Learning Higher-Order Structure from Incomplete Spatiotemporal Data: Multi-Scale Hypergraph Laplacians with Neural Refinement

TL;DR

This paper introduces a novel multi-scale hypergraph Laplacian framework for learning higher-order structures from incomplete spatiotemporal sensor data, improving imputation accuracy in complex missing data scenarios.

Contribution

The authors propose MSHL, a two-stage hypergraph-based method that captures higher-order relations and adapts to the optimal interaction scale, outperforming pairwise methods in real traffic network data.

Findings

MSHL represents group-conservation patterns inaccessible to pairwise graph priors.

MSHL adapts to the best fixed scale up to a logarithmic factor.

MSHL improves imputation over pairwise baselines when higher-order structure is present.

Abstract

Sensor networks increasingly govern modern infrastructure, yet the data they lose are rarely missing in the uniform-random patterns assumed by standard imputation benchmarks. Loop detectors go offline during calibration, roadside cabinets silence clusters of nearby sensors, and newly installed instruments provide no history. Such failures create structured absences whose values are constrained by higher-order relations among groups of sensors, not merely by pairwise proximity. Existing low-rank and graph-based methods often miss this collective structure and can fail when missingness becomes coherent. We introduce Multi-Scale Hypergraph Laplacians (MSHL), a two-stage framework for learning higher-order structure from incomplete spatiotemporal observations. The Discovery stage builds a multi-scale hypergraph from complementary topology and residual-correlation evidence, with an observation-only selector that adapts to the supported interaction scale. The Refinement stage adds a small hypergraph-conditioned residual network that is safe by construction: it learns nonlinear corrections where informative residual features exist and defers to the linear estimate where they do not. We prove that MSHL represents group-conservation patterns inaccessible to pairwise graph priors, adapts to the best fixed scale up to a logarithmic factor, transfers this advantage to held-out imputation error, and admits a one-sided refinement guarantee. On two real traffic networks evaluated across scattered cell missingness, contiguous block outages, and whole-sensor blackouts at five rates, MSHL improves over a pairwise-graph baseline whenever higher-order structure is identifiable and otherwise matches it within sampling noise. The results point to a broader principle for reliable infrastructure learning: missing data should be treated not as isolated entries to fill, but as evidence of structure to discover.