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FAST-Sync: Fast Group Synchronization for any Matrix Lie Group
Shane Holmes, Yiran Luo, Firat Taxpulat, David M. Rosen, Frank Dellaert
TL;DR
Group synchronization is computationally difficult because its maximum-likelihood formulations are often high-dimensional and non-convex, while local optimization depends on initialization. Fast-Sync generalizes linear initialization to arbitrary matrix Lie groups and exploits problem structure and graph topology. Across synthetic and real datasets, its initializations generally compare favorably with spanning-tree initialization and help local optimization reach high-quality solutions.
Problem
Maximum-likelihood group synchronization is often high-dimensional and non-convex, making important instances computationally hard, while local-search outcomes depend on initialization.
Source: §II · p. 2Method
Fast-Sync extends linear initialization to arbitrary matrix Lie groups, exploiting Kronecker structure and graph-based gauge selection.
Source: §I · p. 1, §II · p. 2, B. Fast Approximate Synchronization · p. 3Results
Across real datasets and synthetic group tests, Fast-Sync generally provides better or comparable refinement than spanning-tree initialization and reaches optimal solutions faster in most real-data cases.
Source: §V · p. 7, §VI · p. 8Takeaways & Limitations
Fast-Sync provides high-quality initializations across matrix Lie groups, with a simple Frobenius surrogate and structured linear-algebra design.
Source: §VII · p. 8The theoretical basis for when and why Nested-Dissection-based gauge selection succeeds remains an open challenge.
Source: §VII · p. 8
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Group synchronization recovers absolute group elements from noisy relative measurements and underlies tasks in robotics, computer vision, and cryo-electron microscopy. Fast-Sync extends linear initialization to arbitrary matrix Lie groups, using structural and graph-based enhancements to improve scalability and initialization quality.
I. INTRODUCTION
- GS recovers absolute group elements such as SE(d) poses or SO(d) rotations from noisy relative measurements, across robotics, vision, and cryo-electron microscopy.Applications include pose graph optimization, rotation averaging, and structure from motion.Source: §I · p. 1
- On the MIT SE(2) PGO example, the spanning-tree start lies outside the global optimizer’s basin, while local optimization from Fast-Sync reaches the global optimum.Source: Figure 1 · p. 1
- Fast-Sync generalizes linear initialization to arbitrary matrix Lie groups and targets high-quality initial estimates for downstream optimization.The method adds Kronecker-structure reduction and Nested-Dissection-based gauge fixing.Source: §I · p. 1
- Kronecker-product block reduction improves scalability, while Nested-Dissection-based gauge fixing improves initialization quality.Source: §I · p. 1
II. RELATED WORK
Group synchronization is commonly formulated as maximum-likelihood estimation, but its high-dimensional, non-convex instances can be computationally hard. Existing approximation and certification strategies trade off initialization quality, scalability, and computational cost; Fast-Sync extends linear initialization to arbitrary matrix Lie groups.
II. RELATED WORK
- Maximum-likelihood group synchronization is often high-dimensional and non-convex, and important instances are NP-hard in general.Source: §II · pp. 1–2
- Local optimization scales to large problems, but nonconvexity makes its converged critical point depend on the initialization.Source: §II · p. 2
- Spanning-tree initialization is simple and scalable, but measurement errors accumulate along long paths.Source: §II · p. 2
- Linear initialization relaxes manifold constraints into linear least-squares problems solved by standard matrix factorizations.Prior applications include quaternion and matrix rotation synchronization and special Euclidean group synchronization.Source: §II · p. 2
- Spectral methods apply broadly when the group has an efficient rounding projection, but iterative eigenvalue solvers can converge slowly, especially on sparse graphs.Source: §II · p. 2
- Certifiably correct methods offer global-optimality guarantees under mild conditions, while large-scale convex relaxations remain computationally challenging.Source: §II · p. 2
- Fast-Sync generalizes linear initialization from specific groups to arbitrary matrix Lie groups, with Kronecker exploitation and gauge selection for efficiency and initialization quality.Source: §II · p. 2
III. GROUP SYNCHRONIZATION
The paper formulates group synchronization using group-valued states and noisy relative measurements, then relates its maximum-likelihood objective to a Frobenius-norm surrogate. The surrogate provides the basis for linear initialization when measurement errors are small.
III. GROUP SYNCHRONIZATION
- The unknowns are N group-valued states, with noisy relative measurements between selected state pairs defining the measurement graph.Source: Preliminaries · p. 2
- The adopted isotropic concentrated Gaussian model uses zero-mean Gaussian measurement noise, with each edge weighted by its inverse variance.Source: Preliminaries · p. 2
- The maximum-likelihood formulation assumes errors are small enough for the group logarithm and exponential maps to be inverses on their support.Source: Preliminaries · p. 3
- For small measurement errors, minimizing the Lie-algebra error is first-order equivalent to minimizing a matrix-space Frobenius surrogate.Source: Frobenius error · p. 3
IV. FAST-SYNC
The section motivates a linear initialization because local optimization of the non-convex synchronization cost depends on its starting estimate. Fast-Sync uses a surrogate Frobenius cost to construct that initialization.
IV. FAST-SYNC
- Because the synchronization cost is non-convex, local optimization from an initial estimate can converge only to the nearest minimum.A good initial estimate is crucial for convergence to the global minimum.Source: §IV · p. 3
- A good initial estimate is crucial for local optimization to converge to the global minimum.Source: §IV · p. 3
- Fast-Sync uses the surrogate Frobenius cost to build a fast linear initialization, building on earlier work by Govindu and Martinec and Pajdla.Source: §IV · p. 3
A. The Measurement Matrix
The measurement matrix is assembled from edge-wise block rows and extended with a Kronecker product. Its block structure encodes which group elements participate in each measured pairwise relation.
A. The Measurement Matrix
- The cost is written in compact matricized form after expressing X in vectorized form.The supplied passages introduce this representation before defining the edge-wise block rows.Source: A. The Measurement Matrix · p. 3
- Each edge contributes a block row whose columns place the measured relation at vertex i, an identity block at vertex j, and zeros elsewhere.Stacking these rows for all edges forms the graph-derived matrix H.Source: A. The Measurement Matrix · p. 3
- The Kronecker extension H ⊗ I_d has dimensions d^2M × d^2N, where H has dimensions Md × Nd.The extension is defined by H-bar = H ⊗ I_d.Source: A. The Measurement Matrix · p. 3
B. Fast Approximate Synchronization
Fast-Sync forms a linear least-squares approximation of group synchronization, then uses gauge fixing, sparse ordering, and back-substitution to recover estimates. Rounding returns estimates to the group, and local optimization can refine them.
B. Fast Approximate Synchronization
- Factoring H instead of its Kronecker extension solves a d-times-smaller problem because each row shares the same coefficient matrix.The problem matrix has the structure H ⊗ I_d.Source: B. Fast Approximate Synchronization · p. 3
- Gauge fixing removes the system's rank deficiency and the homogeneous least-squares problem's trivial all-zeros solution.The augmented system fixes one group element to the identity; solution quality can vary with the chosen gauge.Source: B. Fast Approximate Synchronization · p. 3
- Block-preserving Nested Dissection orders graph vertices using small separators, reducing fill-in and tending to limit distance from the identity gauge.The figure shows the Cholesky factor and back-substitution DAG under this ordering.Source: B. Fast Approximate Synchronization · p. 4, Figure 3 · p. 4
- An economy QR factorization of the reduced, gauge-augmented matrix turns the least-squares solve into a triangular solve before recursive back-substitution.The triangular system is R y-hat = Q^T b.Source: B. Fast Approximate Synchronization · p. 4
- Under noise, a branching ambient-space back-substitution can leave the group, so the recovered matrices are projected onto G.The coefficients in the linear combination are matrix-valued; for SO(d), the paper uses polar decomposition, while any matrix Lie group can use a projection Π_G.Source: B. Fast Approximate Synchronization · p. 4
- After rounding, local methods such as Riemannian gradient descent, Gauss-Newton, Levenberg-Marquardt, or Lie-algebra iterations can refine the estimates.These methods operate on the product manifold or use a first-order linearization of the objective.Source: B. Fast Approximate Synchronization · p. 4
V. SYNTHETIC ANALYSIS
Across the tested groups, Fast-Sync’s more costly initialization generally reduces downstream optimization time and achieves better or comparable results to MST. Nested Dissection preserves or slightly improves solution quality while substantially reducing runtime.
V. SYNTHETIC ANALYSIS
- The synthetic evaluation varies graph rewiring probability and measurement noise across six groups, sampling 10 random instances per setting.The groups are SO(3), SE(3), SU(2), SL(4), Gal(3), and Sim(3); higher noise and lower rewiring probability make instances more challenging.Source: §V · p. 6
- Fast-Sync outperforms MST on nearly all SO(3), SE(3), and SU(2) instances, with higher high-noise success and lower total runtime despite greater initialization cost.The heatmap analyses cover SO(3), SE(3), and SU(2).Source: Figure 4 · p. 5, Figure 5 · p. 5, Figure 6 · p. 5
- Across SL(4), Gal(3), and Sim(3), Fast-Sync achieves better or comparable refinement and lower total computational cost than MST.Its higher initialization cost is offset by reduced subsequent optimization time.Source: Figure 7 · p. 5, Figure 8 · p. 6, Figure 9 · p. 6
- Nested Dissection preserves or slightly improves solution quality while substantially reducing runtime through shallower elimination structures, with little effect on local optimizer convergence.The ablation summarizes 10 trials and reports median success in the high-noise regime, runtime, iteration count, and elimination-DAG height.Source: §V · p. 7, Table I · p. 7
- On SO(3) and SE(3), Fast-Sync is approximately 35%–40% faster than Chordal while achieving nearly the same solution quality.Table I compares Fast-Sync with off-the-shelf Chordal initialization.Source: §V · p. 7, Table I · p. 7
VI. RESULTS ON REAL DATASETS
Across four real-world datasets, Fast-Sync has lower initial Frobenius error than MST and reaches the optimal solution faster in most cases, including on noise-corrupted variants. The evaluation spans different Lie groups, problem sizes, connectivity structures, and measurement conditions.
VI. RESULTS ON REAL DATASETS
- The evaluation compares initial error, final refinement error, and computation time because these real-world examples lack ground truth.Each configuration was run 10 times on the same dataset to obtain median runtime estimates.Source: §VI · p. 7
- The four datasets span SE(2), SE(3), SO(3), and Gal(3), with differing problem sizes and graph connectivity.Gal(3) Drone uses synthetic measurements derived from a drone sequence.Source: §VI · p. 7
- Frobenius error is tracked against runtime from initialization through local optimization, with additional measurement noise used to test robustness.Solid and dashed lines distinguish original and corrupted data; dotted lines show optimal-error baselines, using SE-Sync for three datasets and a ground-truth-initialized estimate for Gal(3) Drone.Source: §VI · pp. 7–8
- The difficulty plot places datasets by algebraic connectivity (λ2) and estimated noise level (σ), with filled and unfilled markers indicating original and corrupted data.Low connectivity with high noise marks the more challenging problems; high connectivity with low noise marks the easier ones.Source: §VI · p. 8
- Across all four datasets, Fast-Sync achieves lower initial Frobenius error than MST and reaches the optimal solution faster in most cases, on both original and noise-corrupted data.The datasets span diverse Lie groups, problem scales, and connectivity structures.Source: §VI · p. 8
VII. CONCLUSIONS
Fast-Sync is a group-agnostic initialization method for large synchronization problems in robotics and vision. Its Frobenius surrogate and structured linear-algebra design yield high-quality initializations, while the theoretical basis for ND-based gauge selection remains open.
VII. CONCLUSIONS
- Fast-Sync provides group-agnostic initialization for synchronization problems in large estimation tasks in robotics and vision.The method uses a Frobenius surrogate and linear-algebraic design to produce high-quality initializations across matrix Lie groups.Source: §VII · p. 8
- Its design combines Kronecker reduction, block-preserving ND gauge fixing, stable factorization on the reduced system, and structured back-substitution.Source: §VII · p. 8
- A theoretical foundation explaining when and why ND-based gauge selection succeeds remains an open direction.The paper reports strong practical value for this strategy.Source: §VII · p. 8