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.

本文提出将最小二乘问题转化为凸性的半定规划问题，通过刚体变换及刚体理论来稳定恢复解，同时实验证明该方法在噪声较大时表现显著优于谱松弛法和流形优化方法。

使用半定规划进行多点云的全局配准

Global registration of multiple point clouds using semidefinite  programming

The problem of completing a low-rank matrix from a subset of its entries is
often encountered in the analysis of incomplete data sets exhibiting an
underlying factor model with applications in collaborative filtering, computer
vision and control. Most recent work had been focused on constructing efficient
algorithms for exact or approximate recovery of the missing matrix entries and
proving lower bounds for the number of known entries that guarantee a
successful recovery with high probability. A related problem from both the
mathematical and algorithmic point of view is the distance geometry problem of
realizing points in a Euclidean space from a given subset of their pairwise
distances. Rigidity theory answers basic questions regarding the uniqueness of
the realization satisfying a given partial set of distances. We observe that
basic ideas and tools of rigidity theory can be adapted to determine uniqueness
of low-rank matrix completion, where inner products play the role that
distances play in rigidity theory. This observation leads to an efficient
randomized algorithm for testing both local and global unique completion.
Crucial to our analysis is a new matrix, which we call the completion matrix,
that serves as the analogue of the rigidity matrix.

矩阵完成问题和距离几何问题之间具有类似性质，利用刚性理论的工具，可以设计出用于测试矩阵局部或全局唯一完成性的高效随机算法。