Learning Higher-Order Interactions from Incomplete Spatiotemporal Data with Mixtures of Hypergraph Laplacians

Traffic sensor data are strongly structured in time and across sensors, but reconstructing missing values becomes increasingly difficult as gaps lengthen because missingness removes both local temporal context and the joint observations needed to infer cross-sensor relationships. We introduce MoHL, a mixture of hypergraph Laplacians that learns group-wise relational structure directly from incomplete spatiotemporal data. MoHL discovers overlapping sensor groups from observed co-variation, assigns each interaction order a Laplacian component, and learns their mixture from mask-matched holdouts. This group-wise parameterization pools evidence across related sensors rather than estimating every coupling independently. We show that the learned mixture induces structured weight sharing and characterize when such sharing is beneficial. Across three traffic networks, higher-order groups provide the largest gains under extended gaps, and shared weights consistently outperform freely learned weights on the same support when relational evidence is scarce. MoHL achieves the lowest error among evaluated methods under day-long gaps while using orders of magnitude fewer learned parameters than deep imputers. These results show that higher-order group structure is most useful precisely when missingness makes relational evidence scarce.