowest-magnitude non-harmonic eigenvector of the Hodge Laplacian, a low divergence solution, where harmonic activity is weak. Our approach is eigenvector-only, relying solely on spectral components of the Hodge Laplacian. We compare quasi-harmonic flow options against a harmonic-flow options baseline and state-of-the-art skill-discovery methods from the literature, demonstrating empirical gains and analytical improvements on hitting-time-based metrics.