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REVIEW 3 major objections 5 minor 35 references

Joint State and Noise Covariance Estimation

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

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Pith's one-line read This paper establishes a closed-form update for the noise covariance in joint MAP/ML Gaussian state estimation, and shows that alternating this update with standard least squares recovers trajectories nearly as well as knowing the true…

desk verdict Correct conditional optimality, but the covariance estimate's bias is unquantified and the theory doesn't cover the tested algorithm; still worth refereeing. read the letter →

arxiv 2502.04584 v3 pith:DHDV7VW6 submitted 2025-02-07 cs.RO math.OC

Kasra Khosoussi , Iman Shames This is my paper · ORCID
classification cs.ROmath.OC
keywords jointmaximumaposterioriestimationnoisecovarianceSLAMpose-graphoptimizationblockcoordinatedescentWishartpriorsampleGaussianmeasurement
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

Robotic mapping and structure-from-motion systems usually need the measurement noise covariance in advance, because it sets the weights in least squares, and in practice that covariance is hand-tuned or calibrated offline. This paper asks whether the covariance can be learned online, jointly with the state, and its central result is that for any fixed state the optimal noise information matrix has a closed form: the inverse of a residual sample covariance regularized by a Wishart prior, clipped when the user imposes diagonal or eigenvalue constraints. It also proves that the unconstrained maximum-likelihood version is ill-posed when the sample covariance is singular, so some regularization or constraint is necessary. On this result it builds two algorithms, one that eliminates the covariance and one that alternates a standard weighted-least-squares state step with the closed-form covariance step, and the reported pose-graph and real 3D SLAM experiments show the alternating scheme is accurate and adds only about a millisecond per iteration.

What carries the argument

The load-bearing object is the residual sample covariance $S(x)=\frac{1}{k}\sum_{i=1}^k r_i(x)r_i(x)^\top$. At any fixed state, the covariance subproblem minimizes $-\log\det P+\langle M(x),P\rangle$ over $P\succeq0$, a strictly convex program whose unique solution is $P^\star(x)=M(x)^{-1}$; substituting this solution back into the objective leaves a reduced problem in $x$, which in the unconstrained and diagonal cases is simply minimizing $\log\det M(x)$ (or its diagonal projection). The block-coordinate algorithm carries the argument by alternating the closed-form covariance update with a standard weighted-least-squares update for $x$, so the expensive part of the estimation remains the usual sparse nonlinear least-squares step and the covariance update reuses residuals that step already computes.

What would settle it

Construct residuals whose sample covariance $S(x_0)$ is singular, choose a nonzero nullspace vector $u$, and evaluate the unconstrained ML objective at $P_0+c u u^\top$ for growing $c$; the theory predicts the value decreases without bound. If instead it reaches a finite minimum, the ill-posedness claim fails.

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Extended reading notes

Core claim

Under the joint MAP model with a Wishart prior on the information matrix $P=\Sigma^{-1}$, the conditional negative log-posterior at a fixed state $x$ is $-\log\det P+\langle M(x),P\rangle$, where $M(x)=(kS(x)+V^{-1})/(k+\nu-m-1)$ and $S(x)$ is the sample covariance of the residuals. This inner problem is strictly convex in $P$, and its unique solution is $P^\star(x)=M(x)^{-1}$; for the diagonal variant the solution is $\mathrm{Diag}(M(x))^{-1}$, for the eigenvalue-constrained variant the eigen-decomposition of $M(x)$ is used with the eigenvalues mapped through the clipping in (19), and for the combined variant the same clipping is applied to the diagonal entries. The maximum-likelihood version uses the same formulas with $S(x)$ in place of $M(x)$, except that a singular $S(x)$ makes the unconstrained ML objective unbounded below. Because the inner problem is solved exactly, the joint problem reduces to an outer nonconvex problem in $x$ alone, and the paper's block-coordinate algorithm alternates a weighted least-squares state step with this closed-form covariance step.

Load-bearing premise

The proof of convergence for the main alternating algorithm assumes the state space is bounded and the state step is a specific gradient step on the curved state space, while SLAM states include unbounded translations and the experiments use a different solver step, so the convergence guarantee does not yet cover the setting that was actually tested.

Editorial extensions

If this is right

  • A robot or mapping system can refresh its measurement weights during operation: after each state step, replace the covariance with the closed-form conditional estimate and continue with the same sparse least-squares solver.
  • No ground-truth calibration dataset is needed for the joint MAP variant; the noise model is learned from the measurements themselves, with the Wishart prior absorbing whatever calibration knowledge exists.
  • When the residual sample covariance is singular or nearly singular, the theory says the plain ML problem has no solution, so the eigenvalue floor and the Wishart prior are not optional refinements but necessary safeguards.
  • Structural side information—independent noise components, known sensor variance bounds—can be enforced exactly through the diagonal and eigenvalue constraints without changing the alternating structure.
  • The conditional covariance step costs $O(k m^2)$ with small $m$ in typical SLAM, so the joint estimator's per-iteration overhead stays negligible relative to the nonlinear least-squares state step.

Reading between the lines

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

  • A natural extension is to feed this closed-form update into a robust estimator: reweighted residuals would enter both the state least-squares step and the sample covariance, yielding an outlier-resilient joint estimator with the same alternation.
  • The ill-posedness theorem marks an identifiability boundary for graph-structured SLAM: in poorly connected graphs the number of independent residuals per covariance block is small, so the prior and eigenvalue floor do real statistical work rather than just encoding prior knowledge.
  • The same conditional MAP/ML update could serve as a fast warm-start or proposal for full-Bayesian treatments of unknown covariance, since it gives the conditional mode in closed form at negligible cost.
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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 studies joint estimation of states x and noise covariance (information) matrix P from Gaussian measurements. It formulates joint MAP and ML problems, shows that the inner covariance subproblem is convex for fixed x, and gives analytical solutions (Theorem 1) under diagonal and eigenvalue constraints. It also proves that the unconstrained joint ML problem is unbounded below when the sample covariance is singular (Theorem 2). On the algorithmic side, it proposes a variable-elimination scheme and block-coordinate-descent (BCD) schemes (Algorithms 2-4), with convergence theorems (Theorems 3-4), and validates the approach on linear measurement models, pose-graph optimization benchmarks, and the RIM 3D PGO dataset.

Significance. The conditional optimality result is clean and useful: Theorem 1 correctly characterizes the exact minimizer of the stated objective for each variant, and Theorem 2 correctly identifies the ML ill-posedness due to singular sample covariance. The proposed BCD update has negligible per-iteration cost and can be integrated into existing least-squares solvers, which is a practical strength. If the convergence and bias concerns below are addressed, the framework would be a valuable reference for online covariance estimation in SLAM and related problems. The paper is also commendably honest about several limitations, including the compactness restriction in Section VII.

major comments (3)
  1. [§V-B, Theorem 3, §VI-B] Theorem 3, as stated, does not apply to the algorithm actually tested in §VI-B. The theorem assumes that M and P are compact submanifolds and that Step 1 of Algorithm 3 is a Riemannian gradient step with η=1/eL, leading to Eq. (29). The PGO experiments instead use g2o's Powell's Dog-Leg method on SE(2), whose translational component is noncompact, and the constraint sets for P in Problems 3 and 4 are compact convex sets with nonempty boundary, not smooth submanifolds. The acknowledgment in Section VII that "Theorem 3 requires M to be compact" is accurate, but the gap remains load-bearing because the paper's stated contribution includes convergence analysis for the BCD algorithm and the experimental validation is precisely in the regime not covered by Theorem 3. I request either a convergence result for the actual descent update used in the experiments or a clearly scoped statement that Theorem 3 applies only to an idealized setting, with the PGO experiments presented as heuristic validation.
  2. [§VI-B, Eq. (3), Fig. 5] The evaluation of covariance estimates does not separate bias from variance. Since S(x) in Eq. (3) is the sample covariance of residuals, evaluating it at the jointly estimated x̂ instead of at x_true systematically reduces the residuals; in a linear model with n estimated parameters and K=km scalar residuals, the expected residual covariance is approximately ((K−n)/K)Σ_true under a balanced projection. For the homoscedastic PGO setting in §VI-B, K=16,794 and n≈10,500, giving a naive factor of about 0.375, so the small 2-Wasserstein errors reported in Fig. 5 could be consistent with a substantial downward bias rather than an accurate covariance estimate. Because the motivation in Section I is uncertainty-critical decision making, the paper should quantify this bias, for example by reporting the expected ratio of estimated to true eigenvalues or the calibration of confidence ellipsoids, and should discuss how the eigenvalue constraints and Wishart prior mitigate it.
  3. [Appendix A, Eq. (41), Remark 7] The mode-matching procedure in Algorithm 1 sets ν = w_prior k + m + 1, which makes the prior strength grow with the number of measurements. Consequently, Eq. (41) shows that the conditional MAP estimator converges to (w_prior Σ0 + Σ_true)/(1+w_prior) as k→∞, not to Σ_true, whenever Σ0 ≠ Σ_true. Remark 7 discloses this non-vanishing influence of the prior, but the abstract and Section I present the result as "the optimal noise covariance estimate" without this qualification, and the experiments in §VI-B use exactly this prior construction with w_prior=0.1 and a deliberately inaccurate Σ0. I recommend either using a fixed-ν prior so that the data dominate as k grows, or explicitly rephrasing the optimality and covariance-accuracy claims as being relative to the chosen objective and prior, and studying the resulting bias in the reported error metrics.
minor comments (5)
  1. [Abstract and Section I] The phrase "maximuma posteriori" appears with a missing space in the abstract and in Section I; it should be "maximum a posteriori".
  2. [§V-B] In the sentence beginning "In this section, present two types of algorithms", the word "we" is missing before "present".
  3. [§VI-C] The RIM experiment is only qualitative; the claim that BCD trajectories are "significantly more accurate" would be stronger with a quantitative comparison, such as RMSE or ATE relative to the fixed-covariance baseline.
  4. [Fig. 5] Figure 5 does not show error bars or confidence intervals even though 50 Monte Carlo runs are used; adding them would make the comparisons consistent with Figure 3 and would help assess statistical significance.
  5. [Theorem 3] The phrase "compact submanifold" is imprecise for the constraint sets P in Problems 3 and 4, which are closed convex sets with boundary rather than smooth manifolds; the statement should say "compact constraint sets" or justify the manifold-with-boundary treatment.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the covariance update is the exact minimizer of the stated objective, the Wishart prior is a disclosed modeling input, and the convergence theory is imported from an external theorem.

full rationale

The derivation chain is self-contained. The joint MAP/ML objective (Problem 1 and Remark 1) is the negative log posterior/likelihood; Theorem 1 obtains P*(x) by solving the convex inner subproblem (14) via KKT conditions (Appendix C), so the covariance update is the exact minimizer of the stated objective, not an assumed conclusion. The Wishart prior enters as a modeling input through Algorithm 1 with explicit mode-matching, and its asymptotic behavior is disclosed in Remark 7. M(x) is defined from residuals and prior, and P*(x)=M(x)^-1 follows algebraically from first-order optimality; no equation in the derivation is equivalent to the target result. The convergence guarantees in Theorems 3-4 are imported from an external reference [25] (Peng and Vidal), not from the authors' own prior work; the only self-citation, [19], concerns reliable graphs for SLAM and is not load-bearing for the covariance-estimation claims. The use of Powell's Dog-Leg in experiments while Theorem 3 analyzes Riemannian gradient descent, and the compactness assumption in Theorem 3, are validity/scope limitations, not circularity. No fitted parameter is renamed as a prediction; the experimental Wasserstein/RMSE comparisons are external evaluations against simulated ground truth.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The method rests on standard Gaussian measurement models and a Wishart prior; no new physical entities are introduced. The load-bearing hand-chosen hyperparameters are the prior weight w_prior, the prior covariance Sigma_0, and the eigenvalue bounds lambda_min and lambda_max, which are necessary to prevent collapse of the unregularized estimator. The convergence theorem adds a compactness assumption that is not met by SLAM's translational states.

free parameters (5)
  • lambda_min = 1e-4
    Lower bound on the eigenvalues of the noise covariance Sigma; set by hand in all PGO/RIM experiments to prevent collapse of the ML estimate (Section VI-B).
  • lambda_max = 1e4
    Upper bound on the eigenvalues of the noise covariance, set by hand in experiments; no principled default is given.
  • w_prior = 0.1
    Weight of the Wishart prior relative to measurement likelihood in Algorithm 1; chosen by hand, affecting how strongly the prior regularizes the covariance estimate.
  • Sigma_0 = 0.002I (PGO), 0.01I (RIM)
    Prior covariance estimate fed to Algorithm 1; chosen ad hoc for the experiments and stated to be far from the true covariance.
  • Number of outer BCD iterations = 13 (PGO), 10 (RIM)
    Hand-chosen to ensure convergence across Monte Carlo runs; the paper does not use a data-driven stopping rule.
assumptions (4)
  • domain assumption Measurement noise is zero-mean Gaussian with covariance Sigma_true shared across all measurements of a given type.
    Equation (1) in Section III; this is the foundation of the ML/MAP objectives and of the least squares reformulation.
  • domain assumption A Wishart prior W(P; V, nu) is placed on the noise information matrix P.
    Section III, Eq. (2); chosen for conjugacy, with parameters set by Algorithm 1 in experiments.
  • domain assumption The state manifold M and measurement functions h_i are smooth; M is a product manifold containing R^d and SO(d) components.
    Stated in Section III; needed for the nonlinear least squares formulation and for manifold optimization.
  • ad hoc to paper Compactness of M and P, and Lipschitz smoothness of R, as required by Assumptions 1 and 2 for Theorem 3.
    These assumptions make the convergence proof of Algorithm 3 work, but they are not satisfied by SLAM's translational states; the authors acknowledge this in Section VII.

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Cite this review

Pith. "Pith review of Joint State and Noise Covariance Estimation." pith.science (2026). https://pith.science/paper/DHDV7VW6

@misc{pith2026250204584,
  author       = {Pith},
  title        = {Pith review of: Joint State and Noise Covariance Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHDV7VW6}},
  note         = {Machine review of arXiv:2502.04584}
}
read the original abstract

This paper tackles the problem of jointly estimating the noise covariance matrix alongside states (parameters such as poses and points) from measurements corrupted by Gaussian noise and, if available, prior information. In such settings, the noise covariance matrix determines the weights assigned to individual measurements in the least squares problem. We show that the joint problem exhibits a convex structure and provide a full characterization of the optimal noise covariance estimate (with analytical solutions) within joint maximum a posteriori and likelihood frameworks and several variants. Leveraging this theoretical result, we propose two novel algorithms that jointly estimate the primary parameters and the noise covariance matrix. Our BCD algorithm can be easily integrated into existing nonlinear least squares solvers, with negligible per-iteration computational overhead. To validate our approach, we conduct extensive experiments across diverse scenarios and offer practical insights into their application in robotics and computer vision estimation problems with a particular focus on SLAM.

Figures

Figures reproduced from arXiv: 2502.04584 by the authors.

Figure 1
Figure 1. Confidence ellipses for 10 samples drawn from the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The confidence ellipses corresponding to [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Results of experiments with linear measurement models (Section [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Average RMSE obtained by variants of BCD and g2o [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Average 2-Wasserstein error achieved by variants of BCD and as a function of information level [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: RIM Dataset information matrix component associated with the z coordinate was almost twice as large as those of x and y. This likely reflects the fact that the trajectory in this dataset is relatively flat. This finding highlights that in this case, in addition to the …

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