  Consider $N$ points in $\mathbb{R}^d$ and $M$ local coordinate systems that
are related through unknown rigid transforms. For each point we are given
(possibly noisy) measurements of its local coordinates in some of the
coordinate systems. Alternatively, for each coordinate system, we observe the
coordinates of a subset of the points. The problem of estimating the global
coordinates of the $N$ points (up to a rigid transform) from such measurements
comes up in distributed approaches to molecular conformation and sensor network
localization, and also in computer vision and graphics.
  The least-squares formulation of this problem, though non-convex, has a well
known closed-form solution when $M=2$ (based on the singular value
decomposition). However, no closed form solution is known for $M\geq 3$.
  In this paper, we demonstrate how the least-squares formulation can be
relaxed into a convex program, namely a semidefinite program (SDP). By setting
up connections between the uniqueness of this SDP and results from rigidity
theory, we prove conditions for exact and stable recovery for the SDP
relaxation. In particular, we prove that the SDP relaxation can guarantee
recovery under more adversarial conditions compared to earlier proposed
spectral relaxations, and derive error bounds for the registration error
incurred by the SDP relaxation.
  We also present results of numerical experiments on simulated data to confirm
the theoretical findings. We empirically demonstrate that (a) unlike the
spectral relaxation, the relaxation gap is mostly zero for the semidefinite
program (i.e., we are able to solve the original non-convex least-squares
problem) up to a certain noise threshold, and (b) the semidefinite program
performs significantly better than spectral and manifold-optimization methods,
particularly at large noise levels.
