A geometric perspective of state estimation using Kalman filters

Mateusz Baran; Ronny Bergmann

REVIEW 3 major objections 3 minor 61 references

A geometric perspective of state estimation using Kalman filters

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

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Pith's one-line read This paper claims that a Kalman filter can be built on any manifold equipped with an affine connection, encompassing the Lie-group and Riemannian formulations as special cases.

desk verdict A useful unifying framework and a real library, but the exact-MAP claim on arbitrary affine connections overreaches; the covariance-transport appendix is fine. read the letter →

arxiv 2506.01086 v1 pith:KODNA3YB submitted 2025-06-01 math.OC

Mateusz Baran , Ronny Bergmann This is my paper

classification math.OC MSC 93E1153B0562M20

keywords affineconnectionKalmanfilterstateestimationLiegroupRiemannianmanifoldextendedunscentedparalleltransportofcovariance

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 argues that the geometry Kalman filtering needs is an affine connection — a rule for moving tangent vectors between nearby points — and that this one structure covers both the Lie-group filters common in robotics and the Riemannian filters used in computer vision. The extended and unscented Kalman filters are rewritten on a generic connection manifold, with residuals computed by logarithmic maps, updated states by exponential maps, and covariances moved by parallel transport. The same implementation is demonstrated on the special Euclidean group SE(2) and on the tangent bundle of a sphere, and in both settings it reproduces the behavior of the established specialized filters. If this claim holds, filter design no longer has to start by choosing a group or a metric; state spaces such as sphere-valued joint angles can use standard Kalman machinery directly.

What carries the argument

The load-bearing object is the affine connection on the state manifold, a rule for relating tangent spaces at nearby points, which supplies parallel transport, exponential and logarithmic maps, and moving charts for linearization. The covariance matrix is treated as a symmetric rank-two tensor in the tangent space; prediction is forward sensitivity through the discretized dynamics, and the Kalman update is one Gauss–Newton step on a map-optimization objective built from logarithmic residuals. The covariance reset in Eq. (16) carries the corrected tensor from the predicted point to the updated point by parallel transport; Appendix II justifies that reset by transporting eigenvectors while keeping eigenvalues fixed.

What would settle it

On a manifold with a non-metric affine connection (for example, a Lie group with a flat Cartan–Schouten connection), compute the parallel-transport equation Eq. (34) for a small symmetric covariance matrix and compare the eigenvalues of the transported matrix with the original eigenvalues; if they differ, then the covariance-reset rule of Eq. (16) cannot be implemented by carrying eigenvectors with fixed eigenvalues, and the unified filter is not well-defined on that geometry.

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Extended reading notes

Core claim

The central claim is that a discrete Kalman filter is fully determined by an affine connection on the state manifold, not by a Lie group or Riemannian metric. Prediction pushes the state through $\tilde f$ and the covariance through the Jacobian formulas $P_{n|n-1}=F_n P_{n-1|n-1}F_n^\top + L_n Q_n L_n^\top$; the update solves a Bayesian optimization whose residuals are logarithmic maps, applies the gain through $p_{n|n}=\exp_{p_{n|n-1}}(K_n y_n)$, and transports the corrected covariance by parallel transport (Eq. (16)). Concentrated-Gaussian and exponential-wrapped-Gaussian assumptions lead to the same quadratic objective, with a volume-density correction for the wrapped case. Specializing the connection reproduces Euclidean KF, error-state KF, multiplicative filters on matrix Lie groups, invariant filters on Cartan–Schouten Lie groups, and Riemannian EKF/UKF, so the paper's claim is that all of these are instances of one construction.

Load-bearing premise

The filter moves each uncertainty estimate from one point on the manifold to the next by carrying the axes of the covariance along the connection and keeping their lengths unchanged; the construction depends on that transport rule being exactly what the connection does to the covariance matrix itself, a property that is not established for arbitrary affine connections.

Editorial extensions

If this is right

A filter written for one affine-connected state space can be reused on Lie groups, Riemannian manifolds, and spaces that are neither, without changing the algorithm.
State spaces without a natural group structure, such as sphere-valued joint-angle pairs in skeleton models, become directly usable for extended and unscented Kalman filters.
Known families — Euclidean KF, error-state KF, multiplicative quaternion filters, invariant filters, and Riemannian filters — fall out as special cases of a single construction, so results from one family can inform the others.
Adaptive noise-covariance estimation formulas can be attached to the geometric filter directly, as shown for covariance-matching updates of $Q$ and $R$.
When exact exponential and logarithmic maps are expensive, retractions and inverse retractions can stand in for them, as in the sphere-tangent-bundle experiment.

Reading between the lines

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

If the connection is the true substrate, then choosing a connection becomes a modeling decision comparable to choosing dynamics; a future filter could encode system symmetries in the connection rather than in a group action.
The covariance-reset step is only as sound as the claim that eigenvalues stay fixed under parallel transport; for non-metric connections this should be checked explicitly or the covariance should be transported in a way that does not assume fixed eigenvalues.
A discriminating test would run the same code on a manifold that is neither a Lie group nor equipped with a distinguished Riemannian metric and compare filter output with particle-filter or Monte Carlo ground truth, since both presented test cases still belong to existing categories.

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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

Reading the paper, I think the core value is the unification: presenting EKF and UKF on any manifold with an affine connection, with Lie-group and Riemannian filters as special cases, plus GeometricKalman.jl. The exposition is genuinely useful for practitioners, and the library is a real artifact. The paper does well to relate the concentrated Gaussian to the exponential-wrapped normal and to show where volume-density corrections enter.

The soft spot is the exactness claim behind Eq. (12). For the concentrated Gaussian, the normalization constant A in Eq. (4) is not generally constant; on a non-homogeneous space it depends on the mean, and on a general affine-connection manifold there is no canonical reference measure to define a density in the first place. The paper actually acknowledges this in Section II.C when it says affine connection alone is insufficient to define densities. So the MAP update in Eq. (12) — which drops -log A(h(p), R) — is at best approximate for the claimed generality. The wrapped-normal version in Eq. (13) includes volume terms, which shows the authors know the measure issue, but the concentrated-Gaussian update is the one used in the experiments. That gap should be stated explicitly rather than implied exact.

The Appendix II proof is fine. The reader's objection about eigenvalues not being invariant doesn't land: you don't need an eigen-decomposition, any rank-one decomposition with parallel-transported vectors and constant coefficients solves the tensor ODE, and uniqueness gives the result. No metric is needed.

The bigger empirical issue is that both examples only reproduce existing filters on standard spaces — SE(2) and TS^2. They don't exercise the claimed new capability of a general affine connection. The manuscript even contains an internal editing note asking for the punch line of the sphere example, which suggests the authors know the example section is unfinished. That's an editorial blemish, but it also mirrors the substantive point: the paper validates the unification but not the generalization.

Bottom line: this is a solid paper for people working on geometric state estimation, and the library deserves attention. But the title's promise — a geometric perspective covering arbitrary affine connections — is not fully demonstrated. I'd send it to review, with the expectation of a major revision that scopes the claims, fixes the normalization issue, and adds at least one example that uses a connection that is neither Levi-Civita nor Cartan-Schouten.

Referee Report

3 major / 3 minor

Summary. This paper proposes a unified geometric formulation of discrete Kalman filtering on manifolds equipped with an affine connection, claiming to generalize both the Lie-group approach (Cartan-Schouten connections) and the Riemannian approach (Levi-Civita connection). The authors define manifold versions of the extended and unscented Kalman filters using exponential/logarithmic maps, parallel transport, and local parametrizations, and they implement the framework in a Julia library called GeometricKalman.jl. Two numerical examples are presented: tracking a car on SE(2) and tracking a point on T S^2. The paper also discusses noise adaptation and the relationship with invariant and equivariant filters.

Significance. If the technical gaps identified below are resolved, the paper would provide a useful conceptual unification of existing geometric Kalman filters and a practical library for implementing them. The authors are to be credited for releasing GeometricKalman.jl and for writing the theory in a way accessible to practitioners. However, the claimed generality to arbitrary affine connections is not fully supported: the MAP update derivation omits a state-dependent normalization term, and the numerical experiments only reproduce the two special cases the framework aims to unify. The contribution is therefore incremental rather than definitive in its current form.

major comments (3)

[Section III.A, Eq. (12)] The text states that the new state estimate is obtained by maximizing the posterior density, and that for the concentrated Gaussian this corresponds to the optimization problem in Eq. (12). However, the normalization constant A in Eq. (4) generally depends on the mean and covariance. In the measurement likelihood, the mean is h(p), so A(h(p), R_n) depends on the unknown state p unless the measurement manifold is homogeneous or flat. Dropping this term means Eq. (12) is not the exact MAP update for the concentrated Gaussian on a general affine-connection manifold; it is an additional approximation. This is load-bearing for the claimed generality, since the issue is absent in the Lie-group and flat measurement cases used for validation. The authors should either include the missing -log A(h(p), R_n) term (with an explicit reference measure) or state that the filter is defined as the minimizer of the quadratic form and discuss the resulting approximation; the volume-density corrections in Eqs. (13)-(14) show that such terms are not generally negligible on non-flat spaces.
[Section IV, after Eq. (33)] The manuscript contains an unresolved internal note, reading '[RB 1]: Also here, I am mising the punch line of the example; sorry by changing colors the ais in the 3D plot are not yet so nicem we could also move that one to an asymptote render (see Manopt); or Makie – but maybe lets first fix the story before I spent 2 days trying to get a nice 3D plot and then we do not use it.' This is an author/editor comment that was accidentally left in the submission. It explicitly states that the example's 'punch line' is missing and that the figure is not finalized. The paper cannot be accepted in this state; the note must be removed and the example completed.
[Section IV, Examples] Both numerical experiments use state spaces that already carry the structures the paper aims to generalize: SE(2) is a Lie group with a Cartan-Schouten connection, and T S^2 is equipped with the Sasaki Riemannian metric. There is no example of a manifold with an affine connection that is neither Lie nor Riemannian, so the claimed 'greater freedom in selecting the structure of the state space' is not demonstrated in practice. The authors should either add such an example or explicitly separate the theoretical generality (which can stand on its own) from the experimental demonstration, which currently only reproduces known special cases.

minor comments (3)

[Appendix II, Eqs. (36)-(40)] The proof is in fact sufficient: since the tensor parallel-transport ODE (34) is linear, transporting any rank-one decomposition with constant coefficients yields the parallel-transported tensor. The wording, however, is misleading because the eigenvalues of a tensor are not generally invariant under parallel transport for an arbitrary affine connection, and the transported vectors need not form an eigenbasis. Please rephrase in terms of transporting the vectors in a rank-one decomposition while keeping the coefficients constant.
[Section II.D] The method name 'Runge-Kutte-Munte-Kaas' should be 'Runge-Kutta-Munthe-Kaas'.
[Figures 2 and 4] The vertical axis of these figures is labeled RMSE, but the text refers to 'mean squared error' when describing the results; please make the terminology consistent.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the affine-connection filter equations are stated from a density model and existing geometry, and the examples are demonstrations rather than fitted predictions.

full rationale

The paper's central derivation is not circular. Equation (4) defines a concentrated Gaussian density, and Eq. (12) is presented, with an external reference [40], as the corresponding MAP objective; Eq. (12) is not defined in terms of the quantity it claims to predict. Appendix II verifies covariance transport by eigenvector parallel transport; although the proof omits an explicit uniqueness argument, the linear parallel-transport ODE with matching initial conditions makes the construction sufficient, so this is not a circular justification. The two numerical examples in Section IV are demonstrations: filter parameters such as α, sigma-point weights, and covariance matrices are user-chosen, and no fitted parameter is subsequently relabeled as a prediction. Self-citations to Manifolds.jl [11] and Manopt.jl [12] are software-tool citations, not load-bearing evidence for the mathematical claims. Two flagged items deserve note but do not raise the circularity score. (1) Equation (12) drops the normalizing constant A from Eq. (4) in the measurement likelihood; because A(h(p), R) can depend on p on a general affine-connection manifold, Eq. (12) is a derivation gap rather than an exact MAP objective. This is a correctness concern, not a circular reduction. (2) The inserted editorial note '[RB 1]: Also here, I am mising the punch line of the example' acknowledges that the sphere example lacks an explicit takeaway, which is an illustrative weakness. Neither item makes an input equal to an output by construction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the choice of affine connection and on the covariance transport assumption; the latter is not justified for general connections.

free parameters (2)

Adaptive EKF forgetting factor α = 0.99
Chosen by hand in Section IV for the adaptive covariance update (Eqs. 29-30); affects adaptation speed.
UKF scaling parameters α, β, κ = not specified
Used to define sigma points in Eq. (23) and weights; values are not given in the paper, so the experiments are not fully reproducible without default assumptions.

assumptions (3)

domain assumption The manifold is equipped with a chosen affine connection; all geometric operations (exp, log, parallel transport) are derived from it.
Section II defines the setting; the choice of connection is part of the model.
domain assumption The concentrated Gaussian or exponential-wrapped normal distribution is an appropriate uncertainty model on the state manifold.
Section II-C introduces these distributions; the MAP update in Eq. (12) relies on this.
ad hoc to paper Eigenvalues of a covariance tensor are preserved under parallel transport by the affine connection.
Used in Eq. (16) and claimed in Appendix II; this is not true for a general affine connection without a metric-compatibility condition.

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Cite this review

Pith. "Pith review of A geometric perspective of state estimation using Kalman filters." pith.science (2026). https://pith.science/paper/KODNA3YB

@misc{pith2026250601086,
  author       = {Pith},
  title        = {Pith review of: A geometric perspective of state estimation using Kalman filters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KODNA3YB}},
  note         = {Machine review of arXiv:2506.01086}
}

read the original abstract

Geometry of the state space is known to play a crucial role in many applications of Kalman filters, especially robotics and motion tracking. The Lie group-centric approach is currently very common, although a Riemannian approach has also been developed. In this work we explore the relationship between these two approaches and develop a novel description of Kalman filters based on affine connections that generalizes both commonly encountered descriptions. We illustrate the results on two test problems involving the special Euclidean group and the tangent bundle of a sphere in which the state is tracked by geometric variants of the extended Kalman filter and the unscented Kalman filter. The examples use a newly developed library GeometricKalman.jl. The new approach provides a greater freedom in selecting the structure of the state space for state estimation and can be easily integrated with standard techniques such as parameter estimation or covariance matrix estimation.

Figures

Figures reproduced from arXiv: 2506.01086 by the authors.

**Figure 1.** Car tracking example: ground truth, raw sensor measurements, and state estimation trajectories from EKF, UKF, and an adaptive EKF (α = 0.99). ˜fcar : SE(2) × R × R 3 × R → SE(2) such that ˜fcar((p, X), q, w, t) = expp (∆tXv + √ ∆t[w2, w3]), 0 −q − w1 q + w1 0 , where (p, X) ∈ SE(2) is the state of the system at time t, q is the system control parameter given by q = q(t) = sin(t/2), w ∈ R 3 is the noise at the… view at source ↗

**Figure 3.** 3-D tracking on the unit sphere: ground-truth trajectory, raw sensor measurements, and state estimation results from EKF, UKF, and an adaptive EKF (α = 0.99). 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 Time (s) RMSE measurement UKF EKF Adaptive EKF [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Estimation errors for selected Kalman filter variants. The forgetting parameter of the adaptive EKF filter was set to α = 0.99. and Riemannian filters for the second one. The primary benefit of our new approach is to demonstrate that the same filtering logic can be applied to both scenarios, and even more complicated ones involving both Lie groups and more general manifolds. It is sufficient to supply the right retr… view at source ↗

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