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REVIEW 3 major objections 4 minor 48 references

The Geometry of Extended Kalman Filters on Manifolds with Affine Connection

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The classical EKF on a curved state space is missing two Jacobian corrections, and adding them reduces position and velocity error by 40-60 percent in inertial navigation simulations.

desk verdict Strong geometric filtering theory with a central sign error in the displayed update formulas that contradicts the paper's own simulations; the flaw looks corrigible and the paper deserves peer review. read the letter →

arxiv 2506.05728 v1 pith:3QXQKSSC submitted 2025-06-06 eess.SY cs.SY

Yixiao Ge , Pieter van Goor , Robert Mahony This is my paper
classification eess.SYcs.SY MSC 53B0553C2262M2093E11
keywords extendedKalmanfiltermanifoldstatespaceaffineconnectionconcentratedGaussiandistributionnormalcoordinatesexponentialmapJacobiancovarianceresetinertialnavigation
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

The paper claims that the classical extended Kalman filter, applied to a system whose state space is a curved manifold, is missing two geometric corrections: one when a measurement noise covariance is fused in coordinates anchored at the predicted state, and one when the posterior is re-expressed around the updated estimate. The central object is a concentrated Gaussian distribution written in normal coordinates, and the main technical result (Lemma 3.2) gives the covariance transformation between two normal coordinate charts in terms of the Jacobians $J_1$ and $J_2$ of the exponential map. On the extended pose group $\mathrm{SE}_2(3)$ with direct pose measurements, the geometrically corrected EKF is more accurate and more consistent than the classical EKF and the iterated EKF, cutting position and velocity RMSE by roughly 40 percent in the transient. The ablation study shows the two corrections act synergistically: the reset modification alone degrades performance, and the full benefit appears only when both update and reset are changed.

What carries the argument

The machinery is the normal-coordinate system generated by geodesics of the affine connection, together with the two partial pushforwards of the exponential map: $J_2$ differentiates the exponential with respect to the tangent vector and $J_1$ differentiates it with respect to the base point. $J_2$ carries the covariance transformation in Lemma 3.2 and in the reset step, while $J_1$ and $J_2$ together convert the measurement noise covariance into the state chart in the update step. The appendix supplies computable approximations of these transcendental maps using Jacobi field theory: $J_2(\xi_1,\xi_2)[w] = P^{0,1}_\gamma(w + \tfrac{1}{6}R(v,w)v) + o(|v|^3)$ and $J_1(\xi_1,\xi_2)[w] = P^{0,1}_\gamma(w - \tfrac{1}{2}R(v,w)v) + o(|v|^3)$, where $P$ is parallel transport along the connecting geodesic and $R$ is the curvature tensor. This turns the geometric corrections into implementable linear-algebra operations.

What would settle it

Re-run the $\mathrm{SE}_2(3)$ Monte-Carlo benchmark with initial covariance scaled up by one or two orders of magnitude and check whether the geometric EKF keeps its ANEES advantage over the classical EKF in the first seconds; if the advantage shrinks or reverses, the filter is operating outside the regime where Lemma 3.2 is proven. A complementary test is to replace the approximate $J_1,J_2$ expressions with Jacobians obtained by numerically integrating the Jacobi equation, and ask whether the reported gains persist.

Watch

Extended reading notes

Core claim

The paper's central claim is that a concentrated Gaussian information state is not invariant under a change of normal coordinate chart in the way the classical EKF implicitly assumes. Moving the reference point from $\xi_1$ to $\xi_2$ while keeping the same underlying distribution over the manifold changes the covariance parameter to $\Sigma_2 = J_2^{-1}(\xi_2,\xi_\diamond)\, J_2(\xi_1,\xi_\diamond)\, \Sigma_1\, J_2^\top(\xi_1,\xi_\diamond)\, J_2^{-\top}(\xi_2,\xi_\diamond)$, and Lemma 3.2 proves this is the concentrated Gaussian closest to the original in Kullback-Leibler divergence up to $\mathcal{O}(\mathrm{tr}(\Sigma_1^2))$. The paper applies the same Jacobian reasoning to the update step, expressing the measurement noise covariance $R_k$ in the normal coordinates of the state estimate, and to the reset step, recentring the posterior covariance at the updated state estimate. In the $\mathrm{SE}_2(3)$ inertial navigation benchmark, the resulting geometric EKF reduces position and velocity RMSE by roughly 40--60 percent relative to the classical EKF, improves the ANEES consistency measure by roughly halving transient overconfidence, and, per the ablation study, only delivers the full gain when update and reset corrections are combined.

Load-bearing premise

The load-bearing premise is that the truncated, second-order coordinate-change formulas of Lemma 3.2 remain accurate when errors and covariances are large, because the paper's largest performance gains occur during the initial transient rather than in the asymptotic regime where the $\mathcal{O}(\mathrm{tr}(\Sigma_1^2))$ error term is small.

Editorial extensions

If this is right

  • Any error-state or boxplus EKF that ignores the covariance change under recentring will be overconfident; the geometric corrections directly address that, so the consistency gain is a general consequence of the theory rather than a tuning artifact.
  • In the benchmark problem the full geometric EKF reduces transient position RMSE from $0.7226\,\mathrm{m}$ to $0.4168\,\mathrm{m}$ and velocity RMSE from $0.6271\,\mathrm{m/s}$ to $0.3602\,\mathrm{m/s}$ relative to the classical EKF, with smaller but persistent gains in the asymptotic phase.
  • The update-step modification alone is beneficial, but the reset-step modification alone is harmful; the two corrections work together, which the paper suggests explains why reset-only schemes in the existing literature have not been adopted by the filter community.
  • For the iterated filter, the geometric update correction supplies nearly all the benefit; the full geometric iterated EKF and an update-only geometric iterated EKF perform almost identically, because the iterated update already ends at the best linearisation point and the reset then does little.
  • The measurement correction matters even when the output function is linear, because the measurement noise is defined in coordinates around the true output rather than the predicted output; this is why the position and velocity errors, emphasised by the inhomogeneous measurement noise, improve substantially.

Reading between the lines

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

  • An implication the paper leaves implicit is that any boxplus-style error-state filter on a Lie group or homogeneous space carries the same coordinate-dependence, so applying the $J_2$ reset and the $J_1,J_2$ measurement correction could cure known overconfidence of multiplicative and error-state filters even when the prediction model is unchanged.
  • The propagation step still approximates the process noise covariance by leaving $Q_{k+1}$ unchanged because the noiseless intermediate state $\xi^\diamond_{k+1}$ is unavailable; a testable extension is to estimate or marginalise over that state and use the full $J_2^{-1}(\hat{\xi}_{k+1|k},\xi^\diamond_{k+1}) Q_{k+1} J_2^{-\top}(\cdot)$ transform, which may give further gains under large process no
  • Since Lemma 3.2 is proved only to second order in the covariance, the theory predicts that at sufficiently large initial errors the Jacobian-only corrections should eventually stop helping; if that happens, a second-order or sigma-point variant would be the natural next step rather than a contradiction of the paper's asymptotic claim.
  • The coordinate-change formula is a statement about concentrated Gaussian distributions themselves, so the same $J_2$ transformation should transfer to smoothing and information-form estimators on manifolds, not only to filtering.
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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 / 4 minor

Summary. The paper develops a framework for extended Kalman filtering on smooth manifolds equipped with an affine connection. It defines concentrated Gaussian distributions in normal coordinates, derives a coordinate-change formula (Lemma 3.2) with Jacobian corrections involving the exponential map, and proposes geometric modifications to the propagation, update, and reset steps of the EKF. An iterated variant is also presented. The method is benchmarked on an SE2(3) inertial navigation example with direct pose measurements, reporting improved RMSE and ANEES relative to classical EKF and iterated EKF, with an ablation study separating the contributions of the update and reset modifications. The central formal contribution is the covariance transformation (14) and the associated Jacobian approximations in the appendix.

Significance. If the proposed update and reset formulas were correct, the paper would make a useful contribution to intrinsic state estimation on manifolds. The coordinate-change Lemma 3.2 is a clean and potentially reusable result, and the ablation study is a strength because it isolates the effect of each geometric correction. The paper is also transparent in admitting that the proposed propagation correction is not implementable in its current form. However, the central algorithmic formulas for the update mean appear internally inconsistent: the displayed equations would diverge in a simple Euclidean limit, and the simulations therefore cannot be trusted as evidence for the claimed performance unless the implementation uses different formulas. Because the load-bearing algorithmic claim is affected, the manuscript needs a substantive technical revision before its conclusions can be accepted.

major comments (3)
  1. [Section 4.4, Eqs. (35)-(37), and Algorithm 1] The posterior-mean formula has the wrong sign and is applied to the wrong innovation. In the Euclidean identity-output limit M = N = R^m, h = identity, one has J1 = J2 = C = I and R-dagger = R; Eq. (35) gives eta(epsilon) = r - epsilon with r = phi_haty(y). Minimizing the quadratic objective (33), (Sigma^{-1} + R^{-1}) epsilon = R^{-1} r, yields epsilon* = K r with K = Sigma (Sigma + R)^{-1}. Equation (37) instead states mu^+ = -K r, which moves the estimate away from the measurement and would cause the filter to diverge. The same sign error appears in Eq. (44) and in the pseudocode of Algorithms 1 and 2. In addition, Eq. (37) applies K to the raw innovation r, whereas consistency with the R-dagger covariance in Eq. (36) requires the transformed innovation J1^{-1}J2 r; the mean and covariance updates therefore do not solve the same quadratic minimization. Since the simulations report good performance, the implementation must differ from the displayed formulas; without code or a corrected derivation, the central algorithmic claim is not verifiable.
  2. [Section 4.3, Eqs. (30)-(32)] The claimed propagation modification is not actually part of the proposed filter. The corrected process-noise covariance Q-diamond in Eq. (30) depends on the virtual noise-free state xi_diamond_{k+1} = F(xi_k, u_{k+1}), which is unknown at the time of propagation. The text states that the authors do not have a way to overcome this difficulty and that the best approximation is to set Q_{k+1} to approximate Q-diamond_{k+1}; Eq. (32) then uses Q_{k+1} unmodified, and Algorithm 1 contains no propagation correction. The abstract's claim of "novel geometric modifications to the propagation" is therefore not supported by the presented algorithm. The paper should either restrict its contribution claims to the update and reset steps or provide an implementable, error-bounded approximation to Q-diamond rather than abandoning the modification.
  3. [Lemma 3.2 and Section 6] The central coordinate-change formula (14) is proved only to second order in the covariance, O(tr(Sigma_1^2)), yet the reported performance gains are largest in the initial transient (Table 1 and Figures 4-5), where the covariance and linearization errors are large. The paper provides no bound or numerical evidence that the truncated formula remains accurate outside its asymptotic regime. This is a load-bearing concern because the update and reset steps rely on Eq. (14); the ablation study in Section 6.4 shows that the reset-only variant, which applies the coordinate change without the update correction, degrades performance dramatically. A discussion of the domain of validity of the approximation, or a simulation with a smaller initial covariance, would be needed to support the transient-phase claims.
minor comments (4)
  1. [Section 5, Eq. (44)] Independently of the sign error, the information term in Eq. (44) appears to use J2^T where the derivation from Eq. (42) and the transformed innovation require J2^{-T}; please verify the matrix transposes and show the intermediate steps.
  2. [Appendix A and Section 6.2] The implementation of the Jacobians J1 and J2 in the simulation is not described: it is unclear whether exact algebraic formulas for SE(3)/SE2(3) are used or the o(|v|^3) approximations of Lemmas A.1 and A.3. Providing the exact formulas used would substantially improve reproducibility.
  3. [Section 6.3] The text says the output function (46) is "linear in the state" and hence the geometric ItEKF terminates on the first iteration, but the projection from SE2(3) to SE(3) is not linear in the exponential coordinates used for the error state; please clarify what is meant by linearity here.
  4. [Throughout] There are several typographical errors and minor inconsistencies, including "Substitude" (Eq. 28), "consistancy" (conclusion), "implmentation" (Section 6.4), and the use of both Q^+ and Q_diamond for the same quantity in Section 4.3; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core coordinate-change and EKF update/reset derivations are self-contained and are benchmarked against external algorithms.

full rationale

The central derivation chain is self-contained. Lemma 3.2 derives the transformed concentrated Gaussian parameters (13)--(14) by minimizing the KL divergence of a Gaussian ansatz in the target normal coordinates; the proof computes the actual stationary conditions and uses Theorem 3.1 for covariance propagation, so the result is not assumed by definition. The update formulas (36)--(37) and reset formula (39) are obtained by applying this coordinate-change machinery and standard Gaussian fusion, not by fitting parameters to the simulation outputs. The reset formula is also explicitly attributed to prior external literature (Markley [21], Muller et al. [33,34]) and is a special case of the paper's own Lemma 3.2, so the self-citations [29,32,42] for reset geometry and parallel-transport computation are not load-bearing for the central claim. The benchmark against a classical EKF and an iterated EKF [44] provides external comparison, and the ablation study compares variants of the proposed filter rather than importing the paper's own prior results as evidence. Section 4.3 candidly states that the propagation-side geometric covariance correction is not implemented because the required Jacobian at the virtual state is unavailable ('At the moment, the authors do not have a way to overcome this difficulty'), which is a limitation of the proposed algorithm but not a circular step. A possible algebraic sign inconsistency in Eq. (37) (and mirrored in Eq. (44)) is a correctness concern, not a circularity concern, because it does not make the derivation equivalent to its inputs. Overall, the paper's self-citations are normal background references and the principal derivation stands independently.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The filter's central derivation rests on standard differential geometry plus the concentrated Gaussian modeling assumption. No free parameters are fitted: all noise covariances are simulation inputs. The virtual propagated state is a bookkeeping device, not a physical new entity.

assumptions (5)
  • domain assumption State and output spaces admit affine connections and normal coordinate charts covering the relevant sets.
    Section 2.1 and Equations (8)-(9): the entire construction relies on exponential maps being diffeomorphisms on neighborhoods.
  • domain assumption The information state is well approximated by concentrated Gaussian distributions with small enough covariance for second-order truncations.
    Lemma 3.2 and Theorem 3.1 use O(tr(Sigma^2)) approximations; the claimed transient benefits assume this holds for large errors.
  • domain assumption Process and measurement noise are zero-mean concentrated Gaussians with known covariances in normal coordinates.
    Equations (18)-(21) define the stochastic model; the filter only tracks these parameters.
  • domain assumption The affine connection is symmetric for the Jacobi-field approximations in Appendix A.
    Lemmas A.1 and A.3 state a symmetric connection; for Lie groups the Cartan-Schouten (0)-connection is used.
  • standard math Standard differential geometry results on geodesics, parallel transport, and Jacobi fields.
    Preliminaries Section 2 and Appendix A invoke references [36,38,48].
invented entities (1)
  • Virtual noise-free propagated state xi^diamond_{k+1} = F(xihat_{k|k}, u_{k+1})
    purpose: Defines the coordinate frame in which the process noise covariance Q is expressed before transforming to the prediction frame
    Section 4.3 calls it virtual and unmeasurable; it is a bookkeeping device, not a physical entity.

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Pith. "Pith review of The Geometry of Extended Kalman Filters on Manifolds with Affine Connection." pith.science (2026). https://pith.science/paper/3QXQKSSC

@misc{pith2026250605728,
  author       = {Pith},
  title        = {Pith review of: The Geometry of Extended Kalman Filters on Manifolds with Affine Connection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3QXQKSSC}},
  note         = {Machine review of arXiv:2506.05728}
}
read the original abstract

The extended Kalman filter (EKF) has been the industry standard for state estimation problems over the past sixty years. The classical formulation of the EKF is posed for nonlinear systems defined on global Euclidean spaces. The design methodology is regularly applied to systems on smooth manifolds by choosing local coordinates, however, it is well known that this approach is not intrinsic to the manifold and performance depends heavily on choosing 'good' coordinates. In this paper, we propose an extended Kalman filter that is adapted to the specific geometry of the manifold in question. We show that an affine connection and the concepts of parallel transport, torsion, and curvature are the key geometric structures that allow the formulation of a suitable family of intrinsic Gaussian-like distributions and provide the tools to understand how to propagate state estimates and fuse measurements. This leads us to propose novel geometric modifications to the propagation and update steps of the EKF and revisit recent work on the geometry of the reset step. The relative performance of the proposed geometric modifications are benchmarked against classical EKF and iterated EKF algorithms on a simplified inertial navigation system with direct pose measurements and no bias.

Figures

Figures reproduced from arXiv: 2506.05728 by the authors.

Figure 1
Figure 1. The relationship between two concentrated Gaussian distributions expressed in different normal coordinates on [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Demonstration of the filter propagation step. The true state [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Demonstration of the filter reset step. The hatched ellipse in green represents the updated covariance [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The estimation error are shown for different EKF implementations. The [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: The ANEES of the different EKF implementations ( [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: The estimation error is shown for different EKF implementations. On the left the non-iterated algorithms: The [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: The ANEES of the different EKF implementations are shown. The black dashed line represents the ideal ANEES [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.