FW-NKF: Frequency-Weighted Neural Kalman Filters

Abstract: Robust state estimation is central to robotic autonomy, yet classical Kalman filters struggle with frequency-dependent disturbances and model mismatch such as sensor vibrations, electromagnetic interference, and periodic noise. Although Deep Kalman Filter (DKF) variants extend the Extended Kalman Filtering (EKF) framework by learning latent transitions, they lack explicit mechanisms to suppress band-limited noise components that typically corrupt sensor measurements in real-world scenarios. We introduce the Frequency-Weighted Neural Kalman Filter (FW-NKF), a unified hybrid approach that embeds a causal spectral-shaping operator into the Kalman measurement residual and jointly learns observation, and transition networks. By adapting both the filter spectrum and the latent state representation, FW-NKF attenuates the noise-dominated frequency bands while capturing complex residual structures. We conduct extensive experiments on four heterogeneous benchmarks, including chaotic systems such as multi-dimensional Lorenz systems and full-body inertial pose estimation, and find a reduction in localization error of up to 10% as well as marked improvements in orientation accuracy. Our ablation studies confirm that frequency weighting and deep latent-state modeling contribute to overall performance.

• The paper introduces FW-NKF, which combines spectral reconstruction loss with a stable, learnable causal IIR filter for Kalman innovations, reducing estimation error across nonlinear, chaotic, and real-world sensor-fusion benchmarks.
• The method outperforms six baselines on pendulum, Lorenz-63, UIP-DB, and EuRoC MAV tasks, achieving reported MAE reductions of 28.6%, 88.2%, 19.9%, and 82.4%, respectively, while running at 0.686 ms per step.
• The results show that moderate spectral-loss weighting near λΦ = 0.01 is effective, but larger values can sharply degrade accuracy, highlighting the need for careful tuning and validation of the learned filter’s information-preserving behavior.

Overview

The paper introduces the Frequency-Weighted Neural Kalman Filter (FW-NKF), a hybrid state estimation framework that injects explicit spectral awareness into the Kalman predict–update cycle. The authors' central observation is that classical Kalman filters and their neural extensions (KalmanNet and its descendants) treat measurement noise as spectrally flat, whereas real sensor disturbances—IMU vibration jitter, low-frequency bias drift, periodic electromagnetic interference, and UWB multipath—are concentrated in specific frequency bands. FW-NKF addresses this gap with two complementary mechanisms: a spectral-domain reconstruction loss that supervises the observation model, and a learnable causal IIR filter applied to the Kalman innovation. The paper reports error reductions of up to 10% in localization settings, and substantially larger gains on chaotic benchmarks, while keeping per-step inference cost competitive with neural baselines (2606.02251).

Method

FW-NKF modifies the Kalman filter at two points without altering its recursive structure.

Spectral supervision of the observation model. During training, the observation pathway is supervised with a frequency-domain loss that compares the magnitude spectra of the predicted observation $H(\hat{\boldsymbol{x}}_{t|t-1})$ against a noiseless reference derived by applying $H(\cdot)$ to ground-truth states:

$$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$

The $\ell_1$ norm on magnitude spectra is phase-invariant, robust to temporal misalignment, and encourages the network $H(\cdot)$ to reconstruct noise-free signals by aligning dominant spectral components. This loss is combined with a standard state estimation loss weighted by $\lambda_\Phi$.

Learnable IIR innovation filtering. In the correction step, the raw innovation $\Delta\boldsymbol{y}_t$ is passed through a causal IIR filter $\Phi(z) = B(z^{-1})/A(z^{-1})$ with learnable feedforward coefficients $\{b_i\}$ and feedback coefficients $\{a_j\}$, implemented via a cascaded pole recursion plus FIR stage. The filtered innovation $H(\cdot)$0 replaces $H(\cdot)$1 in the state update. Stability is enforced by constraining the poles of $H(\cdot)$2 to lie strictly inside the unit circle, and covariance propagation uses the Joseph form to preserve positive semidefiniteness under finite precision. The per-step cost is $H(\cdot)$3, which the authors argue is negligible relative to the filter itself.

End-to-end integration. The transition and noise parameters $H(\cdot)$4 are generated by a GRU-conditioned parameterization, an optional mask $H(\cdot)$5 gates updates when measurements are missing, and the full forward pass is differentiable end to end. A supporting proposition, proved via the Cramér–Rao bound, states that if a causal linear filter $H(\cdot)$6 applied to the observations does not reduce Fisher information about the state, then the covariance lower bound for unbiased estimators on the filtered observation is no worse than on the raw observation. This is a conditional result: it holds only when filtering removes noise-dominated content without discarding state-informative components, and the paper does not establish that the learned $H(\cdot)$7 satisfies this condition in practice—it is a motivation rather than a guarantee.

Experimental setup

The method is evaluated on four heterogeneous benchmarks against six baselines: KalmanNet, BayesKNet, Recursive KNet, Recurrent KNet, classical KF, and an autoregressive KF. Notably, the spectral loss is applied to all baselines at matched $H(\cdot)$8 values, since it is a training objective applicable to any differentiable filter—this strengthens the comparison by isolating the architectural contribution of the IIR innovation filter.

The benchmarks are:

• Pendulum: a damped, forced nonlinear pendulum observed via noisy Cartesian tip positions (30,000 trajectories).
• Lorenz-63: the chaotic system with partial (2-of-3 state) observations and injected process perturbations, a deliberately hard setting.
• UIP-DB: 200 minutes of synchronized IMU and UWB range data with optical ground truth for full-body pose estimation, covering acceleration, rotation, and inter-sensor distance tasks.
• EuRoC MAV: 10 sequences of 200 Hz IMU data from a hexacopter, with a 10-dimensional state (position, velocity, quaternion).

Results

The headline results show FW-NKF at $H(\cdot)$9 outperforming all baselines on every benchmark:

Benchmark	FW-NKF (MAE)	Best baseline (MAE)	Reported gain
Pendulum	0.70 ± 0.07	0.98 (Recursive KNet)	28.6% MAE, 45.0% MSE
Lorenz	0.50 ± 0.15	4.24 (Recursive KNet)	88.2% MAE, 98.3% MSE
UIP-DB (acc. RMSE)	4.15 ± 0.20	5.18 (Recursive KNet)	~19.9%
EuRoC MAV	0.12 ± 0.01	0.68 (KalmanNet)	82.4% MAE, 98.3% MSE

The Lorenz and EuRoC gains are striking—FW-NKF's RMSE on EuRoC (0.19) is roughly seven times smaller than KalmanNet's (1.34), with R² of 0.99 versus 0.43. On UIP-DB, orientation MAE improves from 2.77 (classical KF) to 1.34, which the authors attribute to the learned IIR coefficients attenuating multipath-dominated bands before the update.

Sensitivity to $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$0 is a consistent finding. Moderate weighting ($$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$1) is optimal; larger values degrade performance sharply. On Lorenz, $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$2 causes MAE to jump from 0.50 to 8.18, and on the pendulum, MAE rises from 0.708 at $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$3 to 1.205 at $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$4. The authors conclude that excessive spectral emphasis interferes with the temporal objectives of sequential estimation, implying that the method requires careful hyperparameter selection rather than being robust to the loss weight.

Runtime. FW-NKF runs at 0.686 ms/step with 607K parameters—faster than BayesKNet (2.882 ms) and Recursive KNet (1.222 ms), though slower than the classical filter (0.276 ms). Recursive KNet was excluded from the EuRoC evaluation entirely due to exceeding 47 GB of memory, a practical limitation of that baseline rather than of FW-NKF.

Ablations and component analysis

The discussion attributes gains to both components: the spectral loss drives the observation network toward noise-free IMU reconstruction (19.9% RMSE improvement on UIP-DB), while the IIR innovation filter improves noise suppression in the correction step. The analysis of baseline failures is informative—classical KF degrades consistently with white-noise assumption violations, and BayesKNet's variational uncertainty estimation proves insufficient structure for spectrally complex, unstable dynamics, performing competitively on the pendulum but degrading substantially on Lorenz.

Limitations and open questions

Several caveats bear on the results. First, the theoretical justification (Proposition 1) is conditional on the filtered observation retaining at least as much Fisher information as the raw observation; the paper provides no empirical verification that the learned filter satisfies this, and the proposition does not cover the joint learning of $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$5 with the rest of the filter. Second, the noiseless reference targets $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$6 used in the spectral loss are themselves approximations—constructed by applying $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$7 to ground-truth states during training—so the quality of spectral supervision depends on the observation model's fidelity. Third, the sharp performance cliff at larger $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$8 (catastrophic on Lorenz at $$\mathcal{L}_{\Phi} = \Big\lVert \lVert \mathcal{F}(\hat{\boldsymbol{y}}_t) \rVert - \lVert \mathcal{F}(\tilde{\boldsymbol{y}}_t) \rVert \Big\rVert_1$$9) suggests the method's benefit hinges on a narrow hyperparameter regime, and the paper offers no principled procedure for selecting $\ell_1$0 on new domains. Fourth, the IIR filter is shared across measurement channels and applied componentwise; whether channel-specific or cross-channel spectral filtering would help is unexamined. Finally, the Lorenz evaluation uses the partial-observation setting only, and the EuRoC task uses a learned linear observation mapping—how the method behaves with fully nonlinear, high-dimensional observation models remains open.

Conclusion

FW-NKF demonstrates that embedding a learnable causal IIR filter in the Kalman innovation and supervising the observation model with a spectral reconstruction loss yields consistent accuracy improvements across chaotic simulation and real-world IMU/UWB fusion benchmarks, at inference cost comparable to existing neural Kalman filters. The evidence supports the paper's claim that treating measurement noise as spectrally uniform is a substantive modeling deficiency in practical estimation problems. The main open questions are whether the Fisher-information condition underlying the theoretical motivation holds for learned filters in practice, and how to select the spectral loss weight robustly, given the demonstrated sensitivity of performance to this single hyperparameter.