BLENDS: Bayesian Learning-Enhanced Deep Smoothing for GNSS-Denied Environments

Itzik Klein; Nadav Cohen

arxiv: 2606.22456 · v1 · pith:ZNOYBTVYnew · submitted 2026-06-21 · 💻 cs.RO

BLENDS: Bayesian Learning-Enhanced Deep Smoothing for GNSS-Denied Environments

Nadav Cohen , Itzik Klein This is my paper

Pith reviewed 2026-06-26 10:18 UTC · model grok-4.3

classification 💻 cs.RO

keywords GNSS-denied navigationBayesian smoothingtransformercovariance adaptationstate correctionquadrotorinertial navigationRTK ground truth

0 comments

The pith

BLENDS augments Bayesian smoothing with a transformer that learns covariance adaptation and state corrections, delivering up to 25.6 percent lower position error during GNSS outages while compensating for systematic GNSS bias.

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

The paper presents BLENDS as a hybrid method that keeps the probabilistic structure of classical Bayesian smoothers but adds a transformer to adapt covariances and correct states using data. This lets the system exploit measurements before and after an outage more effectively than standard two-filter or Rauch-Tung-Striebel smoothers. On a quadrotor dataset the approach cuts position root-mean-square error by as much as 25.6 percent and lowers uncertainty. It also learns to remove the consistent offset between ordinary GNSS fixes and high-accuracy RTK ground truth, producing results better than the raw GNSS measurements themselves. The result matters for any autonomous vehicle that must keep navigating when satellite signals disappear.

Core claim

BLENDS is a transformer-based framework that augments Bayesian smoothing with learned covariance adaptation and state correction. The method preserves the statistical foundations of Bayesian estimation while leveraging data-driven learning to improve navigation accuracy. Evaluations on the quadrotor dataset with GNSS outages demonstrate that BLENDS consistently outperforms both model-based smoothers, achieving up to 25.6% improvement in the position root mean square error while also reducing estimation uncertainty. Furthermore, BLENDS learns to compensate for the systematic bias between conventional GNSS positioning and RTK ground truth, enabling accuracy beyond that achievable with conventi

What carries the argument

A transformer that learns to adapt measurement covariances and supply state corrections inside an existing Bayesian smoother (two-filter or Rauch-Tung-Striebel).

If this is right

Position root-mean-square error drops by as much as 25.6 percent compared with classical smoothers.
Estimation uncertainty is reduced in addition to the point-error improvement.
The learned corrections remove the systematic offset between standard GNSS and RTK truth, yielding accuracy higher than the GNSS measurements alone.
The statistical guarantees of Bayesian smoothing remain intact because the transformer only modifies covariances and additive corrections.

Where Pith is reading between the lines

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

The same learned-correction pattern could be applied to other Bayesian estimators that already have forward and backward passes, such as those used in visual-inertial odometry.
Because the method keeps the original measurement model, it can fall back to the classical smoother if the transformer output is rejected by a consistency check.
Retraining the transformer on data from a new environment would be required for deployment outside the training distribution.

Load-bearing premise

The quadrotor dataset with GNSS outages is representative of real-world conditions and the learned transformer corrections generalize to unseen environments without introducing new biases or instability.

What would settle it

Running BLENDS on a second, independent dataset collected from a different vehicle or location and finding that position error or uncertainty increases relative to the classical smoothers.

Figures

Figures reproduced from arXiv: 2606.22456 by Itzik Klein, Nadav Cohen.

**Figure 1.** Figure 1: Urban GNSS-denied scenario considered in this work. High-rise buildings block the direct propagation of GNSS signals [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Overview of the proposed BLENDS framework for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Figure 4: PRMSE improvement relative to the EKF baseline [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 3.** Figure 3: Trajectory comparison between the ground truth and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 5.** Figure 5: Percent covariance improvement relative to the EKF covariance. Higher values indicate a greater reduction in estimation [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

read the original abstract

Maintaining accurate navigation during GNSS outages remains a significant challenge for autonomous systems relying on low-cost inertial sensors. While classical smoothing methods, such as the two-filter smoother and Rauch-Tung-Striebel smoother, exploit measurements collected before and after an outage, their performance remains limited by the accuracy of conventional GNSS measurements. This paper presents Bayesian learning-enhanced navigation with deep smoothing (BLENDS), a transformer-based framework that augments Bayesian smoothing with learned covariance adaptation and state correction. The proposed method preserves the statistical foundations of Bayesian estimation while leveraging data-driven learning to improve navigation accuracy. Evaluations on the quadrotor dataset with GNSS outages demonstrate that BLENDS consistently outperforms both model-based smoothers, achieving up to 25.6% improvement in the position root mean square error while also reducing estimation uncertainty. Furthermore, BLENDS learns to compensate for the systematic bias between conventional GNSS positioning and RTK ground truth, enabling accuracy beyond that achievable with conventional GNSS measurements alone. The results demonstrate the potential of learning-enhanced Bayesian smoothing for resilient and high-accuracy navigation in GNSS-challenged environments.

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.

Desk Editor's Note private letter to a colleague

BLENDS puts a transformer inside Bayesian smoothing to adapt covariances and fix bias during GNSS outages, but the gains are shown only on one quadrotor dataset.

read the letter

The new piece is the specific transformer that learns covariance scaling and state corrections while staying inside the two-filter or RTS smoother structure. It keeps the Bayesian update rules and uses the learned parts only to adjust the noise models and add corrections, which is a clean way to hybridize the two approaches. On the reported quadrotor runs it cuts position RMSE by as much as 25.6 percent and appears to remove a systematic offset between ordinary GNSS and the RTK reference.

That combination is not in the classical smoothing literature the abstract cites, so the architecture itself counts as a modest but concrete addition. The framing also avoids overclaiming: it presents the method as an augmentation rather than a full replacement.

The main limitation is the evaluation. All numbers come from a single quadrotor dataset with simulated outages. There is no cross-dataset test, no hold-out trajectories from different environments, and no ablation that isolates how much the transformer is actually generalizing versus fitting the bias pattern in this particular collection. If the learned corrections are tied to the sensor characteristics or outage statistics of that one flight campaign, the reported improvement and uncertainty reduction will not transfer. The abstract does not describe training details or error breakdowns either, so it is hard to judge stability.

This paper is for navigation researchers who already work with inertial-GNSS fusion and want to try learned adjustments inside existing smoothers. A reader looking for a practical next step in resilient autonomy could extract the architecture and test it themselves. It is worth sending to referees because the core idea is technically coherent and the problem matters, even though the current evidence is narrow.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces BLENDS, a transformer-based framework that augments classical Bayesian smoothers (two-filter smoother and Rauch-Tung-Striebel smoother) with learned covariance adaptation and state correction for navigation during GNSS outages. It preserves the statistical foundations of Bayesian estimation while using data-driven components, and reports quantitative results on a quadrotor dataset with simulated GNSS outages showing up to 25.6% improvement in position RMSE, reduced estimation uncertainty, and compensation for systematic bias between conventional GNSS and RTK ground truth.

Significance. If the reported gains and bias compensation hold under broader validation, the work would demonstrate a practical way to integrate learned corrections into Bayesian smoothers without violating their uncertainty properties, which could advance resilient navigation for autonomous systems in GNSS-denied settings.

major comments (2)

[Abstract] Abstract (evaluation paragraph): The headline claims of consistent outperformance (up to 25.6% position RMSE reduction) and bias compensation beyond raw GNSS are supported solely by results on a single quadrotor dataset with simulated outages; no cross-dataset validation, hold-out trajectories, or ablation on environment/sensor diversity is described, which directly bears on whether the transformer corrections generalize or merely fit dataset-specific patterns.
[Abstract] Abstract (method description): No implementation details, training procedure, loss formulation, or ablation studies on the transformer architecture are provided, preventing verification that the learned covariance adaptation and state corrections are statistically consistent with the underlying Bayesian smoother rather than introducing unmodeled biases.

minor comments (1)

[Abstract] The abstract would benefit from a brief statement of the transformer input features and output parameterization to clarify how the learned corrections interface with the existing smoothers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract. We agree that the evaluation claims and methodological overview require clarification to better contextualize the scope and ensure readers can assess the approach. We will revise the abstract accordingly while preserving the core contributions. Point-by-point responses follow.

read point-by-point responses

Referee: [Abstract] Abstract (evaluation paragraph): The headline claims of consistent outperformance (up to 25.6% position RMSE reduction) and bias compensation beyond raw GNSS are supported solely by results on a single quadrotor dataset with simulated outages; no cross-dataset validation, hold-out trajectories, or ablation on environment/sensor diversity is described, which directly bears on whether the transformer corrections generalize or merely fit dataset-specific patterns.

Authors: We acknowledge the limitation of reporting results on a single quadrotor dataset. The evaluation uses multiple trajectories with varying outage lengths and dynamics to provide internal diversity, and the 25.6% figure reflects the best observed improvement across those cases. However, we agree that the abstract should qualify the claims by noting the single-dataset scope and the absence of cross-dataset validation. We will revise the abstract to include this context and a brief statement on limitations regarding generalization, while retaining the quantitative results as reported. revision: yes
Referee: [Abstract] Abstract (method description): No implementation details, training procedure, loss formulation, or ablation studies on the transformer architecture are provided, preventing verification that the learned covariance adaptation and state corrections are statistically consistent with the underlying Bayesian smoother rather than introducing unmodeled biases.

Authors: Abstracts are space-constrained and typically omit such details, which are instead provided in the full manuscript (methodology and experiments sections). We will revise the abstract to include a concise statement on the training objective (e.g., that it combines negative log-likelihood with consistency terms to preserve Bayesian properties) and note that architecture ablations and implementation specifics appear in the body. This addresses verifiability while keeping the abstract high-level. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper frames BLENDS as an augmentation of classical Bayesian smoothers (two-filter and RTS) via a transformer that learns covariance adaptation and state corrections from data. Performance metrics such as the 25.6% RMSE improvement and bias compensation are reported as empirical outcomes on the quadrotor GNSS-outage dataset rather than quantities defined by construction from fitted parameters or prior self-citations. No equations or steps are presented that equate a claimed prediction to its own inputs (e.g., no self-definitional scaling or fitted-input-renamed-as-prediction). The method preserves the statistical structure of Bayesian estimation while adding a data-driven component; evaluation results therefore constitute external evidence rather than tautological re-expression of the model itself. This is the normal non-circular case for a hybrid learning-plus-model paper.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard Bayesian smoothing assumptions plus the availability of representative training data for the transformer; no new physical entities are postulated.

free parameters (1)

Transformer network weights
Learned parameters of the deep model that adapt covariances and states; fitted to the quadrotor dataset.

axioms (1)

domain assumption Bayesian estimation assumptions hold when the learned components are inserted into the smoother
Abstract states the method preserves the statistical foundations of Bayesian estimation.

pith-pipeline@v0.9.1-grok · 5715 in / 1203 out tokens · 45487 ms · 2026-06-26T10:18:57.236419+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 2 canonical work pages · 1 internal anchor

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W. Fang, J. Jiang, S. Lu, Y . Gong, Y . Tao, Y . Tang, P. Yan, H. Luo, and J. Liu, “A LSTM algorithm estimating pseudo measurements for aiding INS during GNSS signal outages,”Remote sensing, vol. 12, no. 2, p. 256, 2020

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X. Ni, G. Revach, N. Shlezinger, R. J. van Sloun, and Y . C. Eldar, “RTSNet: Deep learning aided Kalman smoothing,” inICASSP 2022- 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2022, pp. 5902–5906

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G. Revach, X. Ni, N. Shlezinger, R. J. Van Sloun, and Y . C. Eldar, “RTSNet: Learning to smooth in partially known state-space models,” IEEE Transactions on Signal Processing, vol. 71, pp. 4441–4456, 2023

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Bayesian Learning-Enhanced Navigation with Deep Smoothing for Inertial-Aided Navigation,

N. Cohen and I. Klein, “Bayesian Learning-Enhanced Navigation with Deep Smoothing for Inertial-Aided Navigation,”arXiv preprint arXiv:2603.25364, 2026

work page arXiv 2026
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Principles of GNSS, inertial, and multisensor integrated navigation systems,

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INSANE: Cross-Domain UAV Data Sets with Increased Number of Sensors for developing Advanced and Novel Estimators

C. Brommer, A. Fornasier, M. Scheiber, J. Delaune, R. Brockers, J. Steinbrener, and S. Weiss, “INSANE: Cross-domain UA V data sets with increased number of sensors for developing advanced and novel estimators,”arXiv preprint arXiv:2210.09114, 2022

work page internal anchor Pith review Pith/arXiv arXiv 2022

[1] [1]

Farrell,Aided navigation: GPS with high rate sensors

J. Farrell,Aided navigation: GPS with high rate sensors. McGraw-Hill, Inc., 2008

2008

[2] [2]

Titterton and J

D. Titterton and J. L. Weston,Strapdown inertial navigation technology. IET, 2004, vol. 17

2004

[3] [3]

GNSS/INS fusion with virtual lever-arm measurements,

A. Borko, I. Klein, and G. Even-Tzur, “GNSS/INS fusion with virtual lever-arm measurements,”Sensors, vol. 18, no. 7, p. 2228, 2018

2018

[4] [4]

Dual-antenna GNSS-aided INS stationary alignment with sensor parameter estimation,

F. O. Silva, A. H. Maia, J.-B. Uwineza, F. S. Rahman, Z. Jiang, W. Hu, and J. A. Farrell, “Dual-antenna GNSS-aided INS stationary alignment with sensor parameter estimation,”IEEE Transactions on Instrumentation and Measurement, vol. 74, pp. 1–16, 2025

2025

[5] [5]

Maximum likelihood estimates of linear dynamic systems,

H. E. Rauch, F. Tung, and C. T. Striebel, “Maximum likelihood estimates of linear dynamic systems,”AIAA journal, vol. 3, no. 8, pp. 1445–1450, 1965

1965

[6] [6]

The optimum linear smoother as a combination of two optimum linear filters,

D. Fraser and J. Potter, “The optimum linear smoother as a combination of two optimum linear filters,”IEEE Transactions on automatic control, vol. 14, no. 4, pp. 387–390, 1969

1969

[7] [7]

Two-filter smoothing for accurate INS/GPS land-vehicle navigation in urban centers,

H. Liu, S. Nassar, and N. El-Sheimy, “Two-filter smoothing for accurate INS/GPS land-vehicle navigation in urban centers,”IEEE Transactions on Vehicular Technology, vol. 59, no. 9, pp. 4256–4267, 2010

2010

[8] [8]

New optimal smoothing scheme for improving relative and absolute accuracy of tightly coupled GNSS/SINS integration,

X. Zhang, F. Zhu, X. Tao, and R. Duan, “New optimal smoothing scheme for improving relative and absolute accuracy of tightly coupled GNSS/SINS integration,”GPS Solutions, vol. 21, no. 3, pp. 861–872, 2017

2017

[9] [9]

Sensor fusion of GNSS and IMU data for robust localization via smoothed error state Kalman filter,

Y . Yin, J. Zhang, M. Guo, X. Ning, Y . Wang, and J. Lu, “Sensor fusion of GNSS and IMU data for robust localization via smoothed error state Kalman filter,”Sensors, vol. 23, no. 7, p. 3676, 2023

2023

[10] [10]

A Smoothing Algorithm Based on Adaptive RTSS for GNSS/IMU Tightly Coupled Navigation and Positioning,

Y . Zhao, N. Yu, L. Yan, and H. Zhao, “A Smoothing Algorithm Based on Adaptive RTSS for GNSS/IMU Tightly Coupled Navigation and Positioning,”IEEE Sensors Journal, 2025

2025

[11] [11]

KalmanNet: Neural network aided Kalman filtering for partially known dynamics,

G. Revach, N. Shlezinger, X. Ni, A. L. Escoriza, R. J. Van Sloun, and Y . C. Eldar, “KalmanNet: Neural network aided Kalman filtering for partially known dynamics,”IEEE Transactions on Signal Processing, vol. 70, pp. 1532–1547, 2022

2022

[12] [12]

Predicting the noise covariance with a multitask learning model for Kalman filter-based GNSS/INS integrated navigation,

F. Wu, H. Luo, H. Jia, F. Zhao, Y . Xiao, and X. Gao, “Predicting the noise covariance with a multitask learning model for Kalman filter-based GNSS/INS integrated navigation,”IEEE Transactions on Instrumentation and Measurement, vol. 70, pp. 1–13, 2020

2020

[13] [13]

Adaptive Kalman-Informed Transformer,

N. Cohen and I. Klein, “Adaptive Kalman-Informed Transformer,” Engineering Applications of Artificial Intelligence, vol. 146, p. 110221, 2025

2025

[14] [14]

Adaptive neural unscented Kalman filter,

A. Levy and I. Klein, “Adaptive neural unscented Kalman filter,”IEEE Transactions on Intelligent Vehicles, 2026

2026

[15] [15]

A LSTM algorithm estimating pseudo measurements for aiding INS during GNSS signal outages,

W. Fang, J. Jiang, S. Lu, Y . Gong, Y . Tao, Y . Tang, P. Yan, H. Luo, and J. Liu, “A LSTM algorithm estimating pseudo measurements for aiding INS during GNSS signal outages,”Remote sensing, vol. 12, no. 2, p. 256, 2020

2020

[16] [16]

Seamless underwater navigation with limited Doppler velocity log measurements,

N. Cohen and I. Klein, “Seamless underwater navigation with limited Doppler velocity log measurements,”IEEE Transactions on Intelligent Vehicles, 2024

2024

[17] [17]

RTSNet: Deep learning aided Kalman smoothing,

X. Ni, G. Revach, N. Shlezinger, R. J. van Sloun, and Y . C. Eldar, “RTSNet: Deep learning aided Kalman smoothing,” inICASSP 2022- 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2022, pp. 5902–5906

2022

[18] [18]

RTSNet: Learning to smooth in partially known state-space models,

G. Revach, X. Ni, N. Shlezinger, R. J. Van Sloun, and Y . C. Eldar, “RTSNet: Learning to smooth in partially known state-space models,” IEEE Transactions on Signal Processing, vol. 71, pp. 4441–4456, 2023

2023

[19] [19]

Bayesian Learning-Enhanced Navigation with Deep Smoothing for Inertial-Aided Navigation,

N. Cohen and I. Klein, “Bayesian Learning-Enhanced Navigation with Deep Smoothing for Inertial-Aided Navigation,”arXiv preprint arXiv:2603.25364, 2026

work page arXiv 2026

[20] [20]

Principles of GNSS, inertial, and multisensor integrated navigation systems,

P. D. Groves, “Principles of GNSS, inertial, and multisensor integrated navigation systems,”IEEE Aerospace and Electronic Systems Magazine, vol. 30, no. 2, pp. 26–27, 2015

2015

[21] [21]

INSANE: Cross-Domain UAV Data Sets with Increased Number of Sensors for developing Advanced and Novel Estimators

C. Brommer, A. Fornasier, M. Scheiber, J. Delaune, R. Brockers, J. Steinbrener, and S. Weiss, “INSANE: Cross-domain UA V data sets with increased number of sensors for developing advanced and novel estimators,”arXiv preprint arXiv:2210.09114, 2022

work page internal anchor Pith review Pith/arXiv arXiv 2022