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REVIEW 3 major objections 4 minor 1 cited by

Building Rome with Convex Optimization

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

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Pith's one-line read This paper claims that global bundle adjustment can be solved without initialization by lifting 2D keypoints to 3D with learned depth, solving a convex SDP relaxation with a GPU Burer-Monteiro optimizer that scales to tens of thousands of…

desk verdict Strong engineering paper with an over-scoped headline: the certificates are for a depth-lifted SBA surrogate, not original bundle adjustment, but as a scalable solver for that surrogate it's real and worth engaging. read the letter →

arxiv 2502.04640 v4 pith:ZXEWHJ3J submitted 2025-02-07 cs.RO cs.CVmath.OC

Haoyu Han , Heng Yang This is my paper · ORCID
classification cs.ROcs.CVmath.OC MSC 68T4590C2290C26
keywords bundleadjustmentstructurefrommotionconvexrelaxationsemidefiniteprogrammingBurer-MonteirofactorizationRiemannianoptimizationlearneddepthcertifiableglobaloptimality
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

Global bundle adjustment, the optimization that ties camera poses and 3D landmarks together in structure from motion, is classically hard because it is nonconvex, initialization-dependent, and huge in scale. The paper claims a way around all three obstacles: lift the 2D keypoints to 3D using a learned metric depth model, allow one unknown scaling factor per camera to correct depth errors, and solve the resulting scaled bundle adjustment (SBA) problem through a convex SDP relaxation that is empirically tight. Tightness means the convex problem's global optimum coincides with the global optimum of the original nonconvex SBA, so the solution carries a certificate of global optimality. To make the SDP solvable at extreme scale, the authors factorize it with the Burer-Monteiro method and run a Riemannian trust-region optimizer written directly in C++/CUDA, giving a solver (XM) that is up to 100 times faster than the MANOPT package and scales to over ten thousand camera frames. If correct, this removes the need for careful initialization in global bundle adjustment and makes the convex-optimization route competitive in speed with local solvers, not just in rigor.

What carries the argument

The load-bearing object is the scaled bundle adjustment problem (3), which minimizes the weighted sum over visibility edges of squared 3D distances between the transformed, scaled lift and the landmark position, replacing the 2D reprojection error of classical BA. Translations and landmark positions are eliminated (Proposition 2), leaving a scaled-rotation-only problem (4), which becomes the QCQP (8) over the scaled orthogonal group sO(3) = {sR : s > 0, R in O(3)}; Shor's relaxation turns that into the convex SDP (11) on X = U^T U. The SDP is solved by Burer-Monteiro factorization (19) at increasing ranks r = 3, 4, ..., the staircase that certifies a local solution via the dual optimality certificate Z(y^*) >= 0 and escapes bad local minima by appending the least eigenvector of Z(y^*) as a descent direction. The CUDA implementation of the Riemannian trust-region optimizer, with analytic Hessian-vector products and batched QR retractions, is what makes the dense SDP tractable at thousands of cameras.

What would settle it

Take a real or synthetic image set where the metric depth estimator's error is non-uniform within a single frame (e.g., a scene with strong lighting or material-dependent bias, or deliberately corrupt one half of an image by scaling its depth by a different factor). If running XM-SfM on such inputs produces a certified-global SBA solution whose reconstruction drifts far from the Ceres-on-ground-truth reconstruction, the load-bearing assumption that one scale per camera suffices is falsified. A second, more arithmetic check: on a fresh dataset with N greater than 20,000 cameras, if the staircase repeatedly fails to find Z(y^*) >= 0 within any bounded runtime, the claim of certifiable global optimality at extreme scale would need qualification.

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Extended reading notes

Core claim

The central claim is that the nonconvexity left after 2D keypoints are lifted to 3D by learned depth, with one scale s_i per camera introduced, is benign: the Shor semidefinite relaxation (11) of the equivalent QCQP (8) is empirically tight, so a globally optimal solution of the nonconvex SBA problem (3) can be computed by solving a convex SDP, with the suboptimality gap eta in (16) certifying global optimality. The second claim is that this convex SDP, despite being 3N by 3N and dense, can be solved at extreme scale: because the tight solution has rank three, the Burer-Monteiro factorization (19) plus the Riemannian staircase (Algorithm 1) solves the SDP through a sequence of low-rank nonconvex problems, each certified or escaped via the dual certificate Z(y^*) being positive semidefinite as in Theorem 8; the authors implement the trust-region optimizer directly in C++/CUDA and report solutions for N greater than 10,000 frames, with the BAL-10155 instance certified in four hours when allowed. Their third claim is that wiring XM into a full SfM pipeline yields reconstruction quality comparable to COLMAP and GLOMAP on six benchmark families while being much faster and initialization-free, because the certified SBA solution gives a warm start that lets Ceres converge quickly on the original BA problem.

Load-bearing premise

The framework relies on learned depth being accurate up to a single multiplicative correction per camera: a scalar s_i must be able to absorb all depth errors in camera i's lifted observations, so that the SBA optimum stays close to the true bundle adjustment optimum.

Editorial extensions

If this is right

  • Global bundle adjustment becomes initialization-free: SfM pipelines can replace incremental registration or global initialization with a single convex solve, eliminating the classic failure mode of local minima.
  • The certified SBA solution is a reliable warm start for local BA solvers: the paper shows Ceres then converges quickly on the original reprojection problem, so the convex solver buys accuracy and speed together.
  • Solver time becomes nearly negligible relative to feature matching and depth estimation, so future SfM speedups will come from matching, indexing, and filtering, not from the optimizer.
  • The per-camera scale absorbs model calibration errors across different depth estimators, and the paper demonstrates the pipeline works with four different learned metric depth models.
  • The GPU Riemannian optimizer is up to 100 times faster than MANOPT and handles problems beyond MANOPT's reach, so the convex approach scales where interior-point SDP solvers become unresponsive beyond roughly 2,000 frames.

Reading between the lines

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

  • Because SBA (3) reduces to scaled multi-frame point-cloud registration, outlier-robust point-cloud registration machinery, such as pairwise consistency maximization or graduated non-convexity, could replace the greedy 10-percent residual trimming that the paper flags as a heuristic.
  • The paper's tightness conclusion is empirical; a natural next step is to prove the SDP (11) is exact for structured noise models, such as small per-camera scale errors plus bounded additive noise, which would upgrade the claim from empirically tight to a theorem.
  • A single scale per camera cannot fix depth errors that vary within a frame; extending the formulation to piecewise-affine scale fields, with one extra parameter per region, is a testable modification that would broaden robustness without abandoning the convex relaxation.
  • The method's advantage grows with dataset size, but exhaustive image matching grows quadratically in the number of frames; pairing XM with retrieval-based or learned matching would preserve the reconstruction quality gains while keeping total runtime low.
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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. This paper proposes a scaled bundle adjustment (SBA) formulation that lifts 2D keypoints to 3D using learned depth maps, with a per-camera scaling factor si to absorb depth errors. The authors derive a QCQP reformulation (8) and a convex SDP relaxation (11), then solve the SDP at scale using Burer-Monteiro factorization and a CUDA-implemented Riemannian trust-region optimizer called XM. The paper also assembles a full SfM pipeline, XM-SfM, and evaluates it on BAL, Replica, Mip-Nerf, IMC, TUM, and C3VD datasets, reporting large speedups over Manopt, Ceres, COLMAP, and GLOMAP, with suboptimality and minimum-eigenvalue certificates on many instances.

Significance. If the claims are properly re-scoped, the contribution is significant: a GPU-based solver that empirically solves large SDP relaxations of scaled BA, a concrete demonstration of the Riemannian staircase at N>10,000, and a fast SfM pipeline with learned depth. The propositions in Section II and the LICQ/certification arguments in the appendix are standard and appear correct. The paper is honest about limitations (depth-model dependence, sensitivity to outliers, preprocessing bottlenecks) and about the fact that many real-data results require Ceres refinement. The main weakness is that the headline 'global bundle adjustment to certifiable global optimality' is not supported by the certificates, which concern the SBA surrogate (or a regularized variant), and the largest reported instances do not achieve the claimed certificates.

major comments (3)
  1. [Section II (Eq. (11)) and Conclusion] The optimality certificates are for the convex relaxation (11) of the SBA objective (3), not for the reprojection BA objective (1). The Introduction frames the paper as solving (1), and the Conclusion states that XM 'achieved certifiable global optimality at extreme scales'; both statements overstate what is proved. The link from an (3)-optimal solution to an (1)-optimal solution is an untested modeling assumption: a single positive scalar per camera must absorb the errors of the learned depth map. The paper's own real-data tables (e.g., Tables V and VII) show that XM alone is not accurate and that Ceres refinement is required, and the noise analysis in Appendix F perturbs each observation multiplicatively rather than testing spatially varying per-camera depth bias. Please re-scope the claims to 'certifiable global optimality of the SBA surrogate' and add an experiment with structured (spatially varying) depth errors, or explicitly state that the target problem is (3).
  2. [Tables II and VI; Section V-A] The claim of tightness/global optimality at extreme scale is contradicted by the reported numbers. Table II reports suboptimality-gap 6.2e-1 and min-eig -2.1e1 for BAL-10155, while Table VI reports suboptimality 8.1e-2 for Alameda, yet the caption of Table VI states 'All datasets are solved to global minimum.' The sentence in Section V-A that XM would reach global optimality in four hours is not a substitute for reporting the certificate; no four-hour result is shown. Please either report the certified solutions for these instances or restrict the global-optimality claims to the instances for which a positive certificate was actually obtained (e.g., BAL-93/392/1934 and the other Mip-Nerf scenes).
  3. [Appendix D and Section III] The scale-regularized objective (51) is used without stating whether it was active in the experiments and without reporting lambda. This matters because all of Section III, including Theorem 8 and Algorithm 1 line 9, is developed for the unregularized SDP (17)-(18). If (51) is the actual solved problem, the dual certificate Z(y) in Eq. (21) must include the gradient term lambda grad( sum (X_{3i,3i}-1)^2 ) as in Eq. (52), and the 'min-eig' and 'suboptimality' tables are not directly comparable to the theory. Please state the value of lambda used for each table, and either update Algorithm 1/Theorems 8-9 to the regularized problem or run the main experiments with lambda=0.
minor comments (4)
  1. [Abstract] There are typos: 'certfiable' should be 'certifiable', and 'semidfinite' should be 'semidefinite'.
  2. [Appendix A, Eq. (28)] In the definition of Q1, the second factor appears to be e_i^T \otimes \tilde u_{i,k}^T, not p_{i,k}^T; as written, p is undefined at that point.
  3. [Section V-A] The variant 'CERES-GT-0.1' appears in Table I and Figure 4 but is never defined in the text; please define it explicitly.
  4. [Table III] The header 'XM2 FILTER+ XM2' is ambiguous; consider separating the two configurations (XM2 and FILTER+XM2) or adding a footnote explaining the notation.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the SDP/BM optimality chain is self-contained; the 'global BA' claim is a scoping overstatement, not a circular derivation.

full rationale

Walking the formal chain (3)->(8)->(11)->(19), each step is a genuine transformation or relaxation: Proposition 2 eliminates translations and landmarks exactly via the Laplacian Schur complement, Proposition 3 lifts SO(3) to sO(3), Proposition 5 applies Shor's relaxation, Theorem 8 provides a dual certificate whose LICQ condition is proven for this specific relaxation in Appendix A-D, and Theorem 10 appeals to standard Burer-Monteiro theory [13,27]. The reported tightness is not asserted by citing prior work but is verified by the suboptimality gaps and minimum-eigenvalue certificates in Tables II, IV, and VI, computed from independent primal upper and dual lower bounds via (16) and (73). No fitted parameter is renamed as a prediction: the per-frame scales s_i are optimization variables of (3), and the final real-data numbers are explicitly labeled XM+CERES, meaning Ceres solves (1) from an XM warmstart. Self-citations (SE-Sync/SIM-Sync, Yang's book, TEASER, outlier-robust estimation) provide context and standard tools; Remark 6 explicitly acknowledges the closest prior works, so the SDP argument does not reduce to a self-citation. Scope caveats do exist: the certificates are for SBA (3)/(11), and Appendix D adds a regularizer (51) with an unreported lambda, so calling XM a global solver for the original BA problem (1) is an overstatement; however, overstatement is not equivalence-by-construction, and the derivation chain itself is not circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard SDP/BM theory plus a domain assumption about depth errors being per-camera affine. The free parameters are engineering choices for robustness, not fitted to data. No new physical entities are introduced.

free parameters (3)
  • scale regularization weight lambda = not reported
    Introduced in Appendix D (Eq. 51) to prevent collapse of scales. Changes the objective, so the SDP certificate applies to the regularized problem, not the original SBA. No ablation reported.
  • outlier removal fraction = 10%
    In XM2, the 10% largest-residual measurements are deleted and the solver rerun (Section IV). A heuristic hyperparameter.
  • two-view filter threshold = 3 times the median
    In Section IV 'Filter from two-view estimation', landmarks with errors exceeding 3 times the median are removed. Ad hoc.
assumptions (4)
  • standard math Slater's condition holds for SDP (17) since X=I is strictly feasible
    Used to assert strong duality between primal and dual SDP, Section III.
  • standard math Burer-Monteiro factorization preserves global optimality when rank is sufficient
    Relies on [13,46] to claim that local optima at rank >= r* are globally optimal for the SDP, Section III-A.
  • domain assumption Per-frame scalar scaling is sufficient to correct errors in learned metric depth
    Core modeling choice in Eq. (3). If depth errors are not per-camera affine, the SBA optimum may be far from the true BA solution.
  • domain assumption The determinant condition (9) holds for the solution of (8), or negative-determinant solutions can be projected to SO(3) without loss
    Used in Proposition 3 and the projection step after solving (8); if the projection is far from optimal, the certified optimum of the relaxation is not the true optimum.

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Cite this review

Pith. "Pith review of Building Rome with Convex Optimization." pith.science (2026). https://pith.science/paper/ZXEWHJ3J

@misc{pith2026250204640,
  author       = {Pith},
  title        = {Pith review of: Building Rome with Convex Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXEWHJ3J}},
  note         = {Machine review of arXiv:2502.04640}
}
read the original abstract

Global bundle adjustment is made easy by depth prediction and convex optimization. We (i) propose a scaled bundle adjustment (SBA) formulation that lifts 2D keypoint measurements to 3D with learned depth, (ii) design an empirically tight convex semidfinite program (SDP) relaxation that solves SBA to certfiable global optimality, (iii) solve the SDP relaxations at extreme scale with Burer-Monteiro factorization and a CUDA-based trust-region Riemannian optimizer (dubbed XM), (iv) build a structure from motion (SfM) pipeline with XM as the optimization engine and show that XM-SfM compares favorably with existing pipelines in terms of reconstruction quality while being significantly faster, more scalable, and initialization-free.

Figures

Figures reproduced from arXiv: 2502.04640 by the authors.

Figure 1
Figure 1. Faster, scalable, and initialization-free 3D reconstruction powered by conveX bundle adjustMent ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. A view graph for the bundle adjustment formulation ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. XM-SfM: structure from motion pipeline with XM. TABLE I: Results on the BAL dataset. We report the ATE, RPE and running time for CERES, MANOPT, and our proposed XM solver. The evaluation is conducted on four BAL datasets with varying numbers of frames to demonstrate that our method is both fast and accurate across datasets of different scales (e.g., BAL-10155 indicates there are 10155 camera frames to be reconstruct… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Visualization of BAL datasets. Top: Our XM solver. Middle: CERES-GT-0.01. Bottom: CERES-GT-0.1. Both our XM solver and CERES-GT-0.01 accurately recover the ground truth camera poses and landmarks, whereas CERES-GT-0.1 fails [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Visualization of Replica datasets. Top: Our XM solver. Middle: GLOMAP. Bottom: COLMAP. All methods achieve high accuracy, producing nearly identical reconstruction results. GLOMAP sometimes produce outliers (see column 2 and 3). Takeaway • The SBA problem (3) is easier…
Figure 6
Figure 6. Figure 6: Visualization of Mip-Nerf datasets. Top: COLMAP. Bottom: Our XM solver. 3D-gaussian renderings are the same [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Visualization of IMC2023 datasets. Top: Our XM solver. Bottom: GLOMAP [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Visualization of TUM datasets using our XM solver [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: Visualization of C3VD medical datasets. Top: With ground truth depth. Bottom: With learned depth [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Illustration of local minimum on the Mip-Nerf dataset. [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 13
Figure 13. Figure 13: Breakdown Results on the Mip￾Nerf dataset APPENDIX C BREAKDOWN PLOT OF RUN TIME We present time breakdown plots ( [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Accuracy (ATE-T) and the final Ceres objective value [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.