Kalman Filtering on Cell Complexes

2026-05-15

AI Summary

This work addresses the challenges of modeling high-order interactions in multivariate time series over topological cell complexes—namely, signal coupling, high dimensionality, nonlinear observations, and unknown complex structure—by proposing a topology-aware state-space model. The model characterizes state evolution through a heat-like diffusion process driven by boundary operators and incorporates cellular convolutions combined with nonlinear mappings in the observation mechanism. Innovatively, it applies extended Kalman filtering and online expectation-maximization for state estimation on cell complexes for the first time, and introduces a heuristic algorithm to explicitly infer second-order cells when only lower-order structural information is available. Experiments on synthetic and real-world data from water distribution, sensor, and traffic networks demonstrate that the method simultaneously achieves accurate state estimation and effective recovery of the underlying high-order topology under partial observability.

Abstract

Inferring latent dynamics from multivariate time-series defined over topological cell complexes is crucial for capturing the complex, higher-order interactions inherent in real-world systems such as in water, sensor, and transportation networks. However, reconstructing these latent states is challenging because the signals are coupled across higher-order topologies, while high dimensionality, nonlinear observations, and unknown structures increase the difficulty. To address this, we propose a topology-aware state space framework derived from stochastic partial differential equations on cell complexes. State evolution follows heat-like topological diffusion, with perturbations propagating along boundary operators. Under partial observability, we model observations using a cell complex convolution of latent states coupled with a nonlinear mapping. We perform recursive state estimation via an Extended Kalman Filter, simultaneously learning model parameters and uncertainties through an online Expectation-Maximization algorithm. Finally, for scenarios where only lower-order topological structure is known, e.g., nodes and edges, as in critical infrastructure networks, we introduce a heuristic cell identification algorithm to explicitly infer the second-order cell structures. Validations on synthetic and real datasets from water, sensor and transportation networks demonstrate that our approach yields reliable estimates under partial observability and successfully recovers the underlying topological structures.

Problem

• cell complexes
• latent dynamics
• higher-order interactions
• partial observability
• topological structure

Innovation

• cell complexes
• topological diffusion
• Extended Kalman Filter
• online Expectation-Maximization
• higher-order topology