  We consider the problem of rigid registration, where we wish to jointly
register multiple point sets via rigid transforms. This arises in applications
such as sensor network localization, multiview registration, and protein
structure determination. The least-squares estimator for this problem can be
reduced to a rank-constrained semidefinite program (REG-SDP). It was recently
shown that by formally applying the alternating direction method of multipliers
(ADMM), we can derive an iterative solver (REG-ADMM) for REG-SDP, wherein each
subproblem admits a simple closed-form solution. The empirical success of
REG-ADMM has been demonstrated for multiview registration. However, its
convergence does not follow from the existing literature on nonconvex ADMM. In
this work, we study the convergence of REG-ADMM and our main findings are as
follows. We prove that any fixed point of REG-ADMM is a stationary (KKT) point
of REG-SDP. Moreover, for clean measurements, we give an explicit formula for
the ADMM parameter $\rho$, for which REG-ADMM is guaranteed to converge to the
global optimum (with arbitrary initialization). If the noise is low, we can
still show that the iterates converge to the global optimum, provided they are
initialized sufficiently close to the optimum. On the other hand, if the noise
is high, we explain why REG-ADMM becomes unstable if $\rho$ is less than some
threshold, irrespective of the initialization. We present simulation results to
support our theoretical predictions. The novelty of our analysis lies in the
fact that we exploit the notion of tightness of convex relaxation to arrive at
our convergence results.
