GSP-KalmanNet: Hybrid Graph Kalman Filtering

• GSP-KalmanNet is a hybrid framework that combines graph signal processing with data-driven Kalman filtering for dynamic state estimation over graph-structured data.
• It employs spectral transformations and GRU-based neural networks to compute adaptive Kalman gains, enhancing robustness against model and topology mismatches.
• Empirical results demonstrate significant MSE reduction and scalability improvements in applications like power grids, traffic forecasting, and sensor networks.

GSP-KalmanNet is a hybrid model-based/data-driven framework for tracking dynamically evolving graph signals, integrating the structural advantages of graph signal processing (GSP) with the flexibility of learned Kalman filtering. Addressing the challenges of nonlinear, high-dimensional, and irregular graph domains, the architecture is designed to estimate latent states from noisy, potentially partial graph-based observations, while remaining robust to topology and model misspecification. By restricting inference to the graph spectral domain and parameterizing the Kalman gain with deep recurrent networks, GSP-KalmanNet delivers scalable and accurate inference for spatio-temporal graph systems found in domains ranging from power grids and transportation to sensor and water networks (Buchnik et al., 2023, Sabbaqi et al., 27 Jun 2025).

1. Mathematical Foundations and State-Space Modeling

GSP-KalmanNet builds upon general state-space models (SSMs) for graph signals. Consider a weighted undirected graph $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathbf{W})$$ with $N=|\mathcal{V}|$ nodes and Laplacian $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$. The hidden state $\mathbf{x}_t \in \mathbb{R}^N$ and observation $\mathbf{y}_t \in \mathbb{R}^N$ evolve as: $$\begin{aligned} \mathbf{x}_t &= f_t(\mathbf{L}, \mathbf{x}_{t-1}) + \mathbf{e}_t, \ \mathbf{y}_t &= h_t(\mathbf{L}, \mathbf{x}_t) + \mathbf{v}_t, \end{aligned}$$ where $f_t$ and $h_t$ may be nonlinear, $\mathbf{e}_t \sim \mathcal{N}(0, \mathbf{Q})$, and $\mathbf{v}_t \sim \mathcal{N}(0, \mathbf{R})$ are noises. The Laplacian's eigendecomposition $N=|\mathcal{V}|$0 defines a graph Fourier basis, which enables compact spectral filtering of signals and covariances (Buchnik et al., 2023).

In the discrete-time stochastic partial differential equation (SPDE) variant, the state evolves according to localized edge-driven noise: $N=|\mathcal{V}|$1 with $N=|\mathcal{V}|$2 a diffusion constant and $N=|\mathcal{V}|$3 (Sabbaqi et al., 27 Jun 2025). The observation model applies a $N=|\mathcal{V}|$4-order graph filter: $N=|\mathcal{V}|$5 where $N=|\mathcal{V}|$6 is a sampling matrix and $N=|\mathcal{V}|$7 parameterizes multi-hop aggregation.

2. Graph Signal Processing and Spectral Filtering

GSP-KalmanNet leverages spectral GSP tools by transforming signals to the graph frequency domain. Graph signals are projected via the graph Fourier transform: $N=|\mathcal{V}|$8 enabling compact spectral manipulation. Graph filters are defined as polynomials or general functions of the Laplacian,

$N=|\mathcal{V}|$9

and operate entrywise in the spectral domain, i.e., $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$0 (Buchnik et al., 2023).

The model-based spectral extended Kalman filter (GSP-EKF) reformulates the Kalman recursion in this frequency domain, drastically simplifying high-dimensional updates. The Kalman gain is constrained to be spectral-diagonal, i.e., $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$1, with optimal $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$2 obtained by minimizing the posterior trace.

3. Neural-Aided Spectral Kalman Filtering

GSP-KalmanNet replaces the analytical computation of the Kalman gain with a neural network–driven surrogate in the spectral domain. At each time step, three spectral-domain $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$3-vectors are computed: the innovation ($$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$4), forward-evolution difference ($$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$5), and forward-update difference ($$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$6). These are concatenated and fed to a GRU-based RNN with the following architecture:

• Input: fully connected $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$7, ReLU activation
• GRU layers: two layers, each with hidden size $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$8
• Output: fully connected $$\mathbf{L} = \mathrm{diag}(\mathbf{W}\mathbf{1}) - \mathbf{W}$$9, ReLU, then $\mathbf{x}_t \in \mathbb{R}^N$0 (linear)
• Output vector: interpreted as the spectral-diagonal of the Kalman gain

The full inference loop preserves the spectral prediction step but substitutes the RNN gain estimate in the update step. The RNN parameters are trained end-to-end to minimize the state mean squared error (MSE) plus $\mathbf{x}_t \in \mathbb{R}^N$1 regularization (Buchnik et al., 2023).

4. Learning and Robustness to Model Misspecification

Training uses backpropagation-through-time (BPTT) with stochastic gradient descent over batches of trajectories: $\mathbf{x}_t \in \mathbb{R}^N$2 treating the filter as a differentiable mapping from observations to final hidden states.

Empirical evidence indicates that the RNN-based surrogate gain confers robustness to misspecification in process/measurement covariances, state/observation functions, and even minor perturbations to the graph topology (e.g., dropped or perturbed edges). GSP-KalmanNet achieves higher stability and accuracy under such mismatches compared with both purely model-based (EKF, GSP-EKF) and purely data-driven (GNN+RNN) baselines (Buchnik et al., 2023).

5. Computational Complexity and Scalability

Standard EKF and statistical EM methods require $\mathbf{x}_t \in \mathbb{R}^N$3 operations per step due to matrix inversions. GSP-KalmanNet precomputes the eigenbasis $\mathbf{x}_t \in \mathbb{R}^N$4 offline ($\mathbf{x}_t \in \mathbb{R}^N$5) but thereafter achieves per-step costs of $\mathbf{x}_t \in \mathbb{R}^N$6 (due to matrix-vector multiplications) and $\mathbf{x}_t \in \mathbb{R}^N$7 (RNN forward pass and diagonal gain), exploiting the spectral-diagonal constraint to reduce complexity. This enables scalable inference on graphs with hundreds of nodes, where conventional approaches suffer from prohibitive resource usage or instability (Buchnik et al., 2023).

6. Empirical Performance and Benchmarking

Experiments demonstrate state-of-the-art accuracy and efficiency across synthetic and real-world datasets:

Scenario	Performance Notes	Benchmarks
Synthetic nonlinear SSM	GSP-KalmanNet retains <1 dB loss under topology change, trains more stably, and achieves 10–15 dB MSE reduction in nonlinear cases	EKF, GSP-EKF, KalmanNet, GNN+RNN
IEEE-14 bus power grid	Outperforms model-based by >10 dB, bested KalmanNet/GNN+RNN by 5–8 dB; robust to parameter mismatch and non-Gaussian noise	EKF, GSP-EKF, KalmanNet, GNN+RNN
Traffic/weather/urban/synthetic graph signals	Competitive with cutting-edge GNNs in high-data regime; stable with extreme sparsity; data-driven statistical methods degrade severely	Graph WaveNet, STGCN, DCRNN, direct EM/ML

Latency metrics show GSP-KalmanNet to be the fastest scalable method: $\mathbf{x}_t \in \mathbb{R}^N$8s for $\mathbf{x}_t \in \mathbb{R}^N$9 nodes, $\mathbf{y}_t \in \mathbb{R}^N$0s for $\mathbf{y}_t \in \mathbb{R}^N$1, while model-based EKF fails for $\mathbf{y}_t \in \mathbb{R}^N$2 due to resource limits (Buchnik et al., 2023, Sabbaqi et al., 27 Jun 2025).

7. Extensions, Theoretical Properties, and Application Areas

Recent works generalize GSP-KalmanNet via SPDE-based priors, graph filter–parametrized Kalman gains, and graph convolutional encoder–decoders. The structure is stable to small graph-spectrum perturbations under mild GCNN Lipschitz constraints, and the SPDE formulation induces a non-separable spatio-temporal Gaussian process prior, connecting the state-space model to established kernel methods (Sabbaqi et al., 27 Jun 2025).

Key application areas include traffic forecasting, environmental/weather interpolation, epidemic tracking, water and power networks, and industrial sensor data. GSP-KalmanNet is notable for retaining high accuracy with minimal supervision, extreme sensor sparsity, and adverse model mismatches. Nonetheless, the current architecture is linear in state dynamics, and inference modules are graph-size dependent; open challenges include extending to nonlinear SPDEs, graph-size invariant networks, and time-varying graph topologies. End-to-end training involves standard RNN/GCNN convergence properties, and transferability across domains is achievable by fine-tuning RNN states (Sabbaqi et al., 27 Jun 2025).

GSP-KalmanNet thus provides a computationally efficient, robust, and flexible paradigm for dynamic inference over graph-structured data, unifying signal processing priors and neural network–based adaptivity (Buchnik et al., 2023, Sabbaqi et al., 27 Jun 2025).