Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups

Erlend A. Basso; Finn G. Maurer; Henrik M. Schmidt-Didlaukies; Torleiv H. Bryne

Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read With a full-order covariance reset, left- and right-invariant extended Kalman filters on matrix Lie groups give identical state estimates.

desk verdict The equivalence theorem is exact and the proof is clean; the paper's real weaknesses are in simulation reporting, not the math. read the letter →

arxiv 2506.01514 v1 pith:7CF7AXTB submitted 2025-06-02 eess.SY cs.SY

Finn G. Maurer , Erlend A. Basso , Henrik M. Schmidt-Didlaukies , Torleiv H. Bryne This is my paper

classification eess.SYcs.SY MSC 93E1160H10

keywords extendedKalmanfiltermatrixLiegroupleft-invarianterrorright-invariantcovarianceresetStratonovichSDEinertialnavigationequivalencetheorem

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 derives a continuous-discrete extended Kalman filter for states living on a matrix Lie group, with noise placed on the Lie algebra and the dynamics read in the Stratonovich sense, and calls it the LEKF. Its main result is Theorem 1: the filter built from a left-invariant error and the filter built from a right-invariant error give the same Lie-group-valued state estimate, provided the full-order covariance reset is used after each measurement update. The two filters are connected by the change of variables $\bar g_o = g_o$ and $\bar P = \mathrm{Ad}_{g_o}P\mathrm{Ad}_{g_o}^T$, so only the covariance changes coordinates. This matters because practitioners previously had to choose a left or right error convention; the paper argues that under the full-order reset the choice has no effect on the estimate and can be driven by implementation convenience. Monte Carlo simulations of aided inertial navigation confirm the equivalence and show it fails for zero- and first-order resets.

What carries the argument

The machinery consists of the extended concentrated Gaussian distribution on a Lie group, written $g = g_o\exp(\xi)$ or $g = \exp(\bar\xi)g_o$ with the local error Gaussian on the Lie algebra, together with the full-order covariance reset $P^+ = J_\zeta(I-KC)PJ_\zeta^T$ and its right-invariant analogue. The group Jacobians $J_\zeta$ and $\bar J_\zeta$, defined as derivatives of the logarithm, carry the effect of moving the reference point after the Kalman update; Lemma 1 provides the Kullback-Leibler-minimizing formulas for that re-parameterization. The change of variables $\bar P = \mathrm{Ad}_{g_o}P\mathrm{Ad}_{g_o}^T$ is the transport that carries a left-filter trajectory onto a right-filter trajectory, and Proposition 1's adjoint identities are what make the update step commute with this transport. In short, the equivalence is a symmetry of the filter equations under conjugation by $g_o$.

What would settle it

Run the two full-order LEKFs on a Lie-group estimation problem with a strongly non-Gaussian or multimodal posterior, for example a very large initial attitude uncertainty, and check whether $\bar g_o(t) = g_o(t)$ and $\bar P(t) = \mathrm{Ad}_{g_o(t)}P(t)\mathrm{Ad}_{g_o(t)}^T$ hold to numerical precision throughout the run; a clear violation would show that Theorem 1 does not survive outside its second-order Gaussian regime.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the full-order covariance reset is the ingredient that makes the left- and right-invariant LEKFs equivalent. Theorem 1 states that Algorithm 1 and Algorithm 2 are related by the smooth change of variables $\bar g_o = g_o$ and $\bar P = \mathrm{Ad}_{g_o}P\mathrm{Ad}_{g_o}^T$, which is preserved by both the propagation and the measurement-update equations. Consequently the Lie-group-valued estimate $g_o$ is invariant to whether the estimation error is defined left-invariantly or right-invariantly. The covariance reset is not optional decoration; it is the exact re-parameterization that moves the posterior's reference point after an update, and the proof shows why truncating it to low order destroys the symmetry. The simulation then demonstrates the theorem numerically and shows that reduced-order resets produce different estimates for the two error conventions.

Load-bearing premise

The load-bearing premise is that the Gaussian re-parameterization step used for the covariance reset is accurate enough, since the paper's imported Lemma 1 (Section II-D) is proved only up to second-order linearization error and the left/right equivalence inherits that approximation.

Editorial extensions

If this is right

Under the full-order covariance reset, the left- and right-invariant LEKFs produce identical state estimates, so a user can choose either error convention based on implementation ease.
The two filters' covariance matrices are linked by $\bar P = \mathrm{Ad}_{g_o}P\mathrm{Ad}_{g_o}^T$, so results from one filter can be converted into the other's coordinates.
Zero-order and first-order covariance resets break this equivalence; the Monte Carlo results show nonzero error between the left and right variants when the reset is truncated.
In the simulated aided inertial navigation problem, the full-order reset filters had the lowest total mean absolute error, supporting the full-order reset as the appropriate re-parameterization step.
For a group-affine system with the reset omitted, the LEKF reduces to an IEKF, which connects the equivalence result to the existing invariant-extended-Kalman-filter literature.

Reading between the lines

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

An unstated consequence is that the equivalence is not tied to the particular $SE_2(3)\times\mathbb{R}^3\times\mathbb{R}^3$ navigation example; the proof uses only the group structure, so the same invariance should hold for any matrix Lie group and smooth measurement function covered by Lemma 1.
The simulation shows the left and right reduced-order filters disagree by different amounts in different directions, suggesting that published performance rankings of left- versus right-invariant filters may actually be rankings of their truncated reset approximations.
A practical extension is that the adjoint change of variables allows switching between left and right error conventions mid-run without re-initializing the filter, a possibility the paper does not discuss.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central claim holds up. The paper proves that on a matrix Lie group, a left- and a right-invariant EKF with full-order covariance reset produce identical state estimates, and the proof is a clean algebraic change of variables. The reader's conditional verdict was tied to a caveat about Lemma 1, but that caveat doesn't land: the KL-optimality error in Lemma 1 affects whether the reset is the best Bayesian reparameterization, not whether the two filters are equivalent. The stress-test note got this right.

What is actually new: a Stratonovich-based derivation of the continuous-discrete Lie group EKF, which is a useful simplification over the Itô version in Bourmaud et al., plus the explicit equivalence theorem. The paper also shows that truncating the reset to zero- or first-order breaks the equivalence, which matches practice and clarifies some of the left/right debate in the IEKF/MEKF literature. The simulation confirms the theorem and shows that full-order reset gives the lowest mean error.

Soft spots, in proportion: the simulation section is thin. One scenario, 100 trajectories, no error bars or confidence intervals on the MAE tables, and the measurement noise is inflated by a hand-picked factor of 3 to compensate for linearization. That is a free parameter, and the results could be sensitive to it. There is also no code or data. These are reporting gaps, not flaws in the mathematics. The paper is honest about the infinite series in the full-order reset and explicitly notes that finite truncation falls back to reduced-order behavior. Lemma 1 from [28] is imported and unproved here, but it is external prior work and the equivalence theorem does not depend on its Bayesian optimality claim.

Overall: the algebraic core is solid, the theorem is the real contribution, and the practical implication — full-order reset makes the left/right choice irrelevant for the state estimate — is genuinely useful for navigation and robotics practitioners. With a more careful simulation section, this is a strong paper.

Recommendation: send it to peer review. I would referee it myself and ask for run-to-run variance and a robustness check on the noise inflation factor.

Referee Report

0 major / 5 minor

Summary. The paper derives a continuous-discrete extended Kalman filter for systems on matrix Lie groups, using the Stratonovich interpretation of the system SDE and discrete-time Euclidean measurements. The derivation yields a filter with a full-order covariance reset after each update. The main contribution is Theorem 1, which proves that the left-invariant and right-invariant variants of this filter (Algorithms 1 and 2) produce identical Lie-group-valued state estimates under the change of variables \bar g_o = g_o and \bar P = Ad_{g_o} P Ad_{g_o}^T. The proof is algebraic and uses standard identities for adjoint operators and group Jacobians. A Monte Carlo simulation of an aided inertial navigation problem confirms the equivalence for full-order resets and shows that reduced-order resets break it.

Significance. The result is significant because it eliminates a design choice (left- vs right-invariant error) when a full-order covariance reset is used. The proof is rigorous and general for matrix Lie groups, and the paper also provides a Stratonovich-based derivation of the continuous-discrete Lie-group EKF that simplifies existing Ito-based derivations. The simulation supports the theoretical claim and demonstrates that the common reduced-order resets do not share the invariance. The paper's claims are falsifiable and clearly stated.

minor comments (5)

[Section III.B, Eq. (35)] Equation (35) is written incorrectly: the right-hand side should be multiplied on the right by \bar g_e, so that the time derivative is an element of the tangent space at \bar g_e. As written, the right-hand side is an element of the Lie algebra. The subsequent derivation in Eq. (36) uses the correct form, so this is likely a typo, but it should be fixed.
[Algorithm 1 and text after Eq. (23)] The notation 'ad_G(a_u(g_o))' appears to be a typo; it should be 'ad_{a_u(g_o)}' throughout. The same ambiguity appears in the definition of A(g_n,u) in Section III.B.
[Lemma 1] The phrase 'up second-order linearization error' should read 'up to second-order linearization error'; also, a proof sketch would improve self-containedness, as the lemma is central to the derivation of the right-invariant reset formula.
[Theorem 1] The statement of Theorem 1 should explicitly state the initial condition assumption \bar g_o(0) = g_o(0) and \bar P(0) = Ad_{g_o(0)} P(0) Ad_{g_o(0)}^T, since the proof implicitly relies on this initialization.
[Section IV] The sentence 'Theorem 1 does not hold when using a zero-order or a first-order covariance reset approximation' is imprecise; it would be clearer to say that the equivalence property is lost when the assumptions of Theorem 1 (full-order reset) are not satisfied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equivalence theorem is an exact algebraic identity between two filter algorithms, with no fitted input renamed as a prediction and no load-bearing self-citation.

full rationale

Theorem 1 is proven by exhibiting the smooth change of variables \bar g_o = g_o and \bar P = Ad_{g_o} P Ad_{g_o}^T, then checking propagation (Eqs. 46-47) and update (Eqs. 48-50) using only Proposition 1 identities; the proof does not invoke the theorem's conclusion. The filter derivation uses a classical EKF linearization and a KL-minimizing covariance reset imported from external references [27], [28]; even though Lemma 1 is stated to hold only 'up to second-order linearization error', that caveat bears on whether the reset is the optimal Bayesian reparameterization, not on whether the left and right algorithms are algebraically equivalent under the same approximate reset rule. The cited reset formulas are external prior results (Ge, Van Goor, Mahony), not self-citations by the present authors, and they are load-bearing only insofar as they determine the explicit P^+ formulas, which the equivalence proof treats as given operations. The Monte Carlo study sets the right filter's initial covariance by (45b), a consistency requirement rather than a fitted parameter, and the agreement of L-FO and R-FO is a confirmation of the algebra, not a construction of it. No equation was found in which a defined quantity is implicitly defined by the predicted quantity, and no 'prediction' reduces to a fit.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central theorem is proven from the filter equations; its main external inputs are the KL-minimization lemma from [28] and the Stratonovich modeling choice. The only hand-tuned parameter in the simulation is the measurement noise inflation factor, which does not affect the equivalence proof.

free parameters (1)

Measurement noise inflation factor = 3
In Section V, the filter's measurement covariance N is set to 3 sigma_y^2 I to 'compensate for errors generated by the linearization step'. This is a hand-chosen constant that affects the reported MAE of all filters; it does not enter the equivalence theorem, but it influences the performance comparison between full-order and reduced-order resets.

assumptions (4)

standard math Lemma 1 (spatial counterpart of [28, Lemma 1]): the KL-minimizing re-parameterization of an extended concentrated Gaussian is given by equations (14)-(15), up to second-order linearization error
Used in Section III-B to derive the covariance reset for the right-invariant filter (equation (42)). Proved in [28] for the body-frame case; the spatial version is stated without proof.
domain assumption The system noise is modeled on the Lie algebra and the SDE (16a) is interpreted in the Stratonovich sense
Modeling choice discussed in the introduction with references [11]-[15]. It ensures the state remains on the Lie group and simplifies covariance propagation. The equivalence theorem is proved for this filter, not for an Itô-based filter.
domain assumption The error state xi is assumed small enough that higher-order terms O(|(xi,mu)|^2) can be neglected in equations (23)-(25)
Standard EKF linearization. The theorem is a statement about the linearized filters; it does not cover large-error regimes.
standard math Smoothness of the mappings a, B, c, D and the availability of right/left derivatives on the matrix Lie group
Standard regularity assumptions for deriving the filter and its linearizations.

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Cite this review

Pith. "Pith review of Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups." pith.science (2026). https://pith.science/paper/7CF7AXTB

@misc{pith2026250601514,
  author       = {Pith},
  title        = {Pith review of: Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CF7AXTB}},
  note         = {Machine review of arXiv:2506.01514}
}

read the original abstract

This paper derives the extended Kalman filter (EKF) for continuous-time systems on matrix Lie groups observed through discrete-time measurements. By modeling the system noise on the Lie algebra and adopting a Stratonovich interpretation for the stochastic differential equation (SDE), we ensure that solutions remain on the manifold. The derivation of the filter follows classical EKF principles, naturally integrating a necessary full-order covariance reset post-measurement update. A key contribution is proving that this full-order covariance reset guarantees that the Lie-group-valued state estimate is invariant to whether a left- or right-invariant error definition is used in the EKF. Monte Carlo simulations of the aided inertial navigation problem validate the invariance property and confirm its absence when employing reduced-order covariance resets.

Figures

Figures reproduced from arXiv: 2506.01514 by the authors.

**Figure 1.** The mean and 95-percentile of total, position, and orientation MAE for all filters across 100 trajectories. Future research should investigate the implications of the covariance reset for the convergence results presented for group affine systems using the IEKF. Ultimately, a deeper understanding of Lie group statistics is needed to enhance our comprehension of Lie group estimators. REFERENCES [1] J. M. Mendel, Le… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems
cs.RO 2026-07 accept novelty 3.5 of 10

Four classical and invariant error-state Kalman filters for global aided INS are derived in full, with system matrices, Jacobians and reset rules presented for direct comparison.

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Works this paper leans on

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Pith tools

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

[1] [28]

A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups,

——, “A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups,”IEEE Control Systems Letters, vol. 8, pp. 844–849, 2024

work page 2024

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J. M. Mendel,Lessons in estimation theory for signal processing, com- munications, and control, ser. Prentice Hall signal processing series. Englewood Cliffs, N.J: Prentice Hall PTR, 1995

work page 1995

[3] [2]

P. S. Maybeck,Stochastic Models, Estimation and Control V olume 2, ser. Mathematics in Science and Engineering. Academic Press, Jul. 1982, vol. 141

work page 1982

[4] [3]

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work page 1982

[5] [4]

Discrete extended Kalman filter on Lie groups,

G. Bourmaud, R. M ´egret, A. Giremus, and Y . Berthoumieu, “Discrete extended Kalman filter on Lie groups,” inEuropean Signal Processing Conference, Marrakech, Morocco, Sep. 2013

work page 2013

[6] [5]

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G. Bourmaud, R. M ´egret, M. Arnaudon, and A. Giremus, “Continuous- Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions,”Journal of Mathematical Imaging and Vision, vol. 51, no. 1, pp. 209–228, Jan. 2015

work page 2015

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work page 2003

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Quaternion kinematics for the error-state Kalman filter,

J. Sol `a, “Quaternion kinematics for the error-state Kalman filter,” Nov. 2017, arXiv:1711.02508 [cs]

arXiv 2017