# Topological Kalman Filtering on Cell Complexes

arXiv: https://arxiv.org/abs/2605.15955 · HTML: https://mlsanity.com/paper/2605.15955/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 41 references that are in the map.
- First posted on arXiv 15 May 2026.

## Builds on (most cited first, 20 of 20)

- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [GSP-KalmanNet: Tracking Graph Signals via Neural-Aided Kalman Filtering](https://mlsanity.com/paper/2311.16602/) (2023)
- [Topological Signal Processing over Generalized Cell Complexes](https://mlsanity.com/paper/2201.08993/) (2022)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Weighted simplicial complexes and their representation power of higher-order network data and topology](https://mlsanity.com/paper/2207.04710/) (2022)
- [ResNEsts and DenseNEsts: Block-based DNN Models with Improved Representation Guarantees](https://mlsanity.com/paper/2111.05496/) (2021)
- [Topological Dictionary Learning](https://mlsanity.com/paper/2503.11470/) (2025)
- [Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning](https://mlsanity.com/paper/2409.05135/) (2024)
- [Tracking Network Dynamics using Probabilistic State-Space Models](https://mlsanity.com/paper/2409.08238/) (2024)
- [Matched Topological Subspace Detector](https://mlsanity.com/paper/2504.05892/) (2025)
- [Physics-Informed Topological Signal Processing for Water Distribution Network Monitoring](https://mlsanity.com/paper/2505.07560/) (2025)
- [Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes](https://mlsanity.com/paper/2505.23160/) (2025)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)

## Abstract (arXiv)

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.