The $O(d)$ synchronization problem consists of estimating a set of n unknown orthogonal d× d matrices O₁,…,Oₙ from noisy measurements of a subset of the pairwise ratios OᵢOⱼ⁻¹. We formulate and prove a Cheeger-type inequality that relates a measure of how well it is possible to solve the $O(d)$ synchronization problem with the spectra of an operator, the graph connection Laplacian. We also show how this inequality provides a worst-case performance guarantee for a spectral method to solve this problem.
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Bandeira et al. (2013) studied this question.
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