Depth Descent Synchronization in ${{\,\mathrm{\text {SO}}\,}}(D)$

Tyler Maunu, Gilad Lerman

Abstract: We give robust recovery results for synchronization on the rotation group, $${{\,\mathrm{\text {SO}}\,}}(D)$$ 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 $${{\,\mathrm{\text {SO}}\,}}(D)$$ SO ( D ) . This algorithm, called Depth Descent Synchronization, exactly recovers the underlying rotations up to an outlier percentage of $$1/(D(D-1)+2)$$ 1 / ( D ( D - 1 ) + 2 ) , which corresponds to 1/4 for $${{\,\mathrm{\text {SO}}\,}}(2)$$ SO ( 2 ) and 1/8 for $${{\,\mathrm{\text {SO}}\,}}(3)$$ SO ( 3 ) . In the case of $${{\,\mathrm{\text {SO}}\,}}(2)$$ 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 $${{\,\mathrm{\text {SO}}\,}}(3)$$ SO ( 3 ) and demonstrate that it performs competitively on baseline synthetic data.