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![An illustration of the cycle-edge graph. Cycle node L and edge node i j are connected if i j ∈ L in the original graph G([n], E)](publication/354517541/figure/fig1/AS:1066666179973120@1631324443026/An-illustration-of-the-cycle-edge-graph-Cycle-node-L-and-edge-node-i-j-are-connected-if.png)
An illustration of the cycle-edge graph. Cycle node L and edge node i j are connected if i j ∈ L in the original graph G([n], E)
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We propose a general framework for solving the group synchronization problem, where we focus on the setting of adversarial or uniform corruption and sufficiently small noise. Specifically, we apply a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios and consequently so...
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... define the notion of a cycle-edge graph (CEG), which is analogous to the factor graph in belief propagation. We also demonstrate it in Fig. 1. Given the graph G( [n], E) and a set of cycles C, the corresponding cycle-edge graph G C E (V C E , E C E ) is formed in the following ...
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... G is a Lie group, methods that relax (5) with ν = 2, such as Spectral and SDP, cannot exactly recover the group elements under UCM. Wang and Singer [45] showed that the global minimizer of the SDP relaxation of (5) with ν = 1 and G = SO(d) achieves asymptotic exact recovery under UCM when q ≡ Pr(i j ∈ E b |i j ∈ E) < p c , where p c depends on d (e.g., p c ≤ 0.54 and p c = O (d −1 )). Due to their limited range of q, they cannot estimate the sample complexity when q → 1. ...
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... Figures 10 and 11 show the phase transition plots for Z 2 and SO(2) synchronization, respectively. They include plots for the averaged error S (for CEMP) and averaged error G (for the other algorithms) over ten different random runs for various values of p, q and n ( p appears on the y-axis, q on the x-axis and n varies with subfigures). ...
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... is evident from Fig. 10 that the phase transition plots of Spectral and SDP align well with the red curve. For CEMP and CEMP+GCW, the exact recovery region (dark area) seems to approximately lie in the area enclosed by the red and blue curves. The blue curve of CEMP+GCW is slightly closer to the x-axis than that of CEMP. This suggests that combining CEMP with ...
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... Fig. 11, Spectral and SDP do not seem to exactly recover group elements in the presence of any corruption, and thus a phase transition region is not noticed for them. The phase transition plots of IRLS align well with the red curve. The exact recovery regions of both CEMP and CEMP+GCW seem to approximately lie in the area enclosed by the red ...
Similar publications
Previous partial permutation synchronization (PPS) algorithms, which are commonly used for multi-object matching, often involve computation-intensive and memory-demanding matrix operations. These operations become intractable for large scale structure-from-motion datasets. For pure permutation synchronization, the recent Cycle-Edge Message Passing...
Citations
... The goal is to de-noise local information by exploiting redundancies and global consistency constraints in order to obtain reliable global information. For example, [28] proposes a general group synchronization framework on graphs that exploits a cycle consistency condition. ...
... Further, group synchronization over the permutation group S n aids in establishing globally consistent feature point labels from relative feature point matches [29,34]. See [28] for a more detailed overview of the many applications of group synchronization. ...
... Another global approach is to denoise the data by estimating the edge corruption levels so that severely corrupted measurements can be discarded or (3) can be refined by appropriate edge weights. This can be done, for example, by using a message passing framework [28,35]. ...
Group synchronization is the problem of determining reliable global estimates from noisy local measurements on networks. The typical task for group synchronization is to assign elements of a group to the nodes of a graph in a way that respects group elements given on the edges which encode information about local pairwise relationships between the nodes. In this paper, we introduce a novel higher-order group synchronization problem which operates on a hypergraph and seeks to synchronize higher-order local measurements on the hyperedges to obtain global estimates on the nodes. Higher-order group synchronization is motivated by applications to computer vision and image processing, among other computational problems. First, we define the problem of higher-order group synchronization and discuss its mathematical foundations. Specifically, we give necessary and sufficient synchronizability conditions which establish the importance of cycle consistency in higher-order group synchronization. Then, we propose the first computational framework for general higher-order group synchronization; it acts globally and directly on higher-order measurements using a message passing algorithm. We discuss theoretical guarantees for our framework, including convergence analyses under outliers and noise. Finally, we show potential advantages of our method through numerical experiments. In particular, we show that in certain cases our higher-order method applied to rotational and angular synchronization outperforms standard pairwise synchronization methods and is more robust to outliers. We also show that our method has comparable performance on simulated cryo-electron microscopy (cryo-EM) data compared to a standard cryo-EM reconstruction package.
... The prior examples should hopefully be sufficient to see the breadth of problems modeled and extended by (1). Further examples include: low-rank matrix optimization [53,37], tensor completion [21], and synchronization problems [31,41], among others; however, we do not have the space here to elaborate on these important applications. ...
This paper presents and analyzes the first matrix optimization model which allows general coordinate and spectral constraints. The breadth of problems our model covers is exemplified by a lengthy list of examples from the literature, including semidefinite programming, matrix completion, and quadratically constrained quadratic programs (QCQPs), and we demonstrate our model enables completely novel formulations of numerous problems. Our solution methodology leverages matrix factorization and constrained manifold optimization to develop an equivalent reformulation of our general matrix optimization model for which we design a feasible, first-order algorithm. We prove our algorithm converges to -approximate first-order KKT points of our reformulation in iterations. The method we developed applies to a special class of constrained manifold optimization problems and is one of the first which generates a sequence of feasible points which converges to a KKT point. We validate our model and method through numerical experimentation. Our first experiment presents a generalized version of semidefinite programming which allows novel eigenvalue constraints, and our second numerical experiment compares our method to the classical semidefinite relaxation approach for solving QCQPs. For the QCQP numerical experiments, we demonstrate our method is able to dominate the classical state-of-the-art approach, solving more than ten times as many problems compared to the standard solution procedure.
... We compare our method with the following methods: IRLS-ℓ 1 2 [10], EIG-IRLS [36], CEMP [37], MPLS [11], HARA [23], DMF-SYNCH [15]. The code of these algorithms are released by the authors. ...
Multiple rotation averaging plays a crucial role in computer vision and robotics domains. The conventional optimization-based methods optimize a nonlinear cost function based on certain noise assumptions, while most previous learning-based methods require ground truth labels in the supervised training process. Recognizing the handcrafted noise assumption may not be reasonable in all real-world scenarios, this paper proposes an effective rotation averaging method for mining data patterns in a learning manner while avoiding the requirement of labels. Specifically, we apply deep matrix factorization to directly solve the multiple rotation averaging problem in unconstrained linear space. For deep matrix factorization, we design a neural network model, which is explicitly low-rank and symmetric to better suit the background of multiple rotation averaging. Meanwhile, we utilize a spanning tree-based edge filtering to suppress the influence of rotation outliers. What's more, we also adopt a reweighting scheme and dynamic depth selection strategy to further improve the robustness. Our method synthesizes the merit of both optimization-based and learning-based methods. Experimental results on various datasets validate the effectiveness of our proposed method.
... Given these challenges, theoretical developments have demonstrated the critical role of cycle-consistency information in inferring corrupted measurements [25]. In practice, the consistency constraint on 3-cycles was utilized to estimate the error of each measured relative orientation. ...
... In this work, we propose the first practical method, LongSync, for inferring edge corruption levels from long cycle consistency information. For this purpose, we carefully modify and vectorize the Cycle Edge Message Passing (CEMP) method [25]. This nontrivial modification drastically reduces its computational complexity when using longer cycles. ...
... Instead of employing a robust objective function, Shen et al. [36] and Zach et al. [51] uses the 3-cycle consistency constraint to detect and remove corrupted relative orientations. Lerman and Shi [25] take one step further to estimate the corruption level of each relative measurement by a novel cycle-edge message passing (CEMP) algorithm. They then use the estimated corruption levels to reweigh the graph and solve rotation synchronization using a weighted least squares method. ...
Group synchronization plays a crucial role in global pipelines for Structure from Motion (SfM). Its formulation is nonconvex and it is faced with highly corrupted measurements. Cycle consistency has been effective in addressing these challenges. However, computationally efficient solutions are needed for cycles longer than three, especially in practical scenarios where 3-cycles are unavailable. To overcome this computational bottleneck, we propose an algorithm for group synchronization that leverages information from cycles of lengths ranging from three to six with a time complexity of order (or when using a faster matrix multiplication algorithm). We establish non-trivial theory for this and related methods that achieves competitive sample complexity, assuming the uniform corruption model. To advocate the practical need for our method, we consider distributed group synchronization, which requires at least 4-cycles, and we illustrate state-of-the-art performance by our method in this context.
... All combinations of the three kinds of noise were considered in the literature, both in numerical investigations of estimation methods and in their theoretical analysis [2,5,6,12,13,19,22,26]. The literature cited provides a mere illustrative selection. ...
... exactly as in (13). That it is a minimum follows from the fact F is convex and defined on R 3 , and therefore can have no local maxima. ...
In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or "rounded", onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group , the group of rigid motions of . We based our method on embedding into the algebra of dual quaternions, which has deep algebraic connections with the group . These connections suggest a natural rounding procedure considerably more straightforward than the current state-of-the-art for spectral synchronization, which uses a matrix embedding of . We show by numerical experiments that our approach yields comparable results to the current state-of-the-art in synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of , while yielding estimators of similar quality.
... Later on, similar results are extended to orthogonal group O(d) synchronization [ZWS21,Lin22a]. Recently, there are also alternative models with algorithms utilizing the message-passing procedure [PWBM18,LS22] to solve group synchronization. The summary and comprehensive comparisons of above-mentioned works have been discussed in the literature, and we refer the reader to [LYS20,Lin22a,ZWS21] for details. ...
Rotation group synchronization is an important inverse problem and has attracted intense attention from numerous application fields such as graph realization, computer vision, and robotics. In this paper, we focus on the least-squares estimator of rotation group synchronization with general additive noise models, which is a nonconvex optimization problem with manifold constraints. Unlike the phase/orthogonal group synchronization, there are limited provable approaches for solving rotation group synchronization. First, we derive improved estimation results of the least-squares/spectral estimator, illustrating the tightness and validating the existing relaxation methods of solving rotation group synchronization through the optimum of relaxed orthogonal group version under near-optimal noise level for exact recovery. Moreover, departing from the standard approach of utilizing the geometry of the ambient Euclidean space, we adopt an intrinsic Riemannian approach to study orthogonal/rotation group synchronization. Benefiting from a quotient geometric view, we prove the positive definite condition of quotient Riemannian Hessian around the optimum of orthogonal group synchronization problem, and consequently the Riemannian local error bound property is established to analyze the convergence rate properties of various Riemannian algorithms. As a simple and feasible method, the sequential convergence guarantee of the (quotient) Riemannian gradient method for solving orthogonal/rotation group synchronization problem is studied, and we derive its global linear convergence rate to the optimum with the spectral initialization. All results are deterministic without any probabilistic model.
... Under the RCM and in the full observation case where q = 1, it is shown that the minimizer of the SDR reformulation exactly recovers the underlying Gram matrix (hence the ground-truth rotations) under the conditions that the true observation ratio p ≥ 0.46 for SO(2) (and p ≥ 0.49 for SO(3)) and n → ∞. In [24], the authors established the relationship between cycle-consistency and exact recovery and introduced a message-passing algorithm. Their method is tailored to find the corruption level in the graph, rather than recovering the ground-truth rotations directly. ...
... Let us mention that they also provided guarantees for other compact groups and corruption settings. Following partly the framework established in [24], the work [34] presents an interesting nonconvex quadratic programming formulation of RRS. It is shown that the global minimizer of the nonconvex formulation recovers the true corruption level (still not the ground-true rotations directly) when p 2 q 2 = Ω(log n/n) under the RCM. ...
... We develop our theoretical analysis of ReSync by adopting the random corruption model (RCM). The RCM was previously used in many works to analyze the performance of various synchronization algorithms; see, e.g., [41,20,24,34]. Specifically, we can represent our measurement model (1) on a graph G(V, E), where V is a set of n nodes representing {X ⋆ 1 , · · · , X ⋆ n } and E is a set of edges containing all the available measurements {Y i,j , (i, j) ∈ E}. ...
This work presents ReSync, a Riemannian subgradient-based algorithm for solving the robust rotation synchronization problem, which arises in various engineering applications. ReSync solves a least-unsquared minimization formulation over the rotation group, which is nonsmooth and nonconvex, and aims at recovering the underlying rotations directly. We provide strong theoretical guarantees for ReSync under the random corruption setting. Specifically, we first show that the initialization procedure of ReSync yields a proper initial point that lies in a local region around the ground-truth rotations. We next establish the weak sharpness property of the aforementioned formulation and then utilize this property to derive the local linear convergence of ReSync to the ground-truth rotations. By combining these guarantees, we conclude that ReSync converges linearly to the ground-truth rotations under appropriate conditions. Experiment results demonstrate the effectiveness of ReSync.
... Some important topics in algebraic vision that are omitted include group synchronization (e.g., [RCBL19,LS21]) and uses of polynomial optimization (e.g., [KH07,CAPT22,AAT12]). Readers may consult [ÖVBS17] for a survey that covers numerical and large-scale optimization aspects in 3D reconstruction. ...
In this survey article, we present interactions between algebraic geometry and computer vision, which have recently come under the header of Algebraic Vision. The subject has given new insights in multiple view geometry and its application to 3D scene reconstruction, and carried a host of novel problems and ideas back into algebraic geometry.
... The recent cycle-edge message passing (CEMP) (Lerman & Shi, 2021) overcomes the aforementioned drawbacks of both IRLS and outlier detection methods. It estimates the corruption levels without requiring a good initialization or solving weighted least squares, even when the corruption is high. ...
... However, for Lie group synchronization, it is a challenging open problem to prove that an algorithm can match the information theoretic sample complexity. The best sample complexity bound for SO(2) and SO(3) was established for CEMP: n/ log n = O(p −2 (1 − q) −8 ) (Lerman & Shi, 2021), but there is a clear gap with the desired bound. ...
... When addressing the adversarial corruption model we further assume that for each ij ∈ E, G ij is nonempty. Recall that d is a bi-invariant metric on G and assume WLOG that it is scaled so that d(g 1 , g 2 ) ∈ [0, 1] for any g 1 , g 2 ∈ G. Therefore, s * ij ∈ [0, 1] for all ij ∈ E. We take advantage of the cycle inconsistency d ij,k = d(g ij g jk g ji , e) and its following property (Lerman & Shi, 2021): ...
We propose a novel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency of the group and we thus refer to our method as detection and estimation of structural consistency (DESC). This general framework can be extended to other algebraic and geometric structures. Our formulation has the following advantages: it can tolerate corruption as high as the information-theoretic bound, it does not require a good initialization for the estimates of group elements, it has a simple interpretation, and under some mild conditions the global minimum of our objective function exactly recovers the corruption levels. We demonstrate the competitive accuracy of our approach on both synthetic and real data experiments of rotation averaging.
... The recent theoretically-guaranteed cycle-edge message passing (CEMP) algorithm [11] opens the door for fast, memory efficient, and outlier-robust implementation for compact group synchronization without spectral initialization. Different from the previous cycle-consistency-based methods [1,7,18,27], it uses a fast iterative message passing scheme to globally estimate the corruption levels of the given pairwise measurements. ...
... Different from the previous cycle-consistency-based methods [1,7,18,27], it uses a fast iterative message passing scheme to globally estimate the corruption levels of the given pairwise measurements. It is numerically demonstrated in [11] that CEMP is memory-efficient and fast for SO(d)-synchronization, especially for large d. For permutation synchronization, [20] proposed an efficient implementation of CEMP. ...
... where the weights are updated at each iteration using improved estimates of the corruption levels (we omit their formulas). The bi-invariance property of the distance in (1) implies [11] d CEMP ijk =s * ij whenever ik, jk ∈E g . ...
Previous partial permutation synchronization (PPS) algorithms, which are commonly used for multi-object matching, often involve computation-intensive and memory-demanding matrix operations. These operations become intractable for large scale structure-from-motion datasets. For pure permutation synchronization, the recent Cycle-Edge Message Passing (CEMP) framework suggests a memory-efficient and fast solution. Here we overcome the restriction of CEMP to compact groups and propose an improved algorithm, CEMP-Partial, for estimating the corruption levels of the observed partial permutations. It allows us to subsequently implement a nonconvex weighted projected power method without the need of spectral initialization. The resulting new PPS algorithm, MatchFAME (Fast, Accurate and Memory-Efficient Matching), only involves sparse matrix operations, and thus enjoys lower time and space complexities in comparison to previous PPS algorithms. We prove that under adversarial corruption, though without additive noise and with certain assumptions, CEMP-Partial is able to exactly classify corrupted and clean partial permutations. We demonstrate the state-of-the-art accuracy, speed and memory efficiency of our method on both synthetic and real datasets.
