Tracking Dynamic Simplicial Complexes via Constrained State-Space Estimation

Abstract

Simplicial complexes (SCs) extend graphs to represent higher-order interactions, but couple their simplex levels through the inclusion property. While static SC inference is an emerging research direction, tracking time-varying SCs remains largely unexplored. A central challenge in tracking SCs is to account for the inclusion property. To this end, we build a nonlinear state-space model tailored to SCs. For prediction, we introduce a closure-aware Markov generation model whose conditional mean preserves simplicial inclusion and whose covariance captures edge-triangle dependencies. For correction, we encode inclusion constraints as nonlinear pseudo-measurements, allowing the constraints to inform both the state and covariance updates. We investigate progressively richer treatments of these measurements through a standard extended Kalman filter, an iterated extended Kalman filter, and a Laplace approximation. Experiments then demonstrate the benefits of exploiting the proposed dynamics and constraint information.