The Noise-Sensitivity Phase Transition in Spectral Group Synchronization Over Compact Groups

In Group Synchronization, one attempts to find a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group, up to the rounding error. We identify a Baik-Ben Arous-Péché type phase transition in the noise level, beyond which spectral group synchronization necessarily fails. Below the phase transition, spectral group synchronization succeeds in recovering the unknown group elements, but its performance deteriorates with the noise level. We provide asymptotically exact formulas for the accuracy of spectral group synchronization below the phase transition, up to the rounding error. We also provide a consistent risk estimate, allowing practitioners to estimate the method’s accuracy from available measurements. Acknowledgments. We thank Yoel Shkolnisky and Amit Singer for helpful discussions. ER was partially supported by Israeli Science Foundation grant no. 1523/16 and the HUJI Leibniz center. ER and MG were partially supported by United States – Israel Binational Science Foundation grant no. 2016201. Reproducibility advisory. All the experimental results cited in this paper are fully reproducible. Code to generate all the figures included in this paper and their underlying data can be found in the code supplement [Rom18]. ∗E-mail: [email protected] †E-mail: [email protected] 1 ar X iv :1 80 3. 03 28 7v 1 [ cs .I T ] 8 M ar 2 01 8