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arxiv: 2605.15955 · v2 · pith:R2UIUTTPnew · submitted 2026-05-15 · 📡 eess.SP · stat.ML

Topological Kalman Filtering on Cell Complexes

Chengen Liu , Rohan Money , Ting Gao , Mohammad Sabbaqi , Baltasar Beferull-Lozano , Elvin Isufi This is my paper

Pith reviewed 2026-05-21 08:43 UTC · model grok-4.3

classification 📡 eess.SP stat.ML
keywords topological kalman filtercell complexesstate estimationpartial observabilitytopology recoverynetwork dynamicsextended kalman filter
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The pith

A topology-aware Kalman filter on cell complexes recovers latent states and hidden higher-order structure from partial network observations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to show that a state estimation technique built on cell complexes can track hidden signals in complex networks even when only some measurements are available. It models how those signals spread like heat through higher-order structures, with changes moving along the boundaries between cells. An extended Kalman filter updates the estimates step by step while an online algorithm adjusts the model parameters and noise levels at the same time. When the full cell structure is unknown, a simple rule guesses the missing higher-order connections from the known nodes and edges. If these steps hold, monitoring and prediction in infrastructure systems such as water, sensor, and transportation networks could become more accurate than with ordinary graph-based methods.

Core claim

The authors claim that modeling state evolution as heat-like topological diffusion on cell complexes, with perturbations propagating along boundary operators, permits reliable recursive state estimation via an Extended Kalman Filter under partial observability; an online Expectation-Maximization algorithm simultaneously learns model parameters and uncertainties, while a heuristic cell identification algorithm infers second-order cells from nodes and edges alone, yielding accurate estimates and recovered topological structures on both synthetic and real network data.

What carries the argument

The heat-like topological diffusion process on cell complexes in which perturbations propagate along boundary operators, serving as the state transition model inside the extended Kalman filter.

If this is right

  • State estimates remain reliable under partial observability for signals defined on cell complexes.
  • The heuristic successfully identifies second-order cell structures from lower-order information.
  • The method produces usable results on synthetic data and on real datasets from water, sensor, and transportation networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same diffusion-plus-filter structure might be used for forecasting or anomaly detection once the cells are identified.
  • Explicit cell complexes could outperform ordinary graph models in any system whose interactions involve three or more entities at once.
  • Replacing the heuristic with a data-driven search for cells would be a direct way to test whether better cell recovery improves estimation further.

Load-bearing premise

The state evolves according to heat-like topological diffusion with perturbations propagating along boundary operators, and the heuristic cell identification algorithm accurately infers second-order cells from nodes and edges alone.

What would settle it

Apply the filter to a synthetic network whose true second-order cells are known in advance and check whether the heuristic recovers those cells and whether state estimation error is lower than that of a standard graph Kalman filter; large mismatches would refute the claim.

Figures

Figures reproduced from arXiv: 2605.15955 by Baltasar Beferull-Lozano, Chengen Liu, Elvin Isufi, Mohammad Sabbaqi, Rohan Money, Ting Gao.

Figure 1
Figure 1. Figure 1: A cell complex of order three consisting of ten [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Online 2-cell identification algorithm. Following a Ts-step warm-up, candidate 2-cells are evaluated across N2 discrete windows. In each window, the EKF state is first cached and a candidate 2-cell is activated. If the forecasting NMSEp reduction meets the threshold ϵ, the 2-cell is accepted. If rejected, the system executes a rollback to the cached state and re-runs the EKF using the original topology pre… view at source ↗
Figure 3
Figure 3. Figure 3: One-step-ahead forecasting on synthetic data. The top row shows the true latent states (blue solid lines) versus estimated [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Forecasting comparison between complete and incomplete signals on Wireless Sensor Network dataset. The subplot [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The proposed TKF algorithm is compared with TOPO [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a topology-aware state space framework for Kalman filtering on cell complexes, derived from stochastic partial differential equations. State evolution follows heat-like topological diffusion with perturbations propagating along boundary operators. Observations under partial observability are modeled using cell complex convolution of latent states with a nonlinear mapping. Recursive estimation is performed via an Extended Kalman Filter, with model parameters and uncertainties learned through an online Expectation-Maximization algorithm. A heuristic cell identification algorithm is introduced to infer second-order cells from known lower-order structures (nodes and edges). Validations on synthetic and real datasets from water, sensor, and transportation networks are presented, claiming reliable state estimates under partial observability and successful recovery of underlying topological structures.

Significance. If the central claims hold after addressing validation gaps, this work would provide a principled extension of Kalman filtering to higher-order cell complexes, enabling better modeling of latent interactions in networked systems such as critical infrastructure. The derivation from stochastic PDEs, use of boundary operators, and online EM learning represent strengths that could support practical applications in signal processing on topological domains.

major comments (1)
  1. [Heuristic cell identification algorithm (as described following the EKF/EM framework)] The abstract and framework description claim that the method 'successfully recovers the underlying topological structures' on real networks where only nodes and edges are known. This recovery depends entirely on the heuristic cell identification algorithm to infer second-order cells. However, no quantitative validation is provided, such as precision, recall, overlap metrics, or comparisons to ground-truth cell complexes, nor any analysis of the heuristic's behavior under missing edges or noisy observations. This directly undermines the empirical support for the structure-recovery claim, as the heuristic remains an untested modeling assumption rather than a demonstrated outcome.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment concerning the quantitative validation of the heuristic cell identification algorithm below.

read point-by-point responses

  1. Referee: [Heuristic cell identification algorithm (as described following the EKF/EM framework)] The abstract and framework description claim that the method 'successfully recovers the underlying topological structures' on real networks where only nodes and edges are known. This recovery depends entirely on the heuristic cell identification algorithm to infer second-order cells. However, no quantitative validation is provided, such as precision, recall, overlap metrics, or comparisons to ground-truth cell complexes, nor any analysis of the heuristic's behavior under missing edges or noisy observations. This directly undermines the empirical support for the structure-recovery claim, as the heuristic remains an untested modeling assumption rather than a demonstrated outcome.

    Authors: We agree that the current manuscript relies primarily on qualitative demonstrations and indirect evidence (such as improved state estimation accuracy when inferred cells are used versus lower-order structures alone) to support the recovery claims on real networks. No direct quantitative metrics like precision/recall against ground truth or robustness tests under noise/missing data are included for the heuristic itself. To address this gap, we will revise the manuscript to add: (i) precision, recall, and F1 scores on synthetic cell complexes with known ground truth; (ii) Jaccard overlap and similar metrics on real networks where partial ground truth is available; and (iii) sensitivity analysis of the heuristic under missing edges and noisy observations. These changes will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The framework is derived from stochastic partial differential equations on cell complexes, with state evolution specified as heat-like topological diffusion along boundary operators and observations via cell complex convolution plus nonlinear mapping. Recursive estimation uses an Extended Kalman Filter with online Expectation-Maximization for joint parameter and uncertainty learning. The heuristic cell identification algorithm is introduced separately for inferring 2-cells from nodes and edges. Validations on synthetic and real datasets from water, sensor, and transportation networks supply external empirical checks rather than reducing any prediction or recovery claim to a fitted input or self-definition by construction. No equations or steps in the provided description exhibit self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; full paper likely contains additional modeling choices. The listed items are the explicit assumptions visible in the provided text.

axioms (2)
  • domain assumption State evolution follows heat-like topological diffusion with perturbations propagating along boundary operators
    Directly stated as the basis for the state-space model in the abstract.
  • domain assumption Observations can be modeled via cell complex convolution of latent states plus nonlinear mapping
    Stated as the observation model under partial observability.

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Reference graph

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