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WRAP: Wasserstein-Robust Adaptive Plug-in for Robot Localization

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read WRAP, a Wasserstein-robust plug-in for EKF/ESKF localization, cuts held-out UWB position RMSE by 27.4%.

desk verdict Solid plug-in contribution with honest limitations; the headline DR increment over adapter-only lacks paired statistics and a frozen hyperparameter split, so the unique empirical claim may not survive scrutiny. read the letter →

arxiv 2608.09807 v1 pith:EQSTKRWO submitted 2026-08-10 cs.RO cs.SYeess.SY

Minhyuk Jang , Astghik Hakobyan , Jungjin Lee , Naira Hovakimyan , Insoon Yang This is my paper · ORCID
classification cs.ROcs.SYeess.SY
keywords robotlocalizationextendedKalmanfiltererror-statedistributionallyrobustoptimizationWassersteindistanceadaptivefilteringUWB–IMUGNSS–INSnavigation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether an existing extended or error-state Kalman filter (EKF/ESKF) localization stack can be made robust to changing sensing conditions without replacing the filter. Its answer is WRAP, a plug-in that sits at the noise-statistics interface: a causal adapter supplies time-varying effective process and measurement means and covariances, and a mean-preserving Wasserstein local update then replaces the nominal covariances in the Kalman gain with least-favorable ones. Because the ambiguity sets preserve the adapted means, the robust step cannot correct a wrong mean; it can only reweight propagation against sensing. On 18 UWB–IMU sequences held out from adapter training, adapter-only and WRAP reduce mean 3-D position RMSE by 19.8% and 27.4% relative to the nominal ESKF, and an isotropic ablation reaches 19.5%, which the paper reads as evidence that directional process-covariance redistribution is what adds the extra gain. The authors also report real-time covariance-solve times on embedded hardware and, in an in-sample GNSS–INS study, show that mean adaptation supplies most of the accuracy gain while the robust update improves consistency and rescues over-tightened classical covariance estimates.

What carries the argument

The load-bearing object is the mean-preserving Wasserstein ambiguity set around each adapted nominal noise law, which becomes a Bures–Wasserstein covariance constraint once the means are fixed. The argument runs through the affine–Gaussian reduction of the local minimax estimation problem: the worst-case laws are Gaussian, the minimax estimator is an affine Kalman correction, and the whole problem reduces to a concave covariance maximization. Solving that program yields least-favorable process and measurement covariances, hence the robust gain $K^*_t = \Sigma^{-,*}_{x,t} C_t^{\top} (S^*_t)^{-1}$. This mechanism does the work of separating mean adaptation from covariance robustification and of letting the process and measurement radii be chosen independently.

What would settle it

Run WRAP on a system whose linearization error is known to be large—say, an ESKF with a fast rotation and a nearly unobservable attitude update—and compare the reported posterior covariance with Monte Carlo position-error statistics. If the coverage of the least-favorable covariance degrades monotonically as the linearization residual grows, the affine–Gaussian surrogate is the point of failure.

Watch

Extended reading notes

Core claim

The paper's central claim is that a filter's two failure modes under environmental change—a shifted noise mean and a miscalibrated noise covariance—can be attacked separately through a single adapter-agnostic plug-in. WRAP defines a mean-preserving Wasserstein ambiguity set around the adapted nominal law for process and measurement noise, solves a local minimax mean-square-error problem over those sets, and uses the resulting least-favorable covariances to form a robust Kalman gain. The propagation model, residual, gating statistic, and retraction of the baseline ESKF/EKF are untouched; only the covariance input to the gain changes. The paper reports that this combination improves held-out UWB–IMU trajectory RMSE beyond adapter-only adaptation and, in the GNSS–INS mechanism study, yields better consistency statistics than the nominal filter while leaving persistent mean error unremoved.

Load-bearing premise

The whole construction treats the linearized affine–Gaussian recurrence as a faithful surrogate for the nonlinear EKF/ESKF at every step, so that the least-favorable covariances computed from the local minimax problem remain valid throughout a trajectory.

Editorial extensions

If this is right

  • Any causal adapter—learned, classical, or Bayesian—can be connected to an existing EKF/ESKF through the noise-statistics interface, so the robust gain is available without redesigning the filter core.
  • In bias-dominant regimes, mean adaptation is the main accuracy lever, so WRAP stays close to adapter-only; in low-bias, tail/scale-mismatch regimes, DR-only can help where adaptation hurts, pointing toward regime-aware selection.
  • The observed gap between the full covariance search (27.4%) and the isotropic ablation (19.5%) implies that directional redistribution of process covariance, not simple inflation, is the source of the incremental gain on these sequences.
  • The robust covariance solve is small relative to update budgets (0.05 ms for UWB and 2.92 ms for GNSS), so the plug-in is real-time at the tested update rates on embedded hardware.
  • Because the ambiguity sets preserve the adapted mean, WRAP improves consistency and mitigates over-tightened covariance estimates but cannot remove persistent mean-model error; the GNSS–INS NEES stays above the ideal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is an online regime classifier that triggers adapter-only, DR-only, or WRAP based on a causal proxy for bias dominance and covariance headroom; the paper's post-hoc diagnostics suggest opposite regimes favor different variants, but the authors stop short of an online rule.
  • Because the radius inherits the units and scaling of the noise coordinates, normalizing radii by trace or eigenvalue scale could make them transferable across estimators and datasets, addressing the paper's stated limitation.
  • The robustification is deliberately absent from the gating statistic, so a natural next step is a least-favorable gating threshold that accounts for the worst-case innovation covariance; that would extend WRAP to gross outlier rejection rather than just gain reweighting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces WRAP, a plug-in for EKF/ESKF localization stacks that combines a causal adaptive module for noise means and covariances with a mean-preserving Wasserstein distributionally robust covariance update. The robust update is posed as the local minimax problem in Eq. (11) and reformulated as the Bures–Wasserstein covariance program in Eq. (12), solved online by Frank–Wolfe. Experiments on 18 UWB–IMU sequences and 6 GNSS–INS sequences compare a nominal ESKF/KF-GINS against adapter-only, DR-only, and WRAP variants, reporting RMSE improvements, an isotropic ablation, consistency diagnostics, and embedded runtime. The central claim is that WRAP provides an incremental accuracy gain over adapter-only adaptation, attributable to directional process-covariance redistribution, while remaining adapter-agnostic and real-time.

Significance. If fully supported, WRAP would be a practically useful contribution: it promises a modular, real-time path to improve localization accuracy without altering the core EKF/ESKF machinery, and it cleanly separates mean adaptation from covariance robustification. The paper is unusually transparent about its own limitations, explicitly labeling the GNSS study as in-sample, the minimax result as local, and the diagnostic measures as heuristic; it also provides source code and embedded runtime measurements. However, the central incremental-gain claim is currently underpinned by comparisons without paired statistics or confidence intervals, and the evaluation protocol does not use a frozen hyperparameter split for the Wasserstein radii. The theoretical reduction is inherited from prior work (Nguyen et al. [25] and the same-author preprint [29]) rather than newly proven, so the novelty rests on the integration and the empirical evidence. The manuscript is therefore interesting and likely to be of value to the robotics estimation community, but the evidence for the headline claim needs substantial strengthening before acceptance.

major comments (3)
  1. [§4.3, Table 1, Fig. 2] The headline incremental gain of WRAP over adapter-only (27.4% vs 19.8% mean RMSE reduction) is not supported by paired statistics or confidence intervals. In Table 1, the per-constellation differences are small relative to the reported across-sequence standard deviations: for constellation #1, 0.071±0.010 vs 0.065±0.009; for #2, 0.069±0.006 vs 0.067±0.008; for #4, 0.375±0.075 vs 0.327±0.053. Fig. 2 reports only pooled means. Because the Mamba backbone is the best of the three tested backbones, selection effects may also inflate the observed gap. Please report per-sequence paired differences with confidence intervals or a permutation test, and if the evidence is inconclusive, explicitly present the incremental gain as an observed trend rather than an established result.
  2. [§4.1, Evaluation Protocol; §5, Conclusion] The protocol does not provide a frozen hyperparameter test set: the paper states that each evaluation trial uses radii selected on the other two trials, so the trials are held out from adapter training but do not form a fully frozen hyperparameter test set. This means the absolute 27.4% improvement and the incremental gap over adapter-only can be optimistic relative to a deployment setting where radii would be fixed beforehand. Please either use nested or frozen-split radius selection, or report the distribution of selected radii across trials and the sensitivity of the WRAP-versus-adapter gap to the radius values, and qualify the abstract and conclusion accordingly.
  3. [§4.3, Isotropic ablation] The sentence attributing the observed incremental gain to directional process-covariance redistribution rests on a single pooled comparison between the isotropic ablation (19.5%) and the full covariance search (27.4%), with no uncertainty quantification. Since the measurement-side search and isotropic search coincide for scalar UWB updates, the comparison isolates the process side, but the conclusion would be much stronger if the paper reported per-sequence paired differences between WRAP and the isotropic ablation, including the sign of the difference on each sequence. As written, the ablation is suggestive but not statistically load-bearing.
minor comments (5)
  1. [Abstract] The abstract reports the 19.8% and 27.4% reductions without any caveat about the radius-selection protocol or the absence of confidence intervals; please add a qualifier such as 'observed in this study' or report the uncertainty.
  2. [§4.1, post-hoc diagnostics] The definitions of H_w and T are dense; please state explicitly that q0.99 is the empirical 99th percentile of the standardized residual and that 2.576 is the 99th percentile of a standard normal, so the reader can immediately see that T measures tail deviation from the Gaussian reference.
  3. [§4.2, runtime] The reported inference times of 21–30 ms for the learned adapters are for a 1 Hz refresh, while the DR solve runs at update rate; please clarify whether the 0.05 ms and 2.92 ms figures include the overhead of the warm-started Frank–Wolfe call in Algorithm 1 or only the solver itself, and state how many Frank–Wolfe iterations are typically needed at the tolerance of 10^-4.
  4. [Table 3] The GNSS–INS table reports only pooled mean improvements across six sequences; given the explicitly in-sample nature of this study, please include per-sequence values or a measure of spread so the reader can judge the stability of the 16.1% and 15.7% figures.
  5. [§3.2, Eq. (12)] The lower bounds Σw ⪰ λmin(Σhat_w) I and Σv ⪰ λmin(Σhat_v) I are said to be redundant while preserving the optimum; adding a precise reference to the proposition in [25] or [29] that justifies this redundancy would help readers who are not familiar with the Bures–Wasserstein reformulation.

Circularity Check

1 steps flagged · score 4.0 of 10

Covariance reformulation is inherited from a same-author preprint, but the localization gains are measured against baselines; no empirical circularity found.

  1. self citation load bearing [Section 3.2, around Eq. (12)]
    "With G_{t-1} and D_t mapping the two noise terms, the noise-centric extension [29, Lemma 1 and Remark 3] reduces (11) to the covariance program below. This is a stagewise result for the linearized update, not a global minimax or stability result for the nonlinear recursion."

    The paper's central theoretical object, the covariance reformulation (12), is not proved here; it is asserted to follow from [25] and from [29], a same-author preprint. Since [29] is not machine-checked or otherwise independently validated inside this manuscript, the derivation of the WRAP update is inherited from a self-citation. This does not make the empirical RMSE comparison circular, because the UWB evaluation is on adapter-held-out sequences and the GNSS study is explicitly in-sample; the theoretical claim is imported rather than derived.

full rationale

WRAP's empirical claim is not circular: the UWB adapters are trained on trials 1-3 and 7 and evaluated on trials 4-6, and the GNSS study is explicitly labeled in-sample rather than a generalization test. The reported RMSE numbers therefore are measured against a baseline, not derived from the method's assumptions. The ablation attribution is an ordinary experimental comparison: restricting the covariance search to isotropic inflations and observing a smaller gain than the full search is not an identity; it is a controlled comparison. The one load-bearing self-citation is in Section 3.2: the reduction of the minimax problem (11) to the covariance program (12) is attributed to [25] and to the same-author preprint [29]. Because [29] is not independently verified in this manuscript (no proof, no external checker), the theoretical derivation of the WRAP update is inherited rather than re-derived; this is a 'some self-citation' situation. It does not force the empirical conclusion, so the score is 4 rather than 6+. Statistical concerns (radii cross-validated, no paired tests) are correctness risks, not circularity, and the paper itself notes that the trials 'do not form a fully frozen hyperparameter test set.'

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central method rests on two inputs: the linearized Gaussian surrogate for the filtering recursion and the radii that define the ambiguity sets. The former is a standard domain assumption; the latter is a tuned free parameter. The DR reduction itself is imported from prior work, one external and one same-author preprint, so the paper's novelty is the plug-in integration and empirical study rather than a new derivation.

free parameters (2)
  • Wasserstein radii θw and θv = selected from {0, 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, 5.0}; per-estimator values not reported
    Tuned per estimator on held-out or leave-one-sequence-out trials (Section 4.1). The radii define ambiguity-set size and are a primary tuning input to the robust update.
  • Learned adapter hyperparameters = UWB: d=128, 4 blocks, 100-step window; GNSS: d=256, 2 blocks, 10-step window; 1 Hz refresh
    Hand-chosen architecture settings for the MUSE-based adapters (Section 4.2); needed to reproduce the reported results.
assumptions (4)
  • standard math Wasserstein MMSE reduction of Nguyen et al. [25, Thm 3.1, Cor 4.1] and the noise-centric extension [29, Lemma 1, Remark 3] reduce the minimax problem (11) to the covariance program (12).
    Invoked in Section 3.2; proof not reproduced in this paper and [29] is a same-author preprint.
  • domain assumption Previous estimation error, process noise, and measurement noise are mutually independent, and nominal laws are Gaussian.
    Stated in footnote 1 of Section 3.2; needed for the affine-Gaussian stage and the Bures-Wasserstein formulation.
  • domain assumption D_t has full row rank.
    Required for S_t positive definiteness (Section 3.2).
  • domain assumption The local affine-Gaussian models (5)-(6) faithfully approximate the nonlinear recursion.
    The paper calls the minimax result 'a stagewise result for the linearized update, not a global minimax or stability result' (Section 3.2).

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Cite this review

Pith. "Pith review of WRAP: Wasserstein-Robust Adaptive Plug-in for Robot Localization." pith.science (2026). https://pith.science/paper/EQSTKRWO

@misc{pith2026260809807,
  author       = {Pith},
  title        = {Pith review of: WRAP: Wasserstein-Robust Adaptive Plug-in for Robot Localization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQSTKRWO}},
  note         = {Machine review of arXiv:2608.09807}
}
abstract

Robotic localization under changing sensing conditions can suffer from biased errors and miscalibrated covariances. We present WRAP, an adapter-agnostic Wasserstein-robust plug-in for nonlinear extended Kalman filter (EKF) and error-state Kalman filter (ESKF) stacks. A causal module supplies time-varying effective process and measurement statistics; a mean-preserving Wasserstein local update then computes least-favorable covariances and a robust gain without changing the propagation model, residual, or retraction. This separates mean adaptation from covariance robustification and uses distinct radii for propagation and sensing. On 18 UWB--IMU sequences held out from adapter training, adapter-only and WRAP reduce mean 3-D position RMSE by $19.8\%$ and $27.4\%$ relative to the nominal ESKF; an isotropic ablation reaches $19.5\%$, linking the incremental gain to directional process-covariance redistribution. An in-sample GNSS--INS study shows that mean adaptation provides most of the accuracy gain, while DR improves consistency and mitigates over-tightened classical covariance estimates. The robust solve takes 0.05 ms for UWB and 2.92 ms for GNSS on a Jetson Orin Nano.

Figures

Figures reproduced from arXiv: 2608.09807 by the authors.

Figure 1
Figure 1. Overview of WRAP. The adaptive module supplies nominal process and measurement [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Mean 3-D position RMSE improvement on 18 UWB–IMU evaluation sequences (held out [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Error-regime analysis on the UTIL TDOA2 sequences, with Mamba used for the adaptive [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Works this paper leans on

42 extracted references · 31 canonical work pages

  1. [25]

    Bridging Bayesian and minimax mean square error estimation via Wasserstein distributionally robust optimization,

    V. A. Nguyen, S. Shafieezadeh-Abadeh, D. Kuhn, and P. Mohajerin Esfahani, “Bridging Bayesian and minimax mean square error estimation via Wasserstein distributionally robust optimization,”Math. Oper. Res., vol. 48, no. 1, pp. 1–37, 2023

    2023

  2. [29]

    Distributionally robust Kalman filter,

    ——, “Distributionally robust Kalman filter,”arXiv preprint arXiv:2512.06286, 2025

    arXiv 2025

  3. [1]

    P. D. Groves,Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd ed. Artech House, 2013

    work page 2013

  4. [2]

    UTIL: An ultra-wideband time-difference- of-arrival indoor localization dataset,

    W. Zhao, A. Goudar, X. Qiao, and A. P. Schoellig, “UTIL: An ultra-wideband time-difference- of-arrival indoor localization dataset,”Int. J. Robot. Res., vol. 43, no. 10, pp. 1443–1456, 2024

    work page 2024

  5. [3]

    OpenVINS: A research platform for visual-inertial estimation,

    P. Geneva, K. Eckenhoff, W. Lee, Y. Yang, and G. Huang, “OpenVINS: A research platform for visual-inertial estimation,” inProc. IEEE Int. Conf. Robot. Autom., 2020, pp. 4666–4672

    work page 2020

  6. [4]

    F AST-LIO2: Fast direct LiDAR-inertial odom- etry,

    W. Xu, Y. Cai, D. He, J. Lin, and F. Zhang, “F AST-LIO2: Fast direct LiDAR-inertial odom- etry,”IEEE Trans. Robot., vol. 38, no. 4, pp. 2053–2073, Aug. 2022

    work page 2022

  7. [5]

    Contact-aided invariant extended Kalman filtering for robot state estimation,

    R. Hartley, M. Ghaffari, R. M. Eustice, and J. W. Grizzle, “Contact-aided invariant extended Kalman filtering for robot state estimation,”Int. J. Robot. Res., vol. 39, no. 4, pp. 402–430, Mar. 2020

    work page 2020

  8. [6]

    A new approach to linear filtering and prediction problems,

    R. E. Kalman, “A new approach to linear filtering and prediction problems,”J. Basic Eng., vol. 82, pp. 35–45, 1960

    1960

Show all 42 references
  1. [7]

    A micro Lie theory for state estimation in robotics,

    J. Sol` a, J. Deray, and D. Atchuthan, “A micro Lie theory for state estimation in robotics,” arXiv preprint arXiv:1812.01537, 2018

    2018 arXiv

  2. [8]

    Novel approach to nonlinear/non-Gaussian Bayesian state estimation,

    N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,”IEEE Proc. F Radar Signal Process., vol. 140, no. 2, pp. 107–113, 1993

    1993

  3. [9]

    Unscented filtering and nonlinear estimation,

    S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,”Proc. IEEE, vol. 92, no. 3, pp. 401–422, 2004

    2004

  4. [10]

    The iterated Kalman filter update as a Gauss-Newton method,

    B. M. Bell and F. W. Cathey, “The iterated Kalman filter update as a Gauss-Newton method,” IEEE Trans. Autom. Control, vol. 38, no. 2, pp. 294–297, 1993

    1993

  5. [11]

    On the identification of variances and adaptive Kalman filtering,

    R. Mehra, “On the identification of variances and adaptive Kalman filtering,”IEEE Trans. Autom. Control, vol. 15, no. 2, pp. 175–184, 1970

    1970

  6. [12]

    A new autocovariance least-squares method for estimating noise covariances,

    B. J. Odelson, M. R. Rajamani, and J. B. Rawlings, “A new autocovariance least-squares method for estimating noise covariances,”Automatica, vol. 42, no. 2, pp. 303–308, 2006

    2006

  7. [13]

    Recursive noise adaptive Kalman filtering by variational Bayesian approximations,

    S. S¨ arkk¨ a and A. Nummenmaa, “Recursive noise adaptive Kalman filtering by variational Bayesian approximations,”IEEE Trans. Autom. Control, vol. 54, no. 3, pp. 596–600, 2009

    2009

  8. [14]

    A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices,

    Y. Huang, Y. Zhang, Z. Wu, N. Li, and J. Chambers, “A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices,”IEEE Trans. Autom. Control, vol. 63, no. 2, pp. 594–601, 2018

    2018

  9. [15]

    An adaptive Kalman filter with inaccurate noise covariances in the presence of outliers,

    H. Zhu, G. Zhang, Y. Li, and H. Leung, “An adaptive Kalman filter with inaccurate noise covariances in the presence of outliers,”IEEE Trans. Autom. Control, vol. 67, no. 1, pp. 374–381, 2022

    2022

  10. [16]

    Adaptive Kalman filtering: Measurement and process noise covariance estimation using Kalman smoothing,

    T. Kruse, T. Griebel, and K. Graichen, “Adaptive Kalman filtering: Measurement and process noise covariance estimation using Kalman smoothing,”IEEE Access, vol. 13, pp. 11 863–11 875, 2025

    2025

  11. [17]

    Adaptive Kalman filtering for INS/GPS,

    A. H. Mohamed and K. P. Schwarz, “Adaptive Kalman filtering for INS/GPS,”J. Geodesy, vol. 73, no. 4, pp. 193–203, 1999

    1999

  12. [18]

    Kalman- Net: Neural network aided Kalman filtering for partially known dynamics,

    G. Revach, N. Shlezinger, X. Ni, A. L. Escoriza, R. J. G. van Sloun, and Y. C. Eldar, “Kalman- Net: Neural network aided Kalman filtering for partially known dynamics,”IEEE Trans. Signal Process., vol. 70, pp. 1532–1547, 2022

    2022

  13. [19]

    How to train your differentiable filter,

    A. Kloss, G. Martius, and J. Bohg, “How to train your differentiable filter,”Auton. Robots, vol. 45, pp. 561–578, 2021

    2021

  14. [20]

    TLIO: Tight learned inertial odometry,

    W. Liu, D. Caruso, E. Ilg, J. Dong, A. I. Mourikis, K. Daniilidis, V. Kumar, and J. Engel, “TLIO: Tight learned inertial odometry,”IEEE Robot. Autom. Lett., vol. 5, no. 4, pp. 5653– 5660, 2020

    2020

  15. [21]

    Distributionally robust convex optimization,

    W. Wiesemann, D. Kuhn, and M. Sim, “Distributionally robust convex optimization,”Oper. Res., vol. 62, no. 6, pp. 1358–1376, 2014

    2014

  16. [22]

    Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations,

    P. Mohajerin Esfahani and D. Kuhn, “Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations,”Math. Pro- gram., vol. 171, no. 1, pp. 115–166, 2018

    2018

  17. [23]

    Distributionally robust stochastic optimization with Wasserstein distance,

    R. Gao and A. Kleywegt, “Distributionally robust stochastic optimization with Wasserstein distance,”Math. Oper. Res., vol. 48, no. 2, pp. 603–655, 2023

    2023

  18. [24]

    Villani,Optimal Transport: Old and New

    C. Villani,Optimal Transport: Old and New. Springer, 2009, vol. 338

    2009

  19. [26]

    Distributionally robust Kalman filtering for INS/GPS tightly coupled integration with model uncertainty and measurement outlier,

    K. Si, P. Li, Z.-P. Yuan, K. Qiao, B. Wang, and X. He, “Distributionally robust Kalman filtering for INS/GPS tightly coupled integration with model uncertainty and measurement outlier,”IEEE Trans. Instrum. Meas., vol. 72, pp. 1–13, 2023

    2023

  20. [27]

    Distributionally robust Kalman filtering over finite and infinite horizon,

    T. Kargin, J. Hajar, V. Malik, and B. Hassibi, “Distributionally robust Kalman filtering over finite and infinite horizon,”arXiv preprint arXiv:2407.18837, 2024

    2024 arXiv

  21. [28]

    On the steady-state distributionally robust Kalman filter,

    M. Jang, A. Hakobyan, and I. Yang, “On the steady-state distributionally robust Kalman filter,” inProc. IEEE Conf. Decis. Control, 2025, pp. 2385–2392

    2025

  22. [30]

    A stability guaranteed distributionally robust adaptive Kalman filter for dual uncertainties in model and noise,

    Q. Wang, D. Zhou, Y. Li, and S. Li, “A stability guaranteed distributionally robust adaptive Kalman filter for dual uncertainties in model and noise,”IEEE Trans. Instrum. Meas., vol. 75, 2026

    2026

  23. [31]

    Residual-aware distributionally robust EKF: Absorbing linearization mismatch via Wasserstein ambiguity,

    M. Jang, J. Lee, A. Hakobyan, N. Hovakimyan, and I. Yang, “Residual-aware distributionally robust EKF: Absorbing linearization mismatch via Wasserstein ambiguity,”arXiv preprint arXiv:2604.02749, 2026

    2026 arXiv

  24. [32]

    KF-GINS: An open-sourced software for GNSS/INS integrated navigation,

    X. Niu, L. Wang, Q. Chen, H. Tang, Q. Zhang, and T. Zhang, “KF-GINS: An open-sourced software for GNSS/INS integrated navigation,”GPS Solut., vol. 29, no. 4, Sep. 2025, art. no. 202

    2025

  25. [33]

    Noise covariance matrices in state-space mod- els: A survey and comparison of estimation methods—part I,

    J. Dun ´ ık, O. Straka, O. Kost, and J. Havl ´ ık, “Noise covariance matrices in state-space mod- els: A survey and comparison of estimation methods—part I,”Int. J. Adapt. Control Signal Process., vol. 31, no. 11, pp. 1505–1543, 2017

    2017

  26. [34]

    MUSE: Multimodal uncertainty quantification of state estimation,

    M. Kim, H. Che, B. Chandaka, B. Pramuanpornsatid, C. Yang, S. Cheng, X. Wang, N. Hov- akimyan, and S. Wang, “MUSE: Multimodal uncertainty quantification of state estimation,” arXiv preprint arXiv:2605.17421, 2026

    2026 arXiv

  27. [35]

    On a formula for theL 2 Wasserstein metric between measures on Euclidean and Hilbert spaces,

    M. Gelbrich, “On a formula for theL 2 Wasserstein metric between measures on Euclidean and Hilbert spaces,”Math. Nachr., vol. 147, no. 1, pp. 185–203, 1990

    1990

  28. [36]

    On the Bures–Wasserstein distance between positive definite matrices,

    R. Bhatia, T. Jain, and Y. Lim, “On the Bures–Wasserstein distance between positive definite matrices,”Expo. Math., vol. 37, no. 2, pp. 165–191, 2019

    2019

  29. [37]

    Mamba: Linear-time sequence modeling with selective state spaces,

    A. Gu and T. Dao, “Mamba: Linear-time sequence modeling with selective state spaces,” arXiv preprint arXiv:2312.00752, 2023

    2023 arXiv

  30. [38]

    Attention is all you need,

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” inAdv. Neural Inf. Process. Syst., vol. 30, 2017, pp. 5998–6008

    2017

  31. [39]

    Learning phrase representations using RNN encoder–decoder for statistical ma- chine translation,

    K. Cho, B. van Merri¨ enboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio, “Learning phrase representations using RNN encoder–decoder for statistical ma- chine translation,” inProc. Conf. Empirical Methods Nat. Lang. Process., Doha, Qatar, 2014, pp. 1724–1734

    2014

  32. [40]

    Adaptive filtering with unknown prior statistics,

    A. P. Sage and G. W. Husa, “Adaptive filtering with unknown prior statistics,” inProc. Joint Automatic Control Conf., 1969, pp. 760–769

    1969

  33. [41]

    RoNIN: Robust neural inertial navigation in the wild: Benchmark, evaluations, and new methods,

    S. Herath, H. Yan, and Y. Furukawa, “RoNIN: Robust neural inertial navigation in the wild: Benchmark, evaluations, and new methods,” inProc. IEEE Int. Conf. Robot. Autom., 2020, pp. 3146–3152

    2020

  34. [42]

    i2Nav-Robot: A large-scale indoor–outdoor robot dataset for multi- sensor fusion navigation and mapping,

    H. Tang, T. Zhang, L. Wang, X. Ding, M. Yuan, Z. Xiang, J. Chen, Y. Bian, S. Liu, Y. Wang, G. Wang, and X. Niu, “i2Nav-Robot: A large-scale indoor–outdoor robot dataset for multi- sensor fusion navigation and mapping,”arXiv preprint arXiv:2508.11485, 2025

    2025 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.