Cycle-Sync: Robust Global Camera Pose Estimation through Enhanced Cycle-Consistent Synchronization
Authors: Shaohan Li
Yunpeng Shi
Gilad Lerman
Venue: NeurIPS 2025 spotlight ()
Keywords: Robust Synchronization
Camera Location Estimation
Structure from Motion
Computer Vision
Primary area: applications
TL;DR: We introduce Cycle-Sync, a robust and global framework for estimating camera poses (rotations and locations).

Abstract: We introduce Cycle-Sync, a robust and global framework for estimating camera poses (both rotations and locations). Our core innovation is a location solver that adapts message-passing least squares (MPLS) - originally developed for group synchronization - to the camera localization setting. We modify MPLS to emphasize cycle-consistent information, redefine cycle consistencies using estimated distances from previous iterations, and incorporate a Welsch-type robust loss. We establish the strongest known deterministic exact-recovery guarantee for camera location estimation, demonstrating that cycle consistency alone enables the lowest sample complexity to date. To further boost robustness, we introduce a plug-and-play outlier rejection module inspired by robust subspace recovery, and we fully integrate cycle consistency into MPLS for rotation averaging. Our global approach avoids the need for bundle adjustment. Experiments on synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-from-motion pipelines with bundle adjustment.

Cycle-Sync: Robust Global Camera Pose Estimation through Enhanced Cycle-Consistent Synchronization
Shaohan Li
Yunpeng Shit
Gilad Lerman"
"School of Mathematics, University of Minnesota 'Department of Mathematics, University of California, Davis nicklsh19966 gmail.com, lerman @ umn.edu, ypshi @ @ucdavis.edl
Abstract
We introduce Cycle-Sync, robust and global framework for estimating camera poses (both rotations and locations). Our core innovation is location solver that adapts message- message-passing least squares (MPLS) originally developed for group synchronization to camera location estimation. We modify MPLS to emphasize coclscconsittent information, redefine cycle consistencies using estimated distances from previous iterations, and incorporate a Welsch-type robust loss. We establish the strongest known deterministic exact-recovery guarantee for camera location estimation, showing that cycle consistency sdone-withour access to inter-camera distances-suffice to achieve the lowest sample complexity currently known. To further enhance robustness, we introduce a plug-and-play outlier rejection module inspired by robust subspace recovery, and we fully integrate cycle consistency into MPLS for rotation synchronization. Our global approach avoids the need for bundle adjustment. Experiments on synthetic and real datasets show that Cycle-Syne consistently outperforms leading pose estimators, including full structure- from- motion pipelines with bundle adjustment.
Introduction
Structure- wee---frroootttte (SfM) is a central task in 3D computer vision 211. aimed at reconstructing the 3D structure of a scene from 2D images captured by cameras with unknown poses. It plays critical role in applications such as virtual and augmented reality, robotics, and autonomous driving. core challenge SfM is accurate camera pose estimation, which is also foundational to modern 3D) techniques such as neural radiance fields [17] and Gaussian splatting [ where camera parameters serve priors or inputs for rendering and synthesis.
typical SfM pipeline begins by estimating essential matrices between pairs of cameras, from which local pose information is extracted. It then infers absolute camera orientations from relative ones-a step commonly referred to as rotation synchronization (or averaging). Next, it estimates camera locations from relative direction vectors. Once the camera poses determined, standard final step is to recover the 3D structure of the scene. The aim this work is to revisit both pose estimation tasks and propose global solutions that eliminate reliance the highly incremental and computationally intensive bundle adjustment step (31] which is ubiquitous in current SfM pipelines. We refer to our robust location solver Cycle-S) Synd and use the same name for the full pipeline that incorporates this solver along with robust direction and rotation estimation.
Mathematically, these problems are formulated on graph G(mn), E), where nodes correspond to cameras, and an edge (i,j) E indicates the availability of relative pose information between cameras i and j. The goal of rotation synchronization is to estimate the ground-truth camera rotations
'The Matlab code is released attttpp://gttbbbb.oo///////////3
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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he(x) from noisy relative rotations (Reyyee whose clean counterparts satisfy R;; = R This recovery up to an arbitrary global rotation. The subsequent localization step seeks to estimate the absolute positions {1, ) from noisy unit direction vectors (7k Jises, which approximate the ground-truth directions:
7/ (t; - -1))//111 1-
These locations are only recoverable up global translation and scale.
Camera location estimation is significantly more challenging than rotation estimation, and is therefore the primary focus of our work. First, the direction vectors derived from essential matrices lack scale information. If scale were available, the problem would reduce to a special case of group synchronization over the translation group, making it considerably easier. Second, in modern SfM pipelines, the estimated direction vectors are often highly corrupted due to failures in feature matching or RANSAC, as well error propagation from earlier stages- parttcullrlllly from rotation estimates. Lastly, unlike rotations, the space of camera locations lacks a group structure and is non-compact, which makes location estimation more sensitive and numerically unstable in the presence of corruption and noise.
1.1 Relevant previous works
Since our main contributions focus on camera location estimation, we review the most relevant work on this task, while also briefly noting related advances in rotation synchronization. Early location solvers based on l2 minimization 2 6. 20 32 33 or Loo minimization [8] are highly sensitive to outliers and unsuitable for real-world Structure-froot from--roiiiooo (StM) data. More robust approaches, including Least Unsquared Deviations (LUD)
19] and ShapeFit 5 use convex & objectives, solved via Iteratively Reweighted Least Squares (IRLS) and the Alternating Direction Method of Multipliers (ADMM), respectively. These methods minimize the distance between t; ty and the line defined by Yijy which can overweight long edges and become unstable when edge lengths vary significantly or are corrupted. BATA (36] instead minimizes the sine of the angle between t,- t; and Vij. offering robustness to edge-length variation. However, it treats all edges equally and thus underutilizes information from clean long edges. Fused Translation Averaging (fused-TA) -TA) [T5] alternates between LUD> and BATA-type objectives and merges their outputs using uncertainty estimates, but may underperform both in practice, with no clear winner among the three. Both I-Dimensional SIM (IDSIM) (34) and All-About-that--aae to detect outliers. IDSIM projects directions to ID and applies a heuristic combinatorial algorithm, without theoretical guarantees. AAB enforces coplanarity of Ti.. 75k- and 7h and combines this constraint with message passing. AAB outperforms IDSfM empirically but unstable in near- colinear configurations and only achieves approximate guarantees under a specific probabilistic model.
(AAB) (25) exploit 3-cycle consistency
The only deterministic recovery guarantees for location estimation under adversarial corruption are those for ShapeFit (7] and LUD [12], under Gaussian location priors and Erdōs Rényi measurement graphs with number of nodes approaching infinity and certain bounds on the connection probability and the number of corrupted incident edges per node.
In the related problem of group synchronization, stronger guarantees exist. Cycle-Edge Message Passing (CEMP) [1I handles adversarial and probabilistic corruption in rotation synchronization. was used to create Message-Passing Least Squares (MPLS) (6),, which empirically improves performance camera orientation estimation. However, MPLS progressively downweights cycle information and cannot be extended to location estimation due to the lack of inter-camera distance measurements. Similarly inspired by CEMP, DESC (28) estimates edge corruption levels via quadratic programming, but is computationally slow and its theoretical guarantees do not cover adversarial corruption. DDS [6 is able to address adversarial corruption by exploiting Tukey depth in the tangent space of (d). Other extensions attempt to generalize CEMP or MPLS to permutation synchronization (27 and partial permutation synchronization [14], yet these methods and their theory also do not naturally extend to the location estimation problem.
Finally, several global SfM pipelines incorporate location solvers. These include LUD 19 (used
as a full pipeline), Theia (30] and GL.OMAP (22]. All of them benefit from non-global bundle adjustment [3I], which is explicitly integrated into Theia and GLOMAP.

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1.2 Contributions
Nearly all existing methods for location estimation fall within the IRLS framework and do not fully exploit cycle consistency information. As result, they struggle to handle cycle-consister corruption (i.e., situations when the corrupted edges exhibit cycle-consistent behavior), which frequently arises in real-world SfM datasets.
The goal of this work is propose new location estimation method that is robust and accommodates missing and highly variable edge lengths. We also revisit the global pipeline for pose estimation and improve several of its core components.
corruption
Our contributions to camera location estimation are summarized as follows:
1. We introduce novel formulation with Welsch-type objective function that directly addresses key limitations of LUD-type and BATA-type objectives, particularly the presence large variations in edge distances.
2. We propose new MPLS framework optimize the Welsch objective for location estimation. This framework is designed to fully exploit cycle-consistent information. To handle missing distance data, we redefine cycle consistencies using distances estimated in previous iterations.
We establish the strongest known deterministic exact-recovery guarantee for location estimation under adversarial corruption. Under standard probabilistic models (e.g., i.i.d. Gaussian locations with Erdös Erdös-Rényi connectivity), our theoretical sample complexity improves over all prior work (see Table D..
Our additional contributions to global pose estimation include:
1. We extend the full-cycle MPLS framework to the rotation synchronization problem, yielding significantly reduced orientation error compared to the strongest existing baselines.
2. We introduce a plug-and-pla outlier rejection module inspired by robust subspace recovery. This module significantly improves the performance of existing location estimators.
3. Our global pose estimation pipeline eliminates the need for bundle adjustment. Experiments on both synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-from-mmott pipelines that rely on bundle adjustment.
2 The Cycle-Syne Framework
We first present the Cycle-Sync location solver: 2. introduces its optimization formulation, $2. de- tails its algorithmic implementation with cycle-consistent weighting, and $2.3 establishes theoretical recovery guarantees. Finally, integrrtss the solver into a full camera pose estimation pipeline.
2.1 New optimization formulation for location estimation
We note that all major solutions to the camera location estimation problem are either special cases variants of the following formulation:
Σ p(lle - -s177)ll))
ijEE
subject to aij 2 1, Σ4=0
where p(z) is certain function to be specified later. Here, a.j is interpreted as an auxiliary variable representing the distance between t, and ty. Its constraint, i.e., asj 2 1. aims to avoid the trivial solution f, = 0 for all i's. This general formulation captures several existing methods as special cases. For example, constrained least squares 32
33 uses p(z) _ and LUD [9 uses p(x) |ar]
Furthermore, ShapeFit 7 replaces | f, any Tigll ll by the projection distance from t, - t, the
line of "th- It also switches the Ozj 2 to weaker linear constraint Eijer(t (t, f,, Yop = 1. Let
Ogj denote the angle between the true relative location t; L, and direction Vij- By choosing the optimal Osj, the LUD objective function (with p(x)
= (xr reduces to the ShapeFii one:
p(ltt t, ais Yis I])
p(lit: t, sin 01z),
3

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R Loss Functions
L1
IRLS reweighting
(1 2.)
Residuals Tuoyyee
3.5
Welsch
Solution
by solving WL5
(Wijzzzs) yje.
1 1+1
3
2.5
Est. Distances |I Cer
Esell
Input Directions
(u) uee
V v
p(x)
2
Message-passing reweighting
Vilts
-B(++
dije,
1.5
7kLt
Tiksf
Ex 4yyddgea
Tige
K2
e PC(+)
Eeghxx
0.5
-A(-
0
Rzn
Message-passing
X
Figure 1: Comparison of different losses. Our loss combines the ad- vantages of the Ln and Welsch losses. Specifi- cally, it suppresses the in- fluence of large values (like Welsch), while re- taining a nonsmooth cor- ner at the origin (like L1), which introduces singu- lar point in the reweight- ing function f()) and en- ables exact recovery.
Figure 2: Illustration of Cycle-Syne. The algorithm solves the weighted least squares (WLS) in (3) to estimate the locations {ei The weights for WLS iteratively updated in two ways. The main one is a cycle-edge message passing procedure (bottom unit, where green ele- ments represent cleaner information, and red elements indicate corrupted information). Each cycle weight L' is updated using the two residuals Tik,t and "j,,t- The bar length surrounding Z. reflects its magnitude. The theory indicates higher weights for good (green) cycles. The quantity hij. weighted average of disk,e. defined in 7 and updated each iteration using the estimated locations. The theory guarantees that hig. is a good estimate for the corruption level edge ij. The weights are obtained by applying the function f(x) in ( The weights are also com- puted by IRLS (top unit). The final weights combine the two procedures with X + 1, so the message message-passing unit progressively takes over.
which scales linearly with the length of edges. Therefore, such an objective function may suffer from long and corrupted edges. BATA (36] changes the distance p(sin 0ly) in order make it independent of |tt ty |]. However, it may not sufficiently benefit from long and clean edges that have a stronger effect the global distribution of the locations.
As opposed to previous robust formulation which used p(x) = |æ], we propose minimizing I with
p(x) ea/2] 0|22
(2)
where a is a fixed parameter. Throughout this paper we fix a = 4. We remark that this p(z) is similar to the Welsch objective function p(x) = - e-a="_ The latter objective functions put less emphasis on longer edges, but still grow (with a very small rate) as ||t. - tyll grows. Therefore, they are much less sensitive to large variations of distances. This avoids the limitations of both LUD and BATA. A comparison of these p(z) choices appears in Figure
On the other hand, unlike the traditional Welsch function, our new energy function inherits from 0. This non-differentiabiiity of p(x) at makes the exact recovery
LUD the non-smoothness
of ground truth locations possible under corrupted directions [2
2.2 Solution of the proposed optimization with emphasis on eycles
The formulation proposed in I and (2) can be addressed by an IRLS- approach. Indeed, let Ti = ||t, t, -0 asyYi Il. then our objective function is F [ {asz hijRR) := Eijen P(r:g). As demonstrated in (3), the problem can be tackled with an IRLS scheme:
min {6, [6.)" bISE
Σ WEE
Wig.elle - t, -
(3)

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Typically, wij,+1 is updated by f(rij,)) where residual riga = | tze - Yey. and the reweighting function
(z)=
Here small number to avoid the zero denominator. However, our choice of p(æ) in (2) is nonconvex, making IRLS highly sensitive to weight initialization. Even with good initialization and convex p. IRLS may fail to recover the ground truth under high corruption. Indeed, in the highly corrupted scenario, the residuals Tij, may fail to reflect true corruption levels (defined up to unknown fixed scale):
si = till Iy -
making IRLS easily get stuck in local minima. This limitation motivates a much more robust iterative estimator for s.j- using the cycle-consistency information. Following the MPL.S strategy (26 approximate bi using weighted average of cycle inconsistencies:
Sij.t
Ziz,t
Σ e-ttruaatras dijaatt kEN,
(6)
where 3 > 0 is parameter, Zij,e is the normalization factor ensuring convex combinations, and
disk,t = ||t,, Esiellris + ||tt, tz.el/ + ||tx, tgeel/rss
(7)
measuring the cycle inconsistency. We note that with clean edges and true locations, disk,t = 0. Furthermore, as 3 + oo and ti, - 81j, - sty-
We note that in (6. the corruption level is no longer approximated by a single residual. Instead, it aggregates information from 3-cycles (indexed by ijk) and incorporates residuals from neighboring edges, making Sij.e significantly more stable and robust to outliers.
In early iterations, when distance estimates | ty,|l are unreliable, we blend the residuals r with the corruption scores St,,t to define the edge weights. As estimates improve, the weighting gradually shifts toward $i,.-- This message- nessage-passing passing mechanism enables cycle consistency to refine edge weights, while the weighted least squares updates, in turn, refine cycle inconsistencies. This bi-directional communication between cycles and edges facilitates global information propagation and significantly reduces the risk of getting trapped in local minima. We refer to our method as Cycle-Syne, and its information propagation illustrated in Figure Weight initialization. In the first iteration, when locations are unknown, weights can be initialized as wijo exp( exp(-20 2:g);, where Sij estimates the angular corruption level. Its ground-truth counterpart is defined as
1 ( Yig) E [0, 1].
= π
8
Since this initialization precedes weighted least squares, distances, residuals, and the cycle incon- sistency diyk,t are unavailable. Instead, 87 can be estimated using a variant of the update rule in (6). replacing residuals with previous &ggtt values and substituting digk,t with an AAB-style cycle inconsistency:
dijA = min d)(7))))) E [0,
yE52
where y is constrained to satisfy
anjy + 7f + aga for some aijy Ottee agss > 0.
We refer the reader to 25 for closed-form expression of dus.. When computing āزg. we use only well-shaped triangles-those where the angle between Vi and 75k lies |arcsin(O.6),, x -rrssin since extreme angles can make duj.k unstable, as discussed in our theory. We refer to this modified version of AAB as truncated AAB (T-AAB).
Although this initialization cannot exactly estimate st for corrupted edges without location infor- mation, i, is expected to approach zero clean edges under the assumption of no additive noise. This allows for reliable separation clean and corrupted edges, enabling exact location recovery.
Finally, we remark that Cycle-Sync is not sensitive to initialization. Nonetheless, the AAB-base initialization is lightweight, independent of least squares, and essentially a free improvement. complete description of our location solver is given below.
5

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In all of our experiments, we choose tma = 20, 3 = 20 and X = =t/(t+ 10). We remark that these parameters are not fine-tuned, but the performance of our method is already superior with the suboptimal parameters.
Our MPLS procedure differs from the original version for rotation estimation in several key aspects. First, unlike the rotation setting, we must address missing distance information, and our cycle- inconsistencies depend on iteratively updated distance estimates. Second, we adopt Welsch-type objective, which better handles the large variation in residual scales typical in location estimation, whereas the original MPLS uses f minimization. Lastly, our weighting parameter satisfies X + gradually emphasizing cycle information over residuals, in contrast to the original one where X + 0. Further discussion and experiments on the annealing parameter X are included in Section I.3 in the supplementary material.
Algorithm 1 Cycle-Sync
Input:
(riy hijen, (dijatkeciyy B, (1e121, 8 (default: 10 10-8)
Steps:
Compute {siglifE by T-AAB Initialize edge weights wiy exp(-20s;))
While f S Imazi
t= 1-t+ ++1 (ti, (ain = arg 21 Eijer Wiislle; - t2 - Yis1
Tij.t = || tis Oiji
ij E E
Sij, 1.
Z,z. EREN. e -((luueett))))) ||t.t tallis + 11lj, la,el177 + |RE,
4,, 1
ij E E
hig. (1 >Norreg X1sjf,
E
Wij++1 f((he,) ssp(-4hig,)//00 +8 12 Output: [tr,h=1
E
2.3
Why we need cycle consistency: theory for exact location recovery
To motivate our use of cycle information, present theoretical result showing that AAB-style cycle inconsistency, when combined with iterative reweighting, can exactly separate clean and corrupted directions-eee even without access to inter-camery distances. Intuitively, this holds because corrupted directions tend to violate 3-cycle consistency, while clean directions remain geometrically consistent. Formally, under mild conditions, we show that the estimator Kij,t converges to and remains bounded away from zero on corrupted ones, enabling exact location recovery. Under a probabilistic corruption model, result also yields the lowest known sample complexity for exact recovery.
clean edges
We assume that the edge set E is partitioned into good (clean) edges Eg and bad (corrupted) edges Eg. For ij E Eg. Thj = Tij. and for ij E Es, Yij an arbitrary unit vector distinct from 7/-- Define Ni. = [k E [n]] ik E E}, Gij = {k E No ik,jj E Eg}. and Bij = No \ Gij. Let X = maXIjeæ BBsl//N,s5b = minijEEs EXEG, dig,e and Oig, denote the angle between 7/A and "ijk-
Theorem 2.1. Assume there exists a > O such that for all ij E Eg and k E Nig, a < 0zg < ग and X < 11 V (2 eCa)/u, /,, where C. 2(cos v55 cos a)/sin² 2 Then, for štg, computed by the iteratively reweighted AAB algorithm [25) using Bo < 2A and B r with < u(1 x)2 (2eCad), it holds for all t 0 that
1
Vij E Fig 8ij.
Vij E Fa, Bij, 2 (1 2)
2
In Theorem 2.1 the separation between clean and corrupted edges arises because the upper bound for clean edges, 1/(2 vanishes as iteration / + (note that r > 1), while the lower bound for bad edges remains strictly positive, proportional to their corruption levels. This result makes no assumptions the distribution of corrupted directions, allowing fully adversarial and cycle-consistent

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corruption. The angle condition on 0zy.k addresses instability in dij,k caused by ill-shaped triangles. To ensure this condition, we exclude triangles with 8zj. a or Bij,k π a when computing djj.- In the probabilistic setting with i.i.d. Gaussian locations, Erdōs-Rényí edge probability p. and independent edge corruption probability 9, our result achieves the strongest known recovery guarantee (see Table[I). where neg is the maximum degree of the corrupted subgraph. The proof appears in the supplementary material.
Method
p
cb Pq
ShapeFit
Qn 1/2 log
1/2 n))
O / log n) O /Logooo" 9/2 n)
LUD
12
Ω2 1/3 log
1/3 n)
Our Theory (T-AAB)
Dnn 1/2 log 1/2 log n)
(pp/ log 1/2 n)
Table 1: Phase transition bounds p (lower better) and ep (higher is better) for location recovery.
2.4 Our full pipeline for camera pose estimation
We describe the full pipeline for camera pose estimation. Having covered location estimation, we focus here on the preceding steps: rotation and direction estimation. Given image keypoint matches (e.g., from SIFT or deep features), we estimate essential matrices for camera pairs using RANSAC, assuming calibrated cameras. The relative rotations R.j inferred from the essential matrices.
Rotation averaging via MPLS-cycle. Absolute rotations are recovered from relative ones using the MPLS algorithm. Unlike the original version, we fix X = 1 to emphasize cycle consistency and disable IRLS reweighting. This variant, which we call MPLS-cycle, yields significantly lower orientation error on real SfM datasets. We report these results in the supplementary material.
Direction estimation via robust subspace recovery. Following 19 each ground-truth direction Vij is orthogonal to the vectors v = R,T, x Ryný. where nf. and 77 are the normalized homogeneous coordinates of corresponding keypoints. Thus, 7i lies in the orthogonal complement of the subspace spanned by {ut k- However, many w vectors may be corrupted due to outlier matches. To robustly recover this subspace and estimate direction vectors (a problem reviewed in [2), we use STE [35 [3
which significantly outperforms the REAPER method [00] used in the LUD pipeline [9]
Outlier detection for directions. As an optional filtering step, we use STE to reject direction esti- mates Yi with low inlier numbers. This plug-and and-play module improves the accuracy of downstream location solvers. We observe notable performance gains with this preprocessing. In our real data experiment we use 20 as our minimum number of inliers.
3 Synthetic data experiments
We generate synthetic data under the uniform corruption model (UCM), where n is the number of cameras, q is the corruption probability per edge, a is the noise level, and p is the probability of edge connection in the Erdós-Rén Rényi viewing graph. Ground-truth camera locations t are sampled independently from (0, Iaxa). For each edge ij in the generated graph, the observed direction is given by:
Normalize( - -1 + TeG)), W.p. - 4 Normaliie((sss).
(9)
w.p. g,
where Eij ~ NO, Iaxa) independently.
We also consider an adversarial corruption model where corrupted directions remain cycle-consistent:
(ormalize(t; t + dGij), w.p. 1 - q: Yo Normalize(t tG + oesy), w.p. q,
(10)
where (t5) is set of alternative locations used generate coherent corrupted directions. To avoid ambiguity with the true structure, we require < 0.5 in this model.
We also test the robustness to corruption for different methods. We fix 100. p 0.5. We obtain the estimated absolute locations é, by our Cycle-Syne solver and compare the estimation error with
7

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Unit. Corruption
Corruption
Corruption
=0.05, Adv. Corruption
10
10
1.5
1.5
Median Error
20:5
10
1
1
Ayple-Sync (Ours)
it BATA
10
FusedTA
0.5
10
10
0.5
0.2
0.6 Comupttin Prob.
0.4
0.8
0.6
0.65
0.7
0.75
0.8
0.2
0.4
0.4
0.42
0.44
0.46
0.48
Corrupton Prob.
Corruption Prob.
Corruption Prob.
Figure 3: Median camera location error versus corruption probability. Left right: (1) uniform corruption without noise, (2) uniform corruption with mild noise (a = 0.05), (3) cycle-consistent (adversarial) corruption without noise, and (4) cycle-consistent corruption with mild noise. Error bars indicate standard deviation over 10 independent trials. No error bars are shown in plots with logarithmic scale.
existing works ShapeFit [5]. BATA [6] LUD [9]. FusedTA [S]. Since absolute locations can only be estimated up to global translation and scaling, we remove these ambiguities by computing the minimizer (c*, t²) of min.eR,teRs Eie(s]
||t] (ct, t)ll.
We compute the absolute translation error for camera i as ||t, (c*t, + 12 and report the median error over all cameras Figure 3 We generate 10 instances of the synthetic data and report the standard deviation of these statistics with error bars (no error bars with log scales).
We say method achieves "exact recovery" if it estimates the absolute camera locations with less than 10-4 median error. On UCM with = O, exact recovery is achieved only when S 0.3 except our method and Shapelit,, while our method exactly recovers when g S 0.8. Therefore, our method improves the phase transition threshold for exact recovery by a large margin. For the adversarial setting, Cycle-Sync remains robust up to corruption rates near the theoretical limit (q < 0.5), while all baselines quickly deteriorate. These results validate the effectiveness of our cycle- reweighting and its ability leverage global consistency for accurate location recovery in both stochastic and adversarial scenarios.
4 Real data experiments
We conduct real-world experiments on 13 ETH3D stereo datasets [23] [24] using a personal laptop equipped with an 11th Gen Intel(R) Core(TM) i9- 1900H processor (2.50 GHz, cores, 16 threads) and 16 GB physical memory. This dataset contains 13 sets of undistorted images taken from different indoor and outdoor scenes. It is challenging because its occlusions. viewpoint changes and lighting differences which results in highly corrupted pairwise directions. From raw images, estimate the initial keypoint matches using SIFT feature matching and perform geometric verification with RANSAC. With these initial keypoint matches the input, go through our full camera pose estimation pipeline and compute the output absolute location estimates { Jein) and the absolute rotation estimates {R, heenj- We remark that if the viewing graph contains multiple weakly connected components that cannot be merged into single scene, we retain only the largest component for rotation and location estimation, and exclude the others from computation and evaluation.
We use the callimeter-accuuaall laser-scanned camera poses from the dataset the ground truth. To eliminate the translation and scale ambiguity in camera locations, we preprocess the camera locations by translating them so that the geometric mean of all camera positions is centered at the origin, followed by scaling to ensure that median distance of the cameras from the origin We name the preprocessed ground truth camera locations as {t, ke and camera orientations as R hie())- To evaluate the output, we first align the rotation estimates with Ralign: the minimizer of the L2 rotation alignment enror negaa Σiejo] || RR; | Then, to remove the global scale and translation ambiguities for camera locations, compute the minimizer of the LI alignment

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error as (c), MiD.ER,AERS ElE(m) 1i; - (cRaligat; + +t)l2 For each camera i, we compute the translation error as | - (c" Ralignte t; + + )² 12. For each dataset, we report the median error over cameras. We test our method Cycle-Syne, BATA [36]. FusedTA [[3], ShapeFit [5] and LUD [99] for comparison of different location estimation methods, while the camera orientation method is fixed to MPL.--ycll.. Also, compare with existing global SfM pipelines, we report the translation error of camera poses from LUD, Theia (30] and GLOMAP [22]. In addition, we conduct ablation studies. Starting from LUD+IRLS, the original LUD pipeline, we gradually add each building block of full pipeline to demonstrate the effectiveness of cach block. We summarize the results in Figure 4 We highlight some key comparisons derived from our experiments.
Cycle-Sync Sync
LUD
Glomap
Theia
1
Median Translation
0.8
0.6
0.4
0.2
courtyard
delivery. area
electro
facade
kicker
meadow
office
pipes
playground
reliaf
rellel_2 .2
terrace
terrains
Average
0.23
0.07
2.20
Median Translation Error
Cycle-Sync
LUD
0.25
BATA
ShapeFit
0.2
FusedTA
0.15
0.1
0.05
courtyard
delivery ara
electro
facade
kicker
meadow
office
pipes
playground
rellet
relief 2
terrace
terains
Average
0.8
STEi MIE:CyyleSync
Median Translation Error
0.7
Cycle Sync-MPLS-cyele
0.5
LUD+MPLS-cycle
0.4
LUD+MPLS
LUD+IRLS
0.2
0.1
courtyard
felbeey arpa
elacto
facade
Hoker
mendow
office
Pipes
playground
retiel
mile
terera
tarrain
Average
Figure 4: Median translation errors using ETH3D. Each column represents dataset, the last one shows the average median eтrоr across all datasets. Different methods are presented per column. comparison of all pipelines. Unlike Theia and GLOMAP, Cycle-Sync (ours) estimates locations without bundle adjustment. Middle: comparison for different camera location algorithms; all methods preprocessed by STE and MPLS-cycle for fair comparison. Lower: Ablation studies.
Upper:

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Comparison among SfM pipelines (top panel). We observe that our full pipeline significantly outperforms existing SfM pipelines when averaging across datasets. Here, LUD refers to the original unmodified version proposed in the literature. Our method reduces the median location error to below 0.05, whereas other pipelines yield considerably less accurate results, with median errors exceeding 0.2 on average. Despite bypassing bundle adjustment, our method yields more accurate camera locations on average, whereas Theia and GLOMAP, even with bundle adjustment, suffer from many outliers and large alignment errors on several datasets. Although Theia performs better on the majority of datasets, our method consistently avoids failure cases and achieves the lowest average error across all datasets. Both metrics the number of datasets with superior performance and the average performance across all datasets- are informative: the former reflects robustness across the majority of scenarios, while the latter highlights stability and resilience against more challenging or adversarial datasets.
Comparison of location estimation algorithms (middle panel). In this comparison, we fix all preceding steps across all location solvers for fairness. Specifically, for all baselines, we employ MPLS-Cycle for rotation estimation and STE for direction estimation and filtering. Under this standardized setup, our method consistently outperforms others on 10 out the 14 datasets. In terms of median location error averaged over all datasets, our approach (with STE) achieves reduction of 60.9% compared to LUD (with STE), 66.1% compared to BATA, 89.8% compared to ShapeFit, and 90.0% compared to FusedTA.
Ablation studies (bottom panel). We observe that each component of our method consistently reduces the median location error. Starting from the LUD pipeline (LUD for location + REAPER for pairwise direction + IRLS 33 for rotation averaging). upgrading IRLS to MPLS reduces median location error by 17.8%: upgrading MPLS to MPLS-cycle reduces location error further by 9.0%. This is because better camera orientations improve the initial pairwise direction estimates. We remark that STE is an optional module for Cycle-Sync. Cycle-Sync outperforms all baselines (under the same preprocessing) both with and without STE. Specifically, upgrading LUD (without STE) to Cycle-Sync (without STE) reduces the location error by 66.0%. and the above panel indicates error reduction by 60.9% when using STE for both LUD and Cycle-Sync. Finally, replacing REAPER with STE within Cycle-Syne Synee--yccccne yields a 70.5% improvement. This significant gain stems from STE's effectiveness not just as an algorithm, but also from our specific use of STE in filtering outlying directions.
For other details, we refer the readers to the table in the supplementary material.
5 Conclusion and Limitations
We proposed a global framework for camera pose estimation that fully exploits cycle-consistency information. Our method simultaneously addresses the challenges posed by large variations in pairwise distances and highly corrupted directions. We establish the strongest known exact recovery guarantee for location estimation under adversarial corruption, which also yields improved sample complexity under standard probabilistic models. Empirically, our method is the only algorithm that consistently outperforms all state-of-the-ar baselines on real datasets.
Several limitations remain. Our theoretical guarantees apply only to the initialization phase and do not extend to the convergence of the full nonconvex optimization procedure. Establishing guarantees for the entire location synchronization algorithm- and eventually the full pose estimation pipeline-is an important direction for future work. In addition, the method relies the presence of well-shaped 3-cycles, and performance may degrade on sparse or structured graphs lacking such cycles. All hyperparameters (e.g., reweighting schedules and robust loss parameters) manually selected, and while not overly sensitive, it would be valuable to develop more adaptive or learnable reweighting strategies.
Broader Impact Our work advances the robustness and theoretical understanding of structure- from-motion (SfM) pipelines, with potential applications in robotics, autonomous navigation, digital reconstruction of cultural heritage, and low-cost 3D mapping. By improving pose estimation under high corruption and without bundle adjustment, our method may make SfM more accessible and reliable in challenging cesotrce-conttrained environments.
10

---

Acknowledgement G. Lerman and S. Li were supported by NSF DMS-2152766. Y. Shi was supported by NSF DMS-2514152.
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12

---

Supplementary Material
We provide additional theoretical and experimental details that complement the main paper. In Section A we present full proofs of Theorem 2.1. Section
explains the sample complexity analysis that underlies the bounds for T-AAB reported in Table G additional experimental results: 3D reconstructions (Section C Dn supplementary figures to the ETH3D
dataset [Section[E E runtime comparisons (Section . and extended results rotation synchronization (Section G
and generalization to the IMC-PT dataset (Section
All equation, figure, table, and theorem numbers continue from the main paper.
A Proofs of theory
We first establish Theorem A.1 and then conclude the main theorem.
Theorem A.1. Assume there exists an absolute >> such that for any ij Eg and Nes, a <
Oyj,k a. Then for ij Egg we have dij, C. (Bia + 85k)) where C.. 2(cos a+ v 50-s a
sin a
We note that the triangle is ill-shaped whenever fsj.k k = O or B.j.k R ग. In practice, we want dij k to be small for clean edge ij E Eg whenever the other two edges are relatively clean with small 8 and j... However, in these two ill-shaped cases, Ca in the theorem goes to infinity, and there is no effective upper bound to control dij.k.
Proof. Let 7p be the projected vector of My onto Span( Tkg)- Denote x = -TtjTks s = 7H. 75k T and 7ik 7ki- Since 7/p is in Span Tky), there exists constants a, b such that 7pp a7ki + byisk
By the definition of 7p. we have
(Tu 07Ka bygke 7ks) = 0
(11)
(1i 07
bnjk, 75k) = 0
By linearity of vector inner products, we have
x b =
(12)
0
Solving for and gives
T- y2 T- 2
(13)
Therefore 7/p = 18-13
1- Mks V- 22
Yyke
Case 1: 7pp & 5(1gkk 7kk). In this case, since dij,k S 1, we only need prove 8 55xx > 1/Ca-
Note that by the definition of ((7a,, Tss) we have either < 0 or b < 0, which implies y/ 0 or y xz > On the other hand, since the ground truth directions My. 7iko Thi are cycle consistent, we know that the projection of 7 onto Span()ã.> 75x) is in the set M( 75 Therefore we also have x= 1²²²* z² < 0 and y² -x==" < O, where x² = 2 Tke V² = =n 75 and =* = 7 Tks Without loss of generality, we assume the case x y2 O. If max(5; $76) 2 C. then the lemma is trivial. If max(33 (3 5jx) < C. we first verify two claims.
Claim 1: x- y² By the definition of 3 and Ki we have the following inequality:
sin where a" is the angle such that cos a
2.
8 +
5
2
Irkk 7/ll + lngae 75al = 2sin
sin
<
+
<
(14)
2
C.
13

---

Also, < 0ij.k < ạ - a is equivalent to h3 Ykil
cos 0eg.Al < cos a. Combining this with
equation (14) gives
Mikk Thd Infierss + (15h -73h)²
rrs Yks + 158 (1ks - Vke)
I Iigennal + -1(5RR 796) rasl + 1nfk (7he 7ka1
17gh vaal + -135 Tpal his veal
2
$ cos +
cos
(15)
C.
On the other hand, we know that a"
y²=" and a" = -la0" S - -sin a² This implies that 1-2
x² 3² ain -sin
Claim 2: Let oij = Vig 1i. 65k = "ikk 25 and oe: = -hi- Suppose max 165u1 (6xz)) =
5. Then [x=" - y x²=")) (x -z- S 65: if új E Eg (i.e. dig = 0), then (xx --"=""²² (x yz)l I 48.
In fact, by the definition of x. x,,x, ,=,x,,,--
u.. U. v/' x², we have the following estimate:
10=- - 2²²²) - (x - -yy) = 11 The - 2ed T5 rel) - ((7h) +
+ Ōes)
(rie + )e,)² (15x + 55x) + "xx)"((i + dki))l
$ | stines - - dus - 12 7kk + Vig dyengien
+ Wh o Thi + 765 75k 75 Sxi)l
$ 18g + hi? dvel + 13 nal he 05 rnd
+ hij N5007 57. nkl + hig 7jk" 75
(16)
By the fact that all 7's are unit vectors, the right hand side of the equation above is at most 66 in general; if ij E F, (i.e. 8.z- 0) then it is at most 46.
Combining claim and claim 2, we know that 0 < x y2 S (x- --==-y="*+++ yz) -xx" -y-=")²= <
45 sin a This yields 5 > sina' 4- Note that by ij E Eg. we know that dij = 0. Therefore
8 max ((0zl 55u1, (ōuel) = max(165u], (5x=)) S |dرz| + |óaal 2 sin
f1 z 2ii + S K5 + s
By C. 2(cosn4V/5 ooss sin
know that sin C. therefore the theorem is proved.
Case 2: 7/p E 2(ysk) 7kj)- In this case, let 8sk 7hk 7 and ōzk 7ik 75k- Then
dif.
7ul By the fact that Ng. 75k- Vie coplanar and V/s = %i-- we know that Inn Biy
7ik Vkj Yk = O. We have the following inequalities:
Iree x Yko 7ayl
dey
sin Ozj.1 1(7/2 + 6sx) x (1ă, + 8xz) Tigl
sin Oij.k
1(6u x (7i + 8ks) ) + Tik x ōks) 76s1 sin A.g.
18eulha + Skyl + Iv longl
< sin Oij,k
(17)
< 5,k + dkz sin a
(18)
Zi Note that |5,z] = 2sin
S Si and similarly |5gz1 = sin 85x S 37 Therefore d.y.k S
2
a (3je + +3%k)
Co. (3 + 33 where the latter inequality comes from the fact that Co >
14

---

Proof of the main theorem.
Recall the statement of Theorem 2.1
Main Theorem (Theorem 2.Ip. Assume there exists a > such that for all ij E Fig and
k E Nij. Or < Oij.k < π a, and X < 1 + eCa/ VeCa(2u + +cCa)/n where C. 2(cos + V5 4 cos² a)) /sinn a. Then, for $1g,5 computed by the iteratively reweighted AAB algo- rithm [25] using Bu S 2x and Bu+1 r with 1 < r < p x)2/ (2 it holds for all t 0 that
1
Vij Eb, Suja - (1-
Vij Eg
Sig.t
2,5
Proof. We prove the main theorem by induction. For t 0. the definition of À imply that for all ? E E,
ΣXEN, Jig.k ENEGy dep
Gūyl
(0))
[N1yl 2
2 1 Nesl
8ig 2 p(1 -
[N1sl
Furthermore, by the fact that 0 S dijje S 1 we have for all ij E Egg
ΣXEN,, dij.k ELEB, dij.k
ΣxeB.
1 5 X S 230
S [Nig]
3j |Nal
[No.l
Therefore the theorem is proved when t O.
Next, we assume the theorem holds true for 0, 1.... and show that it also holds true for t By ((++1) and the induction assumption 23 2 maxijeF, Bigar, we have the following the definition of 8ij
inequalities for any ij E Es:
EvEN., e- -0 dey
(1+1)
ENEN. e -2(0 +8(l)
ENEGy e- -Oellh0 +2/0) dig
ENEN.Y e -e(ef(+
EKEGe C -'dij.
(19)
Nesl
|Gyl Nayl
A)
IV
Next we bound 5( (++1) for ij Eg. By the definition of 8ij the fact that d,y = 0 for is E Gij.
(t+1),
and Theorem A.1 we know that
EXENs e -16 +o + +f
ΣKEBi, e +A(e((+08)) desk deyk
(1+1) 1g
(20)
EvEN., e -2116h
EKEN, e -A(55(+
Ca EkEB,, e -966812 -316612+550) +550++55)) (8), + 55x)
S
(21)
ΣKEN, e- -Bi +1(0)
By the induction assumption that slf) p(1-N) 5t for all ij E, we know that
Σ
e (1ff +2(2) (5 + 336) S Σ e -B, " (#t, + +3 (Sit + k5k)
(22)
ke ke
ke
15

---

Note that ze
L for any e > 0 and x > 0. Let
B, الااي and = 8ix + 8j we have
Σ
8 ا (8l++ (5 + fjk) S Bel(1 - A)
1
Byy
(23)
Bep(1 X)
k
Be
Also, by the induction assumption that 2/ z maXijeE, $1 and the nonnegativity of 's ss we have
Σ
B. +o 2 GGyle
(24)
kE Ni5
kEGi
Combining 200
and the definition of X, we have
BBgl
2cCar 1 Beju1 - M1 x)2
eCo
((+1)
(25)
Giyl
28
By the assumption that X < 1 cCo
cCa (2+ =Ca we know that
2nA < 1. Therefore by
A(1
taking < < -()=A)² we guarantee that for any ij E Fig $1 x!(+1) 2eA
This and
23
2T
(19) concludes our theorem.
Comment on the order of μ:: We remark that in the theorem μ = minjjEE, EKEC., /y.h/ (8y). which implies that for all ij E E,
drj, Edg 2 pšوj-
(26)
GGggl
kEGss
We would like to investigate the dependence of μ on n. That is, we estimate the magnitude of μ such
that (26) holds for all edges. First of all, it is safe claim that (26) holds for all ij whose 87, > 0.5 when j is positive constant (i.e., μ O(1). That is, the left-hand side of (26) is lower bounded by a positive constant. Let Ruj.h be the normal vector of the plane Span t t t tÿ where E follows the standard Gaussian distribution. For 87 S 0.5, one can show that
1
Σ kEG
digk 2 GGil
1
Σ kEG.
1/s Yiy
Gij
Vlogn
min (5 1 8)y)
Vlogn 8
(27)
Vlogn
for some absolute constant d', c, which suggests
# c/ logn.
In (2)). the first inequality follows from the definition of dyy.A. the first equality follows from the definition of Bi and the last equality is due to the assumption $ū I 0.5. The second inequality
is commonly assumed for all ij E E in 12 251, which they call the c/ Vo diwssllddisrrbbueeed condition. is proved in ? that the if t i.i.d. with standard Gaussian, then c/ Vlog n-well- distributed condition holds with high probability.
B Explanation of the Order of Complexity for T-AAB in Table 1
We assume the Erdós Erdös-Rényi graph G(n, p), where p is the probability of connecting two nodes, with edge corruption probability We show that the recovery guarantee in Theorem 2.1 holds under this probabilistic model, provided
p = 2(n - 1/2 log 1/2 n)
(28)
and
6 = = logn).
(29)
We note that given
28
22 is equivalent
q= 0 1/0LLL/log))) I)
(30)
16

---

We first verify
330 where we note that it is sufficient to focus on the worst case
= / Vlog for an absolute constant C1-
That is, we show that this choice is sufficient for exact recovery with high probability. We prove exact recovery by establishing with high probability the sufficient condition of Theorem 2.
-ica (2+ eCa
eCo A
x<I+
(31)
We control the ratio of bad cycles as follows. For any fixed edge ij E E, Xi = |B is the average of the Bernoulli random variables X₃ = [kkBBy) where E Bij with probability
1 - (1 - 2) Consequently,
201
E(A,)) = 1 (1- =0(2 3 2q
Vlg n
Next, we investigate the concentration bound for Xij and then for X = maxij Aej. We recall the following one-sided Chernoff bound 4 for independent Bernoulli random variables (X1) with means {pi) P Ei=1 Pi/n, and any 2 1:
2
Pr
(1 7)p
25q
(32)
Applying (32 with the random variables X = [(KE and 17 =
Aci
201 log 181y1
Pr
(33)
Vlg n
To control the size of NoJ1 in above probability bound, we use the following Chernoff bound 4 for i..dd Bernoulli random variables {-x with means μ and any 0 < <
Pr
Σ
xi
np
< 2e pm
(34)
m7
We note that by applying (34) with the random variables [(EEN,) and r7 1/2, obtain that
Pr
No
2 mp²
2e mp
(35)
By combining the bounds in
33 and
(35
we have for sufficiently large
mn
Pr
Aij
4e
amp
+ tnpp
<
+ 2e hmp
(36)
v Vlog
n
By applying a union bound over ij E, we have
4c
EEle- Fnp² + - hpp
Pr
<
(37)
Vlogn
where À maxij Aij. Therefore, with //lloog and sufficiently large n, have
A
4c
w.p.
Ele npp
2 1pmp?
(38)
Vlogn n
Finally, show for a proper constant Ct. concluded, We note that the RHS of 311 is lower bounded by
holds with high probability, and the exact recovery is
eCn
eC,,
cCa
1 +
p
1 +
eCa
+
/ cC (2 +2+
eCu,
P
1
(39)
2,
cCo
(2
+
eCa
2(2 +
cC P
3eC
3cCa
17

---

Combining this estimate of f with 39
we obtain
eCa
cCo
eCa
1
(2
(40)
V
p
p
3cC, Vlog ni
Therefore, to guarantee (31) suffices to RHS of 38 be bounded from above by the RHS of (40). Namely, we require that
4er
Vlog 7
3eC Vlogn
which can be easily satisfied by setting e1
Therefore, with the order of q
I2cCa
(log)) and equivalently c = (cp/ (12pl1 n)) we can guarantee (31) and hence
exact recovery with the probability specified in (38). Consequently, verify that (30) sufficient
for exact recovery with the latter probability.
We finally note that assuming 28 that is, p 2 con 1o2
1/2 n for sufficiently large constant co.
the probability specified in 38 is high. Indeed,
3 np" 2 F npp > 1 - 3n exp(eppppp exp(-min C1 }mp²)
1 - E
(41)
4 12
3n exp( min C1 1
Joo logn) = 1n 2-n--mhh min(
lco
(42)
12
Thus, with high probability, the recovery conditions Theorem 2.1 are satisfied when (88 and 29 hold. We thus verify the bounds reported for T-AAB in Table
C Visualization of 3D sparse models on ETH3D
We compare 3D sparse point cloud reconstructions of Cycle-Syne, GLOMAP and Theia. For their default reconstruction parameters. For the Cycle-Syne pipeline,
GLOMAP and Theia, we feed the resulting camera poses to the 3D point triangulator in COLMAP (it uses bundle adjustment for the triangulations, while fixing camera pose estimators) and return the sparse 3D model. The latter was done quickly without careful tuning of parameters. Table 2 compares the number of triangulated 3D points for different SIM pipelines and some 3D sparse models ETH3D. Tables and 4 demonstrate the actual 3D sparse models by these methods.
We observe from Table that for 9 out of the 13 datasets, Cycle-Sync improves the number of triangulated 3D points. This leads to an improvement of the overall quality of reconstruction for these datasets noticed in Tables and and This is due to the improvement to Cycle-Sync. For the other 4 datasets, Cycle-Sync fails to recover a meaningful 3D) sparse model. For two of these datasets (meadow and office) all three methods dataset (relief) Theia the only one which performs well and for the last dataset (relief_2 2) GLOMAP performs better than the other two methods.
initial camera poses thanks
performing well. For one
Dataset
Cycle-Syne 30851
GLOMAP
Theia
courtyard delivery aarea area
27674
17835
49306
25403
9534
electro
28061
24477
2641
facade
86111
66302
70571
kicker
26649
18685
10232
meadow
647
667
649
office
1351
2686
1395
pipes
8662
3551
1423
playground
16885
11816
416
relief
1642
12617
29588
relief
1695
16902
3212
terrace
25285
13898
7216
terrains
47485
25082
12876
Table 2: Number of triangulated 3D points for each dataset using Cycle-Sync, GLOMAP, and Theia
18

---

Dataset
Cycle-Syne
GLOMAP
Theia
courtyard
delivery_are area
electro
facade
kicker
meadow
office
pipes
playground
Table 3: Triangulated 3D reconstructions (Part 1) using Cycle-Syne, GLOMAP, and Theia.
D Visualization of Camera Pose Estimation
We demonstrate pose estimation results for four scenes: courtyard, meadow, office, and pipes. For each scene, we compare the ground-truth camera poses with those estimated by GLOMAP, Theia, and Cycle-Syne. Figures 8 illustrate these comparisons.
the first three scenes, Cycle-Syne produces camera poses that align more closely with the ground truth, while GLOMAP and Theia exhibit larger misalignments. In the pipes scene, both Cycle-Syne and GLOMAP achieve good alignment, whereas Theia fails to produce meaningful results.
19

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Dataset
Cycle-Syne
GLOMAP
Theia
relief
relief
terrace
terrains
Table 4: Triangulated 3D reconstructions (Part 2) using Sycle-Sync, GLOMAP, and Theia.
E
Supplementary Tables for ETH3D
We provide additional tables and figures to demonstrate the pose estimation quality of Cycle-Sync Table 55 demonstrates the location error for each SfM pipeline. Table demoonstraes the location error for different location estimation algorithms. 7 the effect of each building block of Cycle-Syne by beginning with the LUD pipeline, and gradually adding MPLSS, MPLSS-ycle, Cycle-Syne and STE.
Table 5: Translation Error of each SfM pipeline on ETH3D. Here and denote the mean translation and median translation erro, respectively. BA refers to bundle adjustment.
Scene
Cycle-Sync i
LUD
GLOMAP (with BA)
Theia (with BA)
i
{
courtyard
0.27
0.02
0.85
0.75
0.34
0.03
0.01
0.01
delivery_aree area
0.15
0.04
0.37
0.24
0.01
0.00
0.09
0.00
electro
0.21
0.03
0.30
0.10
0.01
0.01
0.01
0.01
facade
0.25
0.00
0.43
0.18
1.01
1.01
0.01
0.00
kicker
0.02
0.01
0.09
0.02
0.36
0.01
0.01
0.01
meadow
0.02
0.02
0.40
0.28
0.91
1.01
0.68
0.51
office
0.20
0.03
0.17
0.03
0.06
0.01
0.95
0.97
pipes
0.01
0.01
0.06
0.03
0.01
0.01
0.01
0.00
playground
0.12
0.01
0.40
0.13
1.30
0.99
0.01
0.00
relief
0.00
0.00
0.00
0.00
0.90
0.78
0.01
0.01
relief_2
0.01
0.01
0.70
0.73
0.01
0.01
0.81
0.79
terrace
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
terrains
0.01
0.01
0.02
0.01
0.00
0.00
0.42
0.05
Average
0.10
0.01
0.29
0.19
0.38
0.30
0.23
0.18
20

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Figure Visualization of camera location estimations and ground truth on courtyard dataset. Top left: ground truth. Top right: Cycle-Sync. Bottom left: GLOMAP. Bottom right: Theia
Dataset
LUD
BATA
ShapeFit f
FusedTA
Cycle-Syne
f
f
f
courtyard
0.37
0.02
0.41
0.08
0.32
0.02
0.21
0.05
0.27
delivery area
0.25
0.16
0.35
0.23
0.18
0.08
0.96
0.93
0.15
0.04
electro
0.24
0.06
0.23
0.06
0.21
0.02
0.22
0.04
0.21
0.03
facade
0.30
0.01
0.28
0.01
0.98
0.98
0.91
0.86
0.25
0.00
kicker
0.03
0.01
0.03
0.01
0.02
0.01
0.03
0.02
0.02
0.01
meadow
0.06
0.02
0.11
0.05
0.08
0.02
0.08
0.02
0.02
office
0.20
0.03
0.21
0.03
0.20
0.03
0.21
0.03
0.20
0.03
pipes
0.01
0.01
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
playground
0.16
0.03
0.15
0.04
0.08
0.01
0.17
0.08
0.12
0.01
relief
0.00
0.00
0.03
0.01
0.00
0.00
0.00
0.00
relief
0.18
0.19
0.04
0.03
0.90
0.90
0.11
0.08
0.01
0.01
terrace
0.01
0,01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
terrains
0.01
0.01
0.06
0.06
0.01
0.00
0.02
0.01
0.01
0.01
Average
0.14
0.04
0.15
0.05
0.23
0.16
0.23
0.16
0.10
0.01
Table 6: Comparison of mean (f) and median (f) translation error for each ETH3D scene for different
location estimation algorithms.
Runtime
Table compares the runtime of location estimation methods on ETH3D. We observe that STE-based methods are significantly faster than non-STE methods, including LUD+IRLS (the old LUD pipeline). In particular, Cycle-Syne Sync runtime is 48% lower than that of the LUD pipeline. Although Cycle-Sync is slower than common location estimation algorithms such as BATA, ShapeFit, and FusedTA, its
21

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Figure 6: Visualization of camera location estimations and ground truth on the meadow dataset. Top left: ground truth. Top right: Cycle-Sync. Bottom left: GLOMAP. Bottom right: Theia.
Scene
LUDAIRLS
LUD+MPLS
LUD+MPLS-cycke
CydeeSyne+MPPLS-cydee
STE--CydeSSue--MPPS-eyd
E
courtyard
0.85
0.75
0.80
0.71
0.82
0.78
0.76
0.39
0.27
0.02
delivery area
0.37
0.24
0.36
0.23
0.37
0.28
0.16
0.06
0.15
0.04
electro
0.30
0.10
0.31
0.11
0.31
0.10
0.24
0.04
0.21
0.03
0.19
kicker
meadow
0.39
0.28
0.49
0.47
0.48
0.23
0.18
0.06
0.02
0.02
office
0.17
0.03
0.18
0.03
0.22
0.03
0.21
0.03
0.20
0.03
pipes
0.06
0.03
0.05
0.02
0.05
0.02
0.01
0.01
0.01
0.01
playground
0.40
0.13
0.36
0.12
0.42
0.19
0.23
0.01
0.12
0.01
reliet
relicf
0.15
0.01
0.01
terrace
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
temrins
0.02
0.01
0.01
0.01
0.01
0.00
0.01
0.00
0.01
0.01
Averuge
Table 7: Translation errors (7 mean translation error, [ median translation error) across all methods for ablation study.
22

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Figure 7: Visualization of camera location estimations and ground truth on the office dataset. Top
left: ground truth. Top right: Cycle-Sync. Bottom left: GLOMAP. Bottom right: Theia.
Figure 8: Visualization of camera location estimations and ground truth on the pipes dataset. Top left:
ground truth. Top right: Cycle-Sync. Bottom left: GLOMAP. Bottom right: Theia
23

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30
25
MPLSS
15
MPLS cycle onl
10
countard
deliery 301d
eerrro
licher
menatow
office
Pipea
playgmand
relie
mine
Heracaa
twraliss
Amergg
Figure 9: Rotation error (degrees) comparison on ETH3D for different rotation synchronization solvers.
runtime remains within the same order of magnitude, while achieving superior accuracy and stability camera pose estimation.
Scene
LUDHIRLS LUD+MPLS
STE-based (with MPLLS-ycle)
STE-LUD
STE+BATA ShE+ShapoFit STE+FusedTA Cycle-Syne
courtyard
23.60
19.38
6.10
5.83
5.74
6.01
10.05
delivery area
16.99
15.97
4.13
4,06
3.95
4.16
8,38
electro
15.58
13.92
4.21
3.92
3.82
4.19
8.58
facade
100.68
89.37
24.48
23.88
24.47
24.32
30.41
kicker
mendow
0.95
0.92
0.42
0.40
0.41
0,58
4.63
office
1.89
1.70
0.91
0.86
0.87
1.02
5.16
pipes
2.23
2.13
0.73
0.64
0.65
0.77
4,82
playground
10.78
9.60
2.79
2.62
2.57
2.84
7.34
5.49
1.48
relief 2
terrace
8.38
8.53
1,98
1.99
1.89
2.01
6.47
terrains
13.18
12.72
4.37
3.80
3,70
3.96
8.88
Average
17.08
15.50
4.45
4.36
4.43
8.91
Table 8: Runtime comparison (in seconds) of different SfM pipelines on ETH3D.
G Table and Figures for Rotation Synchronization
In this section we show the tables and figures for rotation errors. Figure I demonstrates the rotation errors on ETH3D across different rotation synchronization methods. T demonstrates the rotation enror ETH3D across different pipelines.
We observe that MPLS-cycle (used in our Cycle-Syne) greatly improves rotation accuracy over existing pipelines. On average, MPLS-cycle reduces the median rotation error by 62.8% and mean rotation error by 74.1%. compared the best existing pipeline GLOMAP. Also, our proposed MPLS- cycle reduces the mean rotation error of MPLS by 56.6% and median rotation error by 64.8%. is worth noting that Cycle-Syne outperforms GLOMAP and Theia even without bundle adjustment. This demonstrates that even without bundle adjustment, our approach outperforms baselines that rely on
H Additional Experiment for IMC-PT
In this section we compare camera location estimation results for different location estimation algorithms on IMC-PT. This dataset consists of 9 city-scale image sets, well as ground truth camera poses estimated by aligning COLMAP SfM model with LiDAR scan. We generate image matches using LoFTR 29 deep learning feature matching method instead of SIFT. We use LoFTR since

---

Table 9: Comparison of rotation error (degrees) on ETH3D for different pipelines. Here R. means the rotation error and the median rotation error measured in degrees (0--1809) respectively. BA refers to bundle adjustment.
Scene
Cycle-Syne
LUD
GLOMAP (with BA)
Theia (with BA)
R
R
R
R
R
R
R
courtyard
0.78
0.36
43.85
35.93
3.50
1.61
0.14
0.10
delivery_area
0.76
0.28
0.78
0.43
0.09
0.07
0.41
0.21
electro
1.11
0.18
1.28
0.45
0.11
0.11
0.07
0.06
facade
0.18
0.11
18.10
8.64
0.78
0.35
0.09
0.09
kicker
0.44
0.21
0.25
0.22
0.47
0.22
0.11
0.09
meadow
0.29
0.18
2.94
1.64
21.13
6.05
3.14
3.37
office
3.39
1.69
8.98
0.29
0.69
0.34
7.23
6.34
pipes
0.17
0.18
0.34
0.28
0.12
0.13
0.08
0.09
playground
0.17
0.14
0.34
0.22
0.67
0.23
0.07
0.09
relief
0.11
0.11
0.11
0.12
3.18
1.68
0.14
0.15
relief_2
0.29
0.31
41.78
36.84
0.10
0.10
36.85
34.23
terrace
0.15
0.13
0.27
0.27
0.12
0.12
0.15
0.13
terrains
0.22
0.25
0.27
0.29
0.19
0.21
3.54
1.78
Average
0.62
0.32
9.18
6.59
2.40
0.86
4.00
3.59
Medain Location Error
0.6
0.5
0.3
0.2
besknggee prince
poatt sharal
antiinnn
Prisctaale allny
LUD
BATA
ShapeFit
FusedTA
Cycle-Sync
Figure 10: Median translation error for each IMC-PT scene and their average. The last column
denotes the average median error across all datasets.
is proved to be effective on popular homography estimation, relative pose estimation and visual localization benchmarks. Table an0 and Figure demonstrate the location error of different location estimation algorithms, where rotation synchronization method MPLS-cycle and all methods use STE.
We observe that Cycle-Syne achieves the smallest mean and median location error averaging on all datasets. For 3 out of 9 datasets, Cycle-Syne achieves both the smallest mean and median error For other datasets, Cycle-Syne achieves no significantly larger mean and median error. The largest difference from Cycle-Syne to the best method in mean and median location error is 0.05, which is small compared the average scale of location. The improvement of Cycle-Sync smaller than that in ETH3D. We believe the reason is that LoFTR is not good feature for this dataset, since it tends to overfit repetitive structures such as windows and facades. sum up, while the gains over baselines smaller than on ETH3D, Cycle-Syne still achieves the best mean and median across most scenes.
25

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Table 10: Translation Errors and i) for IMC-PT.
Scene
LUD
BATA
Shapelit
FusedTA
Cycle-Syne
i
i
i
i
i
brandenburg
gate
0.24
0.13
0.25
0.13
0.24
0.12
0.24
0.12
0.24
0.12
buckingham
palace
0.36
0.26
0.37
0.28
0.38
0.26
0.37
0.27
0.35
0.25
colosseum
exterior
0.75
0.56
0.90
0.59
0.67
0.58
0.92
0.58
0.56
0.47
grand place brussels
0.55
0.50
0.54
0.46
0.60
0.53
0.54
0.47
0.54
0.49
palace
westminster
0.43
0.35
0.40
0.32
0.44
0.33
0,41
0.33
0.44
0.35
pantheon
exterior
0.44
0.29
0.44
0.28
0.44
0.29
0.44
0.28
0.44
0.29
taj
mahal
0.12
0.06
0.12
0.05
0.12
0.05
0.12
0.05
0.12
0.06
temple nara japan
0.48
0.31
0.43
0.23
0.45
0.26
0.43
0.24
0.46
0.27
westminster
abbey
0.87
0.35
0.84
0.34
0.83
0.34
0.84
0.34
0.89
0.35
Average
0.47
0.31
0.48
0.30
0.46
0.31
0.48
0.30
0.45
0.29
I Additional Supplementary Tables for ETH3D
L1 Sensitivity to Initialization
In tables 12 and II we report the mean and median location eπгоr on ETH3D data (averaged over different scenes) after several iterations for T-AAB and trivial initialization schemes. While the T-AAB initialization accelerates convergence by providing better starting weights, it does not significantly influence the final accuracy. Even trivial initialization using uniform weights performs similarly after sufficient iterations. Therefore, our method is robust even to trivial initialization, let alone variations in the T-AAB parameter.
Iteration
Mean Error
Median Error
5
0.110
0.020
10
0.100
0.017
15
0.099
0.016
20
0.099
0.016
Table 11: Performance with trivial (uniform) initialization.
Iteration
Mean Error
Median Error
5
0.102
0.018
10
0.099
0.016
15
0.099
0.015
20
0.099
0.014
Table 12: Performance with T-AAB initialization.
1.2 Replacing LUD with BATA in Our Pipeline
In table 13 we show results of replacing LUD with BATA within our reweighting framework. Indeed, integrating LUD under Cycle-Sync's reweighting leads to better performance compared to integrating BATA. This is likely due to the stronger constraint of LUD preventing collapsed trivial solution (the constraint is enforced on every edge). Moreover, our Welsh-type objective function already accounts
26

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for large variations in distances, making angle- based methods such as BATA less advantageous in this case.
Method
Mean Error
Median Error
Ours-L LUD
0.099
0.014
Ours-BATA
0.202
0.079
Table 13: Comparison between LUD and BATA integration under the Cycle-Syne framework on ETH3D data.
1.3 Annealing Schedule A
In tables 14 and 15 we include synthetic experiments for different annealing schedules A in settings with both additive noise and high corruption. Table 14 uses uniform corruption with g = 0.7 and higher noise level = 0.2.
X median err
10+- (ours)
10+f
1+5
0.24
0.36
0.27
0.37
0.29
Table 14: Uniform corruption 9 0.7, noise level 0.2.
Table 15 the adversarial corruption with 0.45 (close to the theoretical limit 0.5) and higher noise level 0.2.
(ours)
10
median err
0.17
0.32
0.18
0.20
0.17
Table 15: Adversarial corruption q = 0.45, noise level a = 0.2.
We observe that our choice of X has the lowest median error for both settings. Therefore, our proposed schedule strikes a good balance between residual-driven updates early on and cycle-consistency emphasis later. This schedule has consistently outperformed alternatives, particularly in settings with both additive noise and high corruption.
We remark that using only the residual for reweighting (i.e., setting A = 0, which corresponds to IRLS) often leads significantly higher errors compared cycle-based reweighting methods. The underlying issue is that, in noisy settings, some bad edges may coincidentally exhibit low residuals. When this occurs, the aggressive reweighting imposed by the Welsch objective can assign them disproportionately large weights, thereby amplifying their adverse impact. In contrast, our cycle- based reweighting effectively overcomes this limitation: it is extremely unlikely for bad edge to exhibit low average cycle inconsistency, unless all cycles it participates in are consistent-an an event that is highly improbable for corrupted measurements. Overall, we find that our annealing strategy consistently achieves the best performance across most scenarios.
We also observe this trend in the context rotation synchronization. Figure G illustrates that emphasizing sele-coonsiteeccy can significantly reduce orientation error. However, for rotation there no need for annealing does not rely on distance estimation.
Additional Experiment on IDSfM Datasets
We report the mean and median location emrors for IDSIM dataset (averaged over different scenes), with all methods consistently preprocessed using STE and MPLS-cycle table [6] Note that the ground truth camera poses in IDSIM are generated by Bundler, which is considered outdated compared to modern tools like COLMAP (and even COLMAP may lack the accuracy of laser scans). We remark that this could be less reliable when benchmarking high-precision location solvers.
27

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Method
Mean Error
Median Error
Ours
0.292
0.115
LUD
0.314
0.132
BATA
0.362
0.130
ShapeFit
0.670
0.410
FusedTA
0.330
0.130
Table 16: Performance comparison on the 1DSfM (photo tourism) datasets, averaged over multiple
scenes. All methods are processed with STE and MPL.S-yycl.
28

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Reviews:
[{"review_id": "Lg1P6NpmfR", "reviewer": "Reviewer_37Zk", "summary": "This paper proposes a framework for robust global location estimation of cameras, given pairwise direction estimates in the context of the Structure-from-Motion (SfM) problem. The approach formulates a LUD-type cost function using a robust Welsch loss and leverages distance-based cycle-consistency over cycles, where the distances are iteratively estimated. In the initial iteration, the weights are derived using angle-based cycle-consistency computed via the AAB method [20], which relies solely on the input relative directions.\n\nTo obtain more accurate global rotations during the rotation averaging step, the authors introduce a variant of the MPLS method that progressively prioritizes cycle-consistency over residual minimization. These improved global rotations are crucial for enhancing the quality of relative translation directions in the subsequent translation averaging stage.\n\nAdditionally, the authors propose a technique called STE, which robustly estimates the relative translation directions using a subspace recovery method [27]. A theoretical guarantee is provided, demonstrating that the proposed method can accurately recover the corruption levels of both inlier and outlier edges.", "questions": "1. When $\\lambda_t$ increases from 0 to 1, the weighting shifts from residuals to cycle-inconsistency. While it’s understandable that cycle-inconsistency may be unreliable in early iterations due to dependence on the initial (and inaccurate) distance estimates, the residuals are also computed from the same initialization and are therefore equally unreliable. This raises a key question: what is the rationale for prioritizing residuals early on if they too are sensitive to initialization? I think that addressing weakness-5 may address this concern also.\n2. It is unclear why the authors specifically advocate for the Welsch loss function. Other parametric robust loss functions, such as Geman-McClure or Huber, can exhibit similar behavior in handling variability in edge lengths when tuned appropriately. Can you correct me if this is wrong?\n3. In MPLS vs MPLS-cycle for rotations averaging, what is meant by \"disabling reweighting\" in Lines 206-207?\n4. Typo: In Algorithm-1, in the first two steps, should it be $\\tilde{s_{ij}}$ instead of $s_{ij}$?\n5. Although the 1DSfM datasets are older, they typically exhibit higher corruption levels than ETH3D. Given that the translation errors in Table 6 (supplementary) for ETH3D are already low, it’s unclear whether improvements in translation averaging significantly impact the final SfM output. As an optional evaluation, it would be valuable to test the method on more challenging datasets like 1DSfM to better highlight its robustness.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The authors propose a comprehensive framework for robust camera translation estimation by:\n\n     (a) Introducing a novel technique based on distance-based cycle-consistency, enabling accurate estimation of camera locations from input translation directions.\n\n     (b) Applying robust modifications to each step that influences the accuracy of translation directions, including (i) a modified cycle-consistency based rotation averaging step, and (ii) the use of matched point coordinates to estimate robust relative translation directions via subspace recovery (STE).\n\n2. The paper is generally clear, well-organized, and accessible to readers familiar with Structure-from-Motion pipelines.\n\n3. Strong theoretical guarantees are provided, demonstrating that the method can accurately recover corruption levels of relative translations even under arbitrary adversarial corruption.\n\n4. The experimental evaluation is extensive, with results reported on both synthetic and real datasets, across a variety of corruption models (uniform and adversarial). The method is compared both as a standalone location solver and within end-to-end SfM pipelines that include bundle adjustment, providing a comprehensive assessment of its practical value.\n\nWeaknesses:\n1. Theorem 2.1 lacks intuitive interpretation, particularly in terms of the values taken by the right-hand side (RHS) of the inequalities involving $s_{ij}^{(t)}$ for inlier and outlier edges. For the recovery guarantee to be meaningful, the RHS for inliers should ideally tend toward zero, while that for outliers should be significantly larger—indicating a clear separation between the two. However, the paper does not provide any insight into the typical magnitudes or range of these RHS terms, making it difficult to assess how strong or practical the theoretical guarantee really is. A small illustrative example or further discussion quantifying the behavior of the RHS terms would greatly improve the clarity and usefulness of this result.\n2. The method relies on a carefully chosen initialization of weights $w_{ij}$ using $\\tilde{s}_{ij}$ from T-AAB, but it’s unclear whether such meticulous design is truly necessary. If so, this should be justified through a sensitivity analysis showing how performance varies under alternative initializations.\n3. The ablation studies are somewhat incomplete, as they do not evaluate the effect of removing or replacing individual components in the pipeline. For example, it remains unclear how a variant such as STE + LUD + MPLS-cycle would perform. Based on the results, it appears that a significant portion of the improvement may be attributable to the STE step rather than the proposed cycle-consistency refinement, which is positioned as the primary contribution. Additionally, it would be informative to evaluate the performance of BATA in place of LUD within the pipeline, especially since BATA is known to yield better solutions in scenarios with high variability in edge lengths.\n4. In the comparison against SfM pipelines such as GloMap and Theia, the results shown in Figure 9 of the supplementary material (a corrected version of upper subfigure of Figure 4 in the main paper) indicate that Theia actually outperforms the proposed method on 9 out of 13 datasets. The average error metric, which favors the proposed method, appears to be disproportionately influenced by a few datasets where Theia performs poorly. Given the variability across datasets, the average may not be a reliable indicator of overall performance. In this context, the frequency of outperformance across datasets would be a more meaningful and fair comparison metric.\n5. The performance of the proposed Cycle-sync method appears to depend on the specific annealing schedule for $\\lambda_t$, which is currently fixed as $\\frac{t}{t+10}$. It would be insightful to evaluate how the method behaves under different trajectories of $\\lambda_t$. In particular, it would be useful to experiment with a schedule where $\\lambda_t \\rightarrow 0$ as $t$ increases—similar to MPLS—thereby placing more emphasis on cycle-consistency in early iterations and gradually shifting focus to the residuals. Additionally, analyzing the performance under a fixed $\\lambda_t$ could help isolate the benefit of annealing itself. Such experiments would clarify the impact of the annealing strategy on the overall robustness and accuracy of the method.", "was_revised": true, "final_justification": "The authors extend the message-passing framework to the non-trivial problem of location synchronization. They provide relevant theoretical guarantees and demonstrate that their method achieves the lowest known sample complexity for exact recovery. Empirically, the approach appears very promising and represents a significant advancement in global location estimation.", "rebuttal": "We are very grateful for your thoughtful and insightful comments. Below, we address each of your concerns in detail.\n\n**1. Intuitive interpretation of Theorem 2.1.**\n\nWe appreciate the suggestion to provide more intuition. In Theorem 2.1, the separation between clean and corrupted edges arises from the fact that the upper bound for clean edges $\\frac{1}{2\\beta_0 r^t}$ vanishes as iteration $t\\to \\infty$ (note that $r>1$), while the lower bound for bad edges remains strictly positive, proportional to their corruption levels. This guarantees exact separation in the asymptotic regime. Although constants such as  $\\mu$ and $C_\\alpha$ can be intricate, they do not affect the eventual separation property. Moreover, our sample complexity result (Table 1) emphasizes the scaling order, which is more meaningful for understanding theoretical guarantees than specific constants.\n\n**2. Sensitivity to initialization**\n\nOur method is quite robust to initialization. While the T-AAB initialization accelerates convergence by providing better starting weights, it does not significantly influence the final accuracy. Even trivial initialization using uniform weights performs similarly after sufficient iterations. Below we report the mean and median location error on ETH3D data (averaged over different scenes) after several iterations for both initialization schemes:\n\nTrivial Initialization:\n| Iteration | mean error | median error |\n|:---------:|:--------:|:----------:|\n|   5     |    0.110    |   0.020  |\n|   10   |    0.100    |   0.017  |\n|   15   |    0.099    |   0.016  |\n|   20   |    0.099    |   0.016  |\n\n\nOur Initialization by T-AAB:\n| Iteration | mean error | median error |\n|:---------:|:--------:|:----------:|\n|   5     |    0.102    |   0.018  |\n|   10   |    0.099    |   0.016  |\n|   15   |    0.099    |   0.015  |\n|   20   |    0.099    |   0.014  |\n\nTherefore, our method is robust even to trivial initialization, let alone variations in the T-AAB parameter.\n\n\n\n**3 (a). The ablation studies STE + LUD + MPLS-cycle**\n\nThis configuration is already included in the middle panel of Figure 4. As stated in lines 276–278, **all location solvers were preceded by the same STE** and MPLS-cycle preprocessing to ensure fairness. Without this common preprocessing, the performance gap between other baselines and Cycle-Sync would be even larger.\n\n**3 (b). Replacing LUD with BATA in our pipeline.**\n\nWe appreciate the suggestion to try BATA within our reweighting framework. Yes, this is another advantage of cycle-sync, that it can incorporate any location solver in its pipeline, which is quite a general framework. Our experiments on ETH3D data show that integrating LUD under Cycle-Sync's reweighting leads to better performance compared to using integrating BATA under Cycle-Sync. It is likely due to the stronger constraint of LUD for anti-collapsing (the constraint is on every edge). Moreover, our Welsh-type objective function has already taken care of the large variation of distances, so angle based method (such as BATA) is not advantageous in this case.\n\n| Method | mean error | median error |\n|:---------:|:--------:|:----------:|\n|   Ours-LUD   |    0.099    |   0.014  |\n|   Ours-BATA   |    0.202    |   0.079  |\n\n\n**4. Frequency of Outperformance vs. Mean Error**\n\nThanks a lot for your suggestion! We agree that frequency of outperforming baselines is a meaningful metric and will include it in our revision. While Theia performs better on 9 of 13 datasets, our method consistently avoids failure cases and achieves the lowest average error. We believe both metrics are informative: the former reflects robustness across diverse scenarios, while the latter captures stability and resilience against more adversarial datasets.\n\n**5. Annealing Schedule $\\lambda_t$.**\n\nThank you for this important observation. We have tested multiple schedules, including fixed, increasing and decreasing ones. Our proposed schedule strikes a good balance between residual-driven updates early on and cycle-consistency emphasis later. This schedule has consistently outperformed alternatives, particularly in settings with both additive noise and high corruption. \n\nPer the request of reviewer jMZZ, we have included experiments with both high corruption and noise. The following experiment uses uniform corruption with $q=0.7$ and higher noise level $\\sigma=0.2$.\n\n| $\\lambda_t$ | $\\frac{t}{10+t}$ (ours) |  $0$ |  $\\frac{10}{10+t}$|  $1$| $\\frac{t}{t+5}$|\n|:-------------------------:|:--------:|:----------:|:----------:|:----------:|:----------:|\n|   median err  |  0.24      |    0.36 |  0.27 | 0.37 | 0.29|\n\n\nThe following experiment test the adversarial corruption with $q=0.45$ (close to the theoretical limit $q=0.5$) and higher noise level $\\sigma=0.2$.\n\n\n| $\\lambda_t$ | $\\frac{t}{10+t}$ (ours) |  $0$ |  $\\frac{10}{10+t}$|  $1$| $\\frac{t}{t+5}$|\n|:-------------------------:|:--------:|:----------:|:----------:|:----------:|:----------:|\n|   median err  |  0.17      |    0.32 |  0.18 | 0.20 | 0.17|\n\nWe remark that using only the residual for reweighting (i.e., setting $\\lambda_t = 0$, which corresponds to IRLS) often leads to significantly higher errors compared to cycle-based reweighting methods. The underlying issue is that, in noisy settings, some bad edges may coincidentally exhibit low residuals. When this occurs, the aggressive reweighting imposed by the Welsch objective can assign them disproportionately large weights, thereby amplifying their adverse impact. In contrast, our cycle-based reweighting effectively overcomes this limitation: it is extremely unlikely for a bad edge to exhibit a low average cycle inconsistency, unless all cycles it participates in are consistent—an event that is highly improbable for corrupted measurements. Overall, we find that our annealing strategy consistently achieves the best performance across most scenarios.\n\nWe also observe this trend in the context of rotation synchronization. Supplementary Figure 10 illustrates that emphasizing cycle-consistency can significantly reduce orientation error. However, for rotation there is no need for annealing as it does not rely on distance estimation.\n\n\n**Q1: Prioritizing Residuals in Early Iterations**\n\nYou are correct that the residual and cycle-inconsistency can both be unreliable in early iterations. However, cycle-based inconsistencies aggregate noise across three edges and are thus more vulnerable to early-stage corruption. As the iterations proceed and the distances become more accurate, the weighted average of cycle inconsistencies provides a more robust estimate of edge corruption. By aggregating information from neighboring edges, it reduces the likelihood of mistakenly identifying a bad edge as good.\n\n\n**Q2: why Welsch loss function?**\n\n\nWhile Geman-McClure and Huber losses also reduce the influence of large residuals, the Welsch loss flattens much faster due to its exponential decay, making it more effective at suppressing large errors.\n\nWe further modify the Welsch loss by replacing $x^2$ with $|x|$ in the exponent, introducing a non-differentiable point at zero. This creates a singularity in the reweighting function, allowing good edges to receive infinite weight (capped in practice) and bad edges to be completely ignored—enabling exact recovery in corruption-only settings. In contrast, standard Geman-McClure, Huber, or Welsch losses always assign finite weights to good edges and nonzero weights to bad ones, preventing exact recovery.\n\nAlthough this weighting scheme appears aggressive, our cycle-based message passing effectively avoids misidentifying bad edges and mitigates the limitations of the objective function.\n\n**Q3: meaning of \"disabling reweighting\" in Lines 206-207?**\n\nThis refers to disabling residual-based IRLS reweighting in rotation averaging. The weighting relies solely on cycle-consistency in MPLS-cycle. We will clarify this in the revised version.\n\n**Q4: Typo in Algorithm 1**\n\nYes, thanks for pointing this out. We will correct it in the revised version.\n\n**Q5: Experiment on 1DSfM datasets?**\n\nIn our earlier submission, we evaluated our method (we did not use STE and MPLS-cycle at the time) on the 1DSfM (photo tourism) datasets and achieved at least a 20% reduction in median error compared to the best baseline. However, reviewers raised concerns about our reliance on SIFT features and recommended switching to deep learning-based features. Another critique was that the ground truth camera poses in 1DSfM are generated by Bundler, which is considered outdated compared to modern tools like COLMAP—and even COLMAP may lack the accuracy of laser scans. This is particularly problematic when benchmarking high-precision location solvers. These limitations raise valid concerns about the suitability of older datasets for evaluating modern methods. Nevertheless, we report the mean and median location errors below (averaged over different scenes), with all methods consistently preprocessed using STE and MPLS-cycle:\n\n| Method | mean error | median error |\n|:-------------------------:|:--------:|:----------:|\n|   Ours   |   0.292    |   0.115  |\n|   LUD   |    0.314    |   0.132  |\n|   BATA   |    0.362    |   0.130  |\n|   ShapeFit   |    0.670    |   0.410  |\n|   FusedTA   |    0.330    |   0.130  |\n\nFor the above reasons, we included an additional experiment in the supplementary material on a city-scale dataset (an extended version of Photo Tourism) using deep learning-based keypoint features (see Table 10 and Figure 11). However, the improvement of Cycle-Sync is much smaller than on ETH3D. We observed that LoFTR tends to overfit repetitive structures like windows and facades on this dataset. Furthermore, the ground truth camera poses are generated by COLMAP, which is likely less accurate than laser scans."}, {"review_id": "2IxzeGjFUy", "reviewer": "Reviewer_SY7H", "summary": "Cycle-Sync is a global camera pose estimation method based on message-passing least squares for cycle consistency and Welsch-like robust loss, instead of L1 or L2 loss, for camera location estimation. With unknown inter-camera distance, Cycle-Sync adapts MPLS on translation estimation. Such a method avoids bundle adjustment in the classical Structure from Motion pipelines while showing superior performance.", "questions": "1. How do you define well-shaped triangles? How do you choose the lower and upper bounds used in truncated AAB (T-AAB)?.\n2. Why does the proposed loss function use $\\alpha$=4? How sensitive is the optimization w.r.t. the choice of $\\alpha$? Or it can be any constant.\n3. It seems the performance decreases a lot when removing the STE outlier filter in some scenes. Have you tried using STE for the baselines?\n4. Others: see the weakness", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths: \n1. A novel global location estimation pipeline based on cycle consistency, showing decent improvement over the SOTA structure-from-motion methods with bundle adjustment. \n2. A new optimization formulation, solved by cycle consistency, leading to reduced corruption level.\n3. The paper is well-written and easy to read. \n\nWeakness: \n1. The contribution claimed to deal with the highly variant edges in distance. In principle, yes. However, there is only one real-world dataset,  ETH3D, used in the evaluation with its precise ground truth from the laser, which is not sufficient to conclude eliminating bundle adjustment for all.  \n2. Missing discussion. It is unclear how fast or how expensive the process of iteratively optimizing the cycle inconsistencies over the estimated locations is.", "was_revised": true, "final_justification": "Most of my concerns are addressed by the detailed rebuttal. I will keep my rating.", "rebuttal": "We are very grateful for your thoughtful and insightful comments. Below, we address each of your concerns in detail.\n\n**1. other real-world dataset**\n\nWe also answered this question in the reply to reviewer 37Zk which we copied here:\n\nIn our earlier submission, we evaluated our method (we did not use STE and MPLS-cycle at the time) on the 1DSfM (photo tourism) datasets and achieved at least a 20% reduction in median error compared to the best baseline. However, reviewers raised concerns about our reliance on SIFT features and recommended switching to deep learning-based features. Another critique was that the ground truth camera poses in 1DSfM are generated by Bundler, which is considered outdated compared to modern tools like COLMAP—and even COLMAP may lack the accuracy of laser scans. This is particularly problematic when benchmarking high-precision location solvers. These limitations raise valid concerns about the suitability of older datasets for evaluating modern methods. Nevertheless, we report the mean and median location errors below (averaged over different scenes), with all methods consistently preprocessed using STE and MPLS-cycle:\n\n| Method | mean error | median error |\n|:-------------------------:|:--------:|:----------:|\n|   Ours   |   0.292    |   0.115  |\n|   LUD   |    0.314    |   0.132  |\n|   BATA   |    0.362    |   0.130  |\n|   ShapeFit   |    0.670    |   0.410  |\n|   FusedTA   |    0.330    |   0.130  |\n\nFor the aforementioned reasons, we conducted an additional experiment in the supplementary material on a city-scale dataset (an extended version of the original Photo Tourism dataset) using deep learning-based keypoint features. See Table 10 and Figure 11 in the supplementary material. However, the performance gain of Cycle-Sync on this dataset is noticeably smaller than that observed on ETH3D. We found that LoFTR is not well-suited for this dataset, as it tends to overfit repetitive structures such as windows and facades. Moreover, the ground truth camera poses are derived from COLMAP reconstructions, which we believe are significantly less accurate than those obtained from laser scans.\n\n\n\n**2. Runtime Cost of Optimizing Cycle Inconsistencies**\n\nThe additional cost of optimizing cycle inconsistencies can be directly inferred from Table 8 in the supplementary material. Specifically, the time difference between the STE+LUD and Cycle-Sync pipelines reflects the added cost of our message-passing-based optimization. We will clarify this in the main text and refer the reader to supplementary material for details.\n\n**Q1. Definition well-shaped triangles? The choice of the lower and upper bounds used in T-AAB?**\n\nWe define well-shaped triangles as those that avoid extreme angle degeneracies. In particular, we require that all angles lie between $\\alpha$ and $\\pi-\\alpha$ for some $\\alpha \\in (0, \\pi/2)$. In practice we choose $\\alpha = arcsin(0.6)\\approx 37 degree$, based on empirical tuning on synthetic datasets. We observed similar performance for any $\\alpha$ between arcsin(0.4) and arcsin(0.7). We note that a larger $\\alpha$ often yields a better T-AAB estimate of corruption levels; however, $\\alpha$ cannot be too large, as this would result in an insufficient number of cycles. We finally remark that our method is robust even to trivial initialization, let alone variations in the T-AAB parameter. Please see our reply to reviewer 37Zk on the comment #2 about the sensitivity to initialization.\n\n**Q2: Sensitivity to Parameters**\n\nWe conducted an ablation study on the key parameters using ETH3D data: the constant $a$  in the robust loss function, the exponential decay constant $\\beta$ in the cycle-consistency updates, and various initialization schemes. The mean and median errors are averaged across different 3D scenes. Our findings show that Cycle-Sync is not sensitive to these choices within reasonable ranges. For the sensitivity to $\\lambda_t$ please see our reply to 37Zk on the comment #5.\n\n| Method | mean error | median error |\n|:-------------------------:|:--------:|:----------:|\n|   Ours   |   0.099    |   0.014  |\n|   $\\beta = 4$   |   0.098    |   0.016  |\n|   $\\beta = 8$   |   0.099    |   0.016  |\n|   $a = 2$   |   0.098    |   0.016  |\n|   $a= 8$   |   0.102    |   0.014  |\n\n**Q3: STE for the baselines?**\n\nThis configuration is already included in the middle panel of Figure 4. As stated in lines 276–278, **all location solvers were preceded by the same STE** and MPLS-cycle preprocessing to ensure fairness. Without this common preprocessing, the performance gap between other baselines and Cycle-Sync would be even larger."}, {"review_id": "fNtQMlEWKt", "reviewer": "Reviewer_jMZZ", "summary": "This paper proposes Cycle-Sync, a robust translation averaging method.\n\nThe key contributions are twofold:\n\n1. A Welsch-type loss function, which is less sensitive to large variations of distances and preserves non-smoothness at the origin.\n2. A message-passing least squares (MPLS) framework that integrates cycle-consistency information across 3-cycles.\n\nExperiments on both synthetic and the ETH3D real datasets show that Cycle-Sync outperforms other translation averaging and SfM baselines.", "questions": "1. Add geometric illustrations or figures to intuitively explain what cycle consistency captures, and how it improves robustness in practice.\n2. Clarify $d_{ij, k}$ in Algorithm 1. Are these precomputed from input directions or estimated during optimization?\n3. Justify hyperparameter choices, especially the value of $a=4$ in the loss, other parameters, and the weight initialization. Including a ablation study would be helpful.\n4. Compare with AAB [20] in the experiments. Since both methods rely on 3-cycle consistency, a direct comparison is necessary.\n5. Consider evaluating on the 1DSfM dataset [26] to improve comparability with prior work.\n6. Evaluate performance under higher noise (e.g., σ = 0.1 or higher) to further support the claim of robustness.\n7. Clarify what is novel in the \"plug-and-play outlier rejection module\" beyond simply using STE.\n\nConsidering the above concerns, I give a borderline reject at this stage. However, I would like to increase my rating if the concerns are addressed.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n1. Translation averaging is indeed an important component in SfM, and the paper tackles the challenging case of cycle-consistent corruption, where other methods underperform.\n2. The proposed translation averaging fully exploits cycle consistency using message-passing least squares.\n3. The Welsch-type loss provides both nonlinearity and robustness, offering advantages over L1/L2 losses.\n\n**Weaknesses:**\n1. The geometric intuition of cycle consistency is not clearly described. It would be helpful to Including intuitive geometric visualizations.\n2. The choice of hyperparameters, such as setting the Welsch loss parameter $a=4$ in line 118, some parameters in line 169, and weight initialization as $w_{ij,0} = exp(-20\\tilde{s}_{ij})$ in line 152, lacks sufficient justification or empirical support. Although the method is claimed to be insensitive to these choices, it would be beneficial to include an ablation study or comparative analysis to support this claim.\n3. Fig. 2 is not self-explanatory and the propagation mechanism in the bottom-left part is unclear. It would be helpful to provide a more detailed textual description.\n4. In Algorithm 1, it is not explained how the 3-cycle inconsistencies $d_{ij, k}_{k\\inC_{ij}}$ are obtained or whether they are known a priori or estimated.\n5. The plug-and-play outlier rejection module is claimed as a contribution, but it simply uses STE in line 215. While effective, the novelty may be overstated unless further integration or modification is described.\n6. The paper only verifies the performance of real data on ETH3D dataset, while ShapeFit, BATA, Fused-TA, LUD and other methods include results on the widely used 1DSfM dataset [26]. It would be beneficial to provide the comparisons on that dataset.\n7. In Fig. 3, the robustness is only demonstrated under mild noise $\\sigma = 0.05$. It would be valuable to explore performance under higher noise levels to validate the claim of robustness.\n8. The All-About-that-Base (AAB) [20] is discussed as a related work that also leverages 3-cycle consistency, but not included in the experiments. Since AAB is most similar in spirit to Cycle-Sync, a direct experimental comparison would be highly relevant.", "was_revised": true, "final_justification": "The authors have addressed my most of the concerns during the rebuttal and follow-up discussion. However, I encourage more precise wording in the final version (e.g., since STE is applied rather than introduced, it should be described accordingly). I also recommend including the additional experiments and clarifications from the rebuttal to further strengthen the paper.", "rebuttal": "We are very grateful for your thoughtful and insightful comments. Below, we address each of your concerns in detail.\n\n**1. The geometric intuition of cycle consistency**\n\nThank you for highlighting this. We will include a geometric visualization in the revision. Intuitively, cycle consistency implies that traversing the triangle formed by three cameras via their estimated displacements (distance $\\times$ direction) should return to the starting point. Deviations from this indicate local inconsistency and are strong indicators of corruption.\n\n\n**2. The choice of parameters.**\n\nWe observed that the method performs stably across a range of settings for $a$, $\\beta$ and initialization, validating its robustness.\n\nPlease refer to our response to Reviewer SY7H on Q2 about the sensitivity to $a$ and $\\beta$. \n\nPlease also see our response to Reviewer 37Zk on the comment #2 (sensitivity to initialization) and comment #5 (sensitivity to $\\lambda_t$). \n\n**3. Fig. 2 is not self-explanatory.**\n\nWe agree that Figure 2 can be made clearer. In the revised version, we will expand the caption to explain that the lower-left schematic illustrates how corruption scores are computed via a weighted average of cycle inconsistencies, with weights derived from residuals via an exponential decay function. This mechanism is central to our cycle-based reweighting strategy.\n\n**4. Explanation of $d_{ij, k}$ in Algorithm 1**\nThis is a typo—it should be $\\tilde d_{ij,k}$. These are not learned online but are precomputed using the AAB-inconsistency measure described just after line 156. They are then used to run the T-AAB procedure and initialize the edge weights. We will clarify this in the revision.\n\n**5. Novelty of STE-Based Filtering Module**\n\nYes, we directly use STE to estimate direction vectors via the equation in line 210. However, to our knowledge, no prior work has used the number of inlier keypoint matches identified by STE—or any other robust subspace recovery (RSR) method—as a statistic for filtering out bad edges. We emphasize that this step (line 210) differs from traditional geometric verification using the epipolar constraint: our proposed statistic is directly tied to the estimated directions and is well-suited for identifying and rejecting bad ones.\n\nThus, while STE itself is not novel, our use of it to construct a new filtering statistic is. In particular, although the LUD paper uses REAPER to estimate directions, they do not leverage inlier counts from REAPER—or any RSR method—as a basis for rejecting unreliable directions.\n\n**6. Experiments on the 1DSfM dataset.**\n\nWe also answered this question in the reply to reviewer 37Zk which we copied here:\n\nIn our earlier submission, we evaluated our method (we did not use STE and MPLS-cycle at the time) on the 1DSfM (photo tourism) datasets and achieved at least a 20% reduction in median error compared to the best baseline. However, reviewers raised concerns about our reliance on SIFT features and recommended switching to deep learning-based features. Another critique was that the ground truth camera poses in 1DSfM are generated by Bundler, which is considered outdated compared to modern tools like COLMAP—and even COLMAP may lack the accuracy of laser scans. This is particularly problematic when benchmarking high-precision location solvers. These limitations raise valid concerns about the suitability of older datasets for evaluating modern methods. Nevertheless, we report the mean and median location errors below (averaged over different 3D scenes), with all methods consistently preprocessed using STE and MPLS-cycle:\n\n| Method | mean error | median error |\n|:-------------------------:|:--------:|:----------:|\n|   Ours   |   0.292    |   0.115  |\n|   LUD   |    0.314    |   0.132  |\n|   BATA   |    0.362    |   0.130  |\n|   ShapeFit   |    0.670    |   0.410  |\n|   FusedTA   |    0.330    |   0.130  |\n\nFor the aforementioned reasons, we conducted an additional experiment in the supplementary material on a city-scale dataset (an extended version of the original Photo Tourism dataset) using deep learning-based keypoint features. See Table 10 and Figure 11 in the supplementary material. However, the performance gain of Cycle-Sync on this dataset is noticeably smaller than that observed on ETH3D. We found that LoFTR is not well-suited for this dataset, as it tends to overfit repetitive structures such as windows and facades. Moreover, the ground truth camera poses are derived from COLMAP reconstructions, which we believe are significantly less accurate than those obtained from laser scans.\n\n\n**7.  performance under higher noise levels**\n\nWe include additional experimental results evaluating Cycle-Sync and baselines under high additive noise $\\sigma=0.2$ and high corruption. Our method continues to remain competitive.\n\nFor example, for adversarial corruption when $q=0.45$ (close to the theoretical limit $q=0.5$) and  $\\sigma=0.2$\n\n| Method | Ours | LUD | BATA| SF| FusedTA\n|:-------------------------:|:--------:|:----------:|:----------:|:----------:|:----------:|\n|   median error  |  0.17      |    0.91 |  0.31 | 1.47 | 0.23|\n\n\nFor example, for uniform corruption when $q=0.7$ and  $\\sigma=0.2$\n\n\n| Method | Ours | LUD | BATA| SF| FusedTA\n|:-------------------------:|:--------:|:----------:|:----------:|:----------:|:----------:|\n|   median error  |  0.24      |    0.63 |  0.27 | 1.44 | 0.52|\n\n\nWe conclude by noting that in real datasets—assuming reasonably accurate keypoint features and ground truth camera poses—the error model is better characterized as corruption plus small noise. In practice, the relative measurements on edges are obtained via fundamental matrix estimation followed by RANSAC, which typically either succeed (correctly identify inlier keypoint match) or fail outright, with much less cases falling in between. \n\n\n**8. Comparison with AAB**\n\nWe conducted an experiment on ETH3D where we removed edges flagged by AAB (based on its statistics), extracted the parallel rigid component, and ran LUD or other solvers on this cleaned graph. Before running AAB, the graph is already preprocessed by STE and MPLS-cycle for fair comparison. We report the mean and median location error averaged over different 3D scenes. While AAB-based preprocessing improves robustness to some extent, Cycle-Sync achieves much better results. This further demonstrates the advantage of our message-passing reweighting over standalone edge pruning.\n\n| Method | mean error | median error |\n|:-------------------------:|:--------:|:----------:|\n|   Ours   |   0.099    |   0.014  |\n|   AAB, remove 30% edges   |   0.153    |   0.092  |\n\n\nRemoving 30% edges keeps 95% of the nodes. We note that our method is evaluated on the full graph, and even though our error is significantly smaller than AAB evaluated on a cleaner graph. Interestingly, AAB+LUD performs slightly worse than LUD alone. A possible reason is that STE has already performed the role of filtering bad edges—arguably more accurately than AAB. Applying AAB afterward may discard additional good edges or nodes, creating sparse regions in the graph and reducing the overall stability of the algorithm."}, {"review_id": "TcKGRrddFg", "reviewer": "Reviewer_e5bW", "summary": "The paper proposes a method for camera location synchronization which is resistant to corrupted data which is present in real-world Structure-from-Motion (SfM) applications. By using redesigned Message Passing Least Squares (MLPS) to minimize a Welsch function the paper shows high robustness, great performance and solves the limitations of Least Unsquared Deviations (LUD) and BATA-type objectives. The paper also establishes a deterministic exact-recovery guarantee under adversarial corruption which outperforms all prior work under standard probabilistic model constraints. The framework is then extended to the rotation synchronization problem to reduce the orientation error. Lastly the proposed model in the paper removes any need for bundle adjustment which are computationally expensive but present in all referenced SfM pipelines.", "questions": "None.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "1. Quality - The paper is well written with all claims being well supported and methods used being increments of previous works. The paper provides a clear overview of the method and how it is used and what its drawbacks are. The two main downsides to the quality is that in some places there are missing references for statements that are clear to the authors but might not be clear to a third party. The second downside is that some graphs (i.e. Fig 3, Fig 4) are a bit illegible due to the absolute differences of values being shown so a better representation of those would make a difference.\n2. Clarity - The paper is clearly written with all theoretical setups providing assumptions and proofs. The introduction, related work and methodology provide clear overviews of what the paper proposes. The experimental setup is properly noted and reproducible.\n3. Significance - The paper provides a method that consistently outperforms referenced methods in the task it was tested for. It advances the topic and provides insights into sensor synchronization, not only for location but for rotation and direction.\n4. Originality - The paper builds upon previous methods' limitations and solves them in a novel way, providing valuable insights into the topic.", "was_revised": true, "final_justification": "The authors have addressed all issues from my review, why I keep the rating of 5 and propose to accept this contibution.", "rebuttal": "We are very grateful for your thoughtful and insightful comments. Below, we address each of your concerns in detail.\n\n**1. missing references for statements**\n\nThank you for pointing this out. We will go through the manuscript carefully and ensure that all statements—especially those not common knowledge—are properly cited with relevant references. We are committed to improving the clarity and completeness of our exposition.\n\n\n\n\n**2. Legibility of Figures 3 and 4**\n\n\nWe agree that the current figures can be made more informative. In the revision, we will normalize axes or apply log/relative scaling where appropriate to improve the legibility and ensure that important trends and differences are clearly visible."}]

Author rebuttal:

Meta-review: The paper received overwhelming positive reviews, three accepts and one borderline accept. AC sees now reason to diverge from the reviewers' recommendation.
Decision: Accept (spotlight)