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REVIEW 1 major objections 6 minor 42 references

Making Rotation Averaging Fast and Robust with Anisotropic Coordinate Descent

T0 review · 1 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that anisotropic rotation averaging reduces to a sequence of single-rotation projections onto $\mathrm{SO}(3)$, making uncertainty-aware averaging as fast as isotropic solvers.

desk verdict The core ACD solver is a solid contribution, but the paper has a false mathematical claim and a robust-variant result that appears to be copy-pasted rather than measured. read the letter →

arxiv 2506.01940 v2 pith:HWJOI7LQ submitted 2025-06-02 cs.CV

Yaroslava Lochman , Carl Olsson , Christopher Zach This is my paper
classification cs.CV
keywords rotationaveraginganisotropicchordaldistanceblockcoordinatedescentstructurefrommotionuncertaintyiterativelyreweightedleastsquaresSO(3)projectionsemidefiniterelaxation
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 tries to make anisotropic rotation averaging—the step in 3D reconstruction that aligns all camera orientations using pairwise rotation estimates with directional uncertainties—both accurate and fast. It claims that the per-camera subproblem of the anisotropic chordal objective has a closed-form solution: update camera $k$ to the projection of a weighted combination of its neighbors' rotations onto $\mathrm{SO}(3)$. That single operation gives a coordinate descent solver, ACD, which the paper reports is orders of magnitude faster than the semidefinite-programming relaxation cSO(3) while matching its accuracy on public structure-from-motion and ETH3D datasets. A robust refinement variant, ACDAIRLS, often outperforms prior robust averaging methods that rely on minimum spanning tree initializations. If the claim holds, uncertainty-aware rotation averaging stops being a small-scale luxury and becomes a practical drop-in for large reconstructions.

What carries the argument

The load-bearing object is the symmetric block matrix $N$, whose $(i,j)$ block is $N_{ij}=M_{ij}\tilde{R}_{ij}$ with $M_{ij}=\tfrac12\operatorname{tr}(H_{ij})I - H_{ij}$; the two-view Hessian $H_{ij}$ is converted into a direction-dependent weight that penalizes rotations more in directions where the relative pose estimate is less certain. The key operation is the coordinate update $R_k \leftarrow \operatorname{proj}_{\mathrm{SO}(3)}(N_k^\top R)$, where $N_k$ is the $k$-th block column of $N$ and $R$ stacks all current absolute rotations, and $\operatorname{proj}_{\mathrm{SO}(3)}$ is the SVD projection $U\operatorname{diag}(1,1,\det(UV^\top))V^\top$. This update is the exact closed-form solution of the one-rotation anisotropic subproblem, and the paper proves that the analogous isotropic update keeps iterates inside $\mathrm{SO}(3)^n$; shuffling the update order at each sweep gives a stochastic coordinate descent that converges to a stationary point with a large basin of attraction.

What would settle it

Run ACD on a real SfM dataset with every Hessian replaced by the identity matrix, so the anisotropic weights carry no information. If the resulting accuracy stays clearly above RCD's, the gain comes from somewhere other than the Hessian-derived weights; if it drops to RCD's level, the dependence on Hessian quality is confirmed. A second check is to increasingly perturb the Hessian eigenvectors and verify that ACD's error reduction over RCD degrades roughly as in the paper's Figure 2.

Watch

Extended reading notes

Core claim

The paper's central claim is that minimizing the anisotropic chordal distance $\min_{R\in\mathrm{SO}(3)^n} -\langle N, RR^\top\rangle$, where the block matrix $N$ encodes two-view Hessian uncertainties, can be solved by block coordinate descent with the update $R_k \leftarrow \operatorname{proj}_{\mathrm{SO}(3)}(N_k^\top R)$. Starting from the isotropic block-coordinate subproblem [13,29], the paper shows that enforcing $R_k\in\mathrm{SO}(3)$ makes the semidefinite constraint redundant, so the costly convex relaxation of [27] is unnecessary for practical optimization. The authors prove that the projected update is exactly the minimizer of the anisotropic objective with respect to one rotation, and that in the isotropic setting the same update reproduces the iterates of rotation coordinate descent [29] with fewer matrix operations. On real SfM and ETH3D data, ACD initialized with zeros reports lower average angular error and a higher share of sub-degree-accurate cameras than the isotropic solvers RCD and Shonan, and roughly matches the accuracy of the certifiable SDP solver while being orders of magnitude faster. The paper does not claim a global optimality certificate for ACD; it claims a high success rate from diverse initializations plus speed.

Load-bearing premise

The two-view Hessian matrices are trusted as the uncertainty model, so if those Hessians are noisy or uninformative, the anisotropic objective may not improve over isotropic rotation averaging and the reported accuracy gains could vanish.

Editorial extensions

If this is right

  • Anisotropic rotation averaging can scale to hundreds of cameras without an SDP solver; the reported runtime is orders of magnitude below cSO(3) and comparable to RCD and Shonan.
  • Because the update is a small SVD on a 3×3 matrix, the solver needs no minimum spanning tree or chaining initialization—zero initialization suffices on the tested datasets.
  • The share of cameras with angular error below 1 degree rises on most tested datasets, which should translate into fewer grossly misplaced cameras in downstream global structure-from-motion.
  • The same solver can be wrapped in iteratively reweighted least squares or in the anisotropic robust refinement of [41]; the resulting ACDAIRLS reports the best average accuracy on the tested SfM datasets.

Reading between the lines

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

  • The same block-coordinate projection argument likely extends to synchronization problems over other matrix Lie groups with a known projection, such as similarity transforms or $\mathrm{SE}(3)$, although the paper does not develop this.
  • ACD's advantage over isotropic averaging is only as good as the Hessians it consumes; coupling it with learned or robustified uncertainty estimates could push the gains further than the two-view bundle-adjustment Hessians used here.
  • The per-block updates are independent given the current R, so a parallel or GPU implementation could make the runtime scale even better than the reported sequential timing suggests.
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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

1 major / 6 minor

Summary. The paper proposes Anisotropic Coordinate Descent (ACD), a block-coordinate descent solver for anisotropic rotation averaging, where per-edge uncertainties from two-view Hessians enter an anisotropic chordal distance objective. The key step is a reformulation of the per-block SDP subproblem (7) into the simple closed-form update R_k = project_SO(3)(N_k^T R) in Eq. (10). The authors also integrate ACD into robust pipelines: ACD IRLS (IRLS initialized with ACD) and ACDAIRLS (anisotropic robust refinement). Experiments on SfM and ETH3D datasets compare ACD against RCD, Shonan, cSO(3), L1IRLS, and Zhang et al. The central derivation leading to Eq. (10) is correct, and the non-robust experiments are extensive and generally favor ACD. However, the reported ACD IRLS results are bit-identical to L1IRLS in all tables, which is unexplained and undermines the claim that ACD initialization benefits robust refinement.

Significance. If the claims hold, ACD is a valuable practical contribution: it is the first dedicated solver for anisotropic chordal distances that avoids the scalability limitations of the SDP-based cSO(3), while providing state-of-the-art accuracy on public structure-from-motion datasets. The derivation of the projection update is clean and self-contained, the comparisons use external baselines, and the synthetic perturbation study in Figure 2 provides a reasonable check on the method's dependence on Hessian quality. The main obstacle is the unexplained exact equality between ACD IRLS and L1IRLS in Tables 3 and 4; resolving it is necessary before the robust-initialization contribution can be accepted.

major comments (1)
  1. [§4.3, Tables 3–4] The ACD IRLS results are exactly identical to L1IRLS for every dataset and every metric (e.g., LU Sphinx: Angl. err. 0.41 vs. 0.41, AUC@1 98.57 vs. 98.57; boulders: 0.11 vs. 0.11, 100.00 vs. 100.00). ACD IRLS is defined as IRLS from [5] applied to an ACD initialization, while L1IRLS is initialized with MST (Section 4.2). The paper does not explain this exact equality. If the IRLS stage is initialization-insensitive, that should be stated and supported, but this would be hard to reconcile with the ablation in Table 5, where different ACD initializations converge to different optima on several datasets. The more plausible reading is that the ACD initialization is not actually consumed by the implemented IRLS routine, or that the ACD IRLS numbers were produced by the MST-initialized baseline. This is load-bearing for the robust claim in the abstract and conclusions, and must be clarified or corrected.
minor comments (6)
  1. [Equation (4), §3.1] The claim that A^T(AA^T)^{1/2}^dagger is always in SO(3) is false: for A = diag(1,1,-1), AA^T = I, so the expression equals A, which has determinant -1. The SVD argument yields VU^T, which is a reflection when det(A) < 0. The statement should be restricted to matrices with positive determinant, or the argument reworked. This does not invalidate the ACD update in Eq. (10), but the proof as written is incorrect.
  2. [Algorithm 1] The pseudocode writes R_k^{(t+1)} <- project_SO(3)(N_k^T R^{(t)}) for each k, but in a sequential coordinate descent the most recently updated blocks should be used immediately. The notation should distinguish the current iterate from the previous one, or state that the update uses the current values of all other blocks.
  3. [§4.2, Tables 1–2] The text says ACD is run 10 times on each dataset due to shuffling, but Tables 1 and 2 report a single number per metric. Please state whether the reported value is the mean, median, or best over the 10 runs, and consider including variance or intervals.
  4. [§4.3] The sentence 'ACD does not require hyperparameter tuning' is slightly overstated, since ACDAIRLS uses the Geman-McClure scale tau = 5 degrees. The paper should clarify that this value is fixed following the author's implementation of [5] and is not tuned per dataset, rather than claiming there is no hyperparameter at all.
  5. [§4.4, Figure 3] The runtime comparison would be more informative with a description of the hardware, the number of repeats, and whether the reported runtimes are medians or means. Also, the runtime curves stop at around 160 cameras; a larger-scale experiment would strengthen the scalability claim.
  6. [§4.2] The real-data accuracy gains rely on the quality of the two-view Hessians used as uncertainty estimates. The synthetic perturbation study in Figure 2 partially addresses this, but a real-data sensitivity analysis (e.g., comparing results with and without Hessian marginalization, or with perturbed Hessians) would help quantify the effect of Hessian estimation errors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ACD update is derived from the anisotropic objective and validated against external baselines; two correctness concerns are noted but are not circular.

full rationale

The paper's central derivation is self-contained. Equation (10), R_k* = project_SO(3)(N_k^T R), is derived by direct manipulation of objective (1): fixing all rotations except R_k turns the sub-problem into maximizing <N_k^T R, R_k> over SO(3), and the paper explicitly notes that this 'exactly coincides with minimizing (1) w.r.t. R_k for a fixed k.' No fitted parameter, learned quantity, or cited theorem is needed for this step. The evaluation is against external baselines (RCD [29], Shonan [11], L1IRLS [5], Zhang [41]) using ground-truth rotation errors, so the reported accuracy gains are not encoded in the method. The anisotropic objective (1) and the stronger SDP (6) are inherited from the authors' prior work [27], but this is a premise rather than a circular derivation: the solver's correctness does not depend on [27]'s SDP tightness findings, and the paper's ablation (Table 5) tests ACD against the SDP solver cSO(3) [27] as an external comparison. Two non-circular concerns are flagged. First, the proof of Eq. (4) is mathematically wrong: for A = diag(1,1,-1), A^T(AA^T)^{1/2 dagger} has determinant -1 and is not in SO(3), so the claimed invariance of RCD iterates is not established by that argument. This is a correctness risk in the motivation, not a circularity. Second, Tables 3-4 report ACD IRLS results exactly equal to L1IRLS on every dataset, which the paper acknowledges ('the solution accuracy of ACD IRLS is identical to L1IRLS [5]'); this suggests the ACD initialization may not change the IRLS trajectory, but the paper does not claim an improvement for ACD IRLS, and the novel robust contribution is ACDAIRLS. These issues lower confidence in some claims but do not make the derivation circular.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No new physical entities or fitted parameters are introduced by the ACD solver itself. The only hyperparameter is the robust kernel scale tau, which is reused from prior work. The main assumptions are domain assumptions about Hessian-based uncertainty and the statistical model inherited from Olsson et al. [27].

free parameters (1)
  • Geman-McClure kernel scale tau = 5 degrees
    Used in the ACDAIRLS robust objective (Eq. 11); taken from the author implementation of Chatterjee and Govindu [5], not fitted to the data in this paper.
assumptions (3)
  • domain assumption The anisotropic chordal distance objective (1) is the maximum likelihood objective under anisotropic Langevin noise.
    Inherited from Olsson et al. [27]; the paper does not re-derive this connection and relies on it to justify the benefit of anisotropic rotation averaging.
  • domain assumption Hessians from two-view bundle adjustment are accurate approximations of the inverse covariance of relative rotations.
    Used to form M_ij in Eq. (1) and in the experimental pipeline described in Section 4.2. If these Hessians are poor, the anisotropic objective may not reflect true uncertainty.
  • standard math Block coordinate descent on SO(3)^n converges to a stationary point.
    Cited from Peng and Vidal [30]; the paper relies on this theorem for the convergence statement in Section 3.2.

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Cite this review

Pith. "Pith review of Making Rotation Averaging Fast and Robust with Anisotropic Coordinate Descent." pith.science (2026). https://pith.science/paper/HWJOI7LQ

@misc{pith2026250601940,
  author       = {Pith},
  title        = {Pith review of: Making Rotation Averaging Fast and Robust with Anisotropic Coordinate Descent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWJOI7LQ}},
  note         = {Machine review of arXiv:2506.01940}
}
read the original abstract

Anisotropic rotation averaging has recently been explored as a natural extension of respective isotropic methods. In the anisotropic formulation, uncertainties of the estimated relative rotations -- obtained via standard two-view optimization -- are propagated to the optimization of absolute rotations. The resulting semidefinite relaxations are able to recover global minima but scale poorly with the problem size. Local methods are fast and also admit robust estimation but are sensitive to initialization. They usually employ minimum spanning trees and therefore suffer from drift accumulation and can get trapped in poor local minima. In this paper, we attempt to bridge the gap between optimality, robustness and efficiency of anisotropic rotation averaging. We analyze a family of block coordinate descent methods initially proposed to optimize the standard chordal distances, and derive a much simpler formulation and an anisotropic extension obtaining a fast general solver. We integrate this solver into the extended anisotropic large-scale robust rotation averaging pipeline. The resulting algorithm achieves state-of-the-art performance on public structure-from-motion datasets. Project page: https://ylochman.github.io/acd

Figures

Figures reproduced from arXiv: 2506.01940 by the authors.

Figure 3
Figure 3. Runtimes of rotation averaging algorithms. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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