FAST-Sync: Fast Group Synchronization for any Matrix Lie Group

Shane Holmes; Yiran Luo; Firat Taxpulat; David M. Rosen; Frank Dellaert

cs.ROSep 29, 2026

FAST-Sync: Fast Group Synchronization for any Matrix Lie Group

Authors: Shane Holmes, Yiran Luo, Firat Taxpulat, David M. Rosen, Frank Dellaert

Organizations: Northeastern University, Boston, USA · School of Interactive Computing, Georgia Institute of Technology, Atlanta, GA, USA

Abstract

Group synchronization (GS) is the problem of estimating a set of $N$ unknown elements $g_1,\ldots, g_N \in \mathcal{G}$ in a group $\mathcal{G}$ , given noisy measurements of a subset of their pairwise ratios $g_i^{-1} g_j$ . GS problems lie at the core of many state estimation tasks in robotics and computer vision, including 3D vision, robotic mapping, inertial navigation, and molecular reconstruction. Unfortunately, GS problems are typically both high-dimensional and non-convex, and therefore hard to solve in general. In this paper, we present Fast-Sync, a fast linear approximation method for GS that is suitable for initializing local manifold-based optimizers or certifiable global methods. Our approach generalizes chordal initialization to arbitrary matrix Lie groups, and additionally proposes two new key algorithmic enhancements: we show how to exploit both the Kronecker-product structure in the problem data matrix and the topology of the synchronization graph to improve speed, scalability, and accuracy. Experimental evaluation across several GS tasks demonstrates that Fast-Sync provides high-quality initializations that enable local optimizers to efficiently recover globally optimal GS solutions, achieving high success rates even with considerable measurement noise.

Figures & tables

Fig. 1: MIT dataset. Top panel shows a spanning tree initialization of an

\mathrm{SE}\!\left(2\right)

PGO problem, which is not in the basin of attraction of the global optimizer; the middle panel shows our linear Fast-Sync initialization; bottom panel shows the subsequent local optimization starting from Fast-Sync , converging to the global optimum.

Fig. 2: Measurement graph for a SfM dataset (Statue of Liberty,

N=17

Panel of Figure 3 — Fig. 3: Statue of Liberty dataset: the $3N\times 3N$ Cholesky factor $R$ (left) corresponding to the back-substitution DAG (right) obtained using a ND ( ND ) ordering heuristic. Compare with Figure 2 . There and in the DAG above we numbered the vertices and the block columns of $R$ according to the ND ordering. The root (16) will be assigned the group identity. The vertices directly connected to the root will lie exactly on the group $\mathcal{G}$ , whereas deeper nodes may move off the manifold in the ambient-space back-substitution process of equation ( 17 ).

Panel of Figure 4 — Fig. 4: Heatmap grid analysis of $\mathrm{SO}\!\left(3\right)$ .

Panel of Figure 5 — Fig. 5: Heatmap grid analysis of $\mathrm{SE}\!\left(3\right)$ .

Panel of Figure 6 — Fig. 6: Heatmap grid analysis of $\mathrm{SU}\!\left(2\right)$ .

Panel of Figure 7 — Fig. 7: Heatmap grid analysis of $\mathrm{SL}\!\left(4\right)$ .

Panel of Figure 8 — Fig. 8: Heatmap grid analysis of $\mathrm{Gal}\!\left(3\right)$ .

Panel of Figure 9 — Fig. 9: Heatmap grid analysis of $\mathrm{Sim}\!\left(3\right)$ .

Panel of Figure 10 — Fig. 10: Cross-group benchmark: speed vs. Frobenius accuracy for representative datasets across multiple matrix Lie groups.

Statistic	Method	SO3	SE3	SIM3	GAL3	SL4
Optimality	Chordal	1.0	1.0	—	—	—
	Fast w/o ND	1.0	1.0	0.8	1.0	1.0
	Fast	1.0	1.0	0.9	1.0	1.0
Speed (ms)	Chordal	46.7	79.4	—	—	—
	Fast w/o ND	33.9	67.1	143	187	534
	Fast	28.6	50.0	72.2	113	333

TABLE I: Comparison of Fast-Sync with and without ND ( ND ), and against off-the-shelf Chordal initialization

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Learning Adaptive Solvers for Distributed Factor Graph Optimization on Matrix Lie Groups

Jaeho Shin, Maani Ghaffari, Yulun Tian

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Riemannian Optimization Decentralized Optimization Lie Group Robust Optimization Robust Trajectory Robotic Perception

Open paper AskPDF arXiv

Modern robotic perception increasingly involves large-scale geometric optimization problems distributed across multiple robots or sessions. However, existing distributed solvers often depend on brittle hand tuning and primarily target rigid body pose graphs. To address this, we present DeepCORD, a learning-augmented framework for distributed factor graph optimization on general matrix Lie groups. By unfolding a parallel and accelerated Riemannian optimizer into differentiable iterations, DeepCORD learns a self-supervised feedback policy that dynamically adapts solver parameters according to the optimization phase and communication status. The resulting method enables adaptive distributed optimization over matrix Lie groups under both synchronous and asynchronous communication regimes. Extensive experiments on real-world

\mathrm{SE}

(3) pose graph optimization and

\mathrm{SL}

(4) projective submap alignment show that our method achieves lower objective values than existing distributed baselines on most benchmarks across realistic operating scenarios.

Abstract

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