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

Traffic sensor data exhibit strong temporal and cross-sensor dependencies, yet long gaps erase both a sensor's local history and the joint observations needed to learn its relationships with other sensors. 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, constructs a Laplacian component for each interaction order, and learns their mixture through holdouts that mimic the missingness pattern. By sharing weights across group-induced couplings, MoHL pools scattered evidence instead of estimating each coupling independently. We characterize the induced weight-sharing structure and analyze the conditions under which sharing reduces estimation risk. Experiments on three traffic networks show that higher-order groups are most beneficial under long gaps, while support-matched comparisons demonstrate the advantage of shared weights when relational evidence is scarce. MoHL achieves the lowest error among the evaluated baseline methods under day-long gaps, with orders of magnitude fewer learned parameters than deep imputers. These results show how group-wise sharing can sustain reliable reconstruction precisely when missingness depletes the evidence needed to learn cross-sensor dependence.