Depth Descent Synchronization in SO(D)

We give robust recovery results for synchronization on the rotation group, SO(D). In particular, we consider an adversarial corruption setting, where a limited percentage of the observations are arbitrarily corrupted. We develop a novel algorithm that exploits Tukey depth in the tangent space of SO(D). This algorithm, called Depth Descent Synchronization, exactly recovers the underlying rotations up to an outlier percentage of 1 / (D(D- 1) + 2) , which corresponds to 1/4 for SO(2) and 1/8 for SO(3). In the case of SO(2), we demonstrate that a variant of this algorithm converges linearly to the ground truth rotations. We implement this algorithm for the case of SO(3) and demonstrate that it performs competitively on baseline synthetic data.