强噪声环境下的广义正交Procrustes问题
The generalized orthogonal Procrustes problem in the high noise regime
摘要：
我们考虑的问题，估计云的点从许多嘈杂的意见，云未知的旋转后，可能的反射。这是群作用下估计的一般问题的一个实例，最初受到3-D成像和计算机视觉中的应用的启发。我们专注于一个政权的噪声水平大于信号的幅度，以至于旋转不能可靠地估计。我们提出了一个简单而有效的程序的基础上不变多项式（有效地：革兰氏矩阵）来恢复信号，我们评估它对我们得出的问题的基本限制。我们表明，我们的方法适应的噪声水平，是统计上最优的（常数）的低和高噪声制度。在研究方差的估计，我们遇到的问题的敏感性的一种类型的薄Cholesky分解，我们提供了一个改进的界限，这可能是独立的利益。
英文摘要：
We consider the problem of estimating a cloud of points from numerous noisy observations of that cloud after unknown rotations, and possibly reflections. This is an instance of the general problem of estimation under group action, originally inspired by applications in 3-D imaging and computer vision. We focus on a regime where the noise level is larger than the magnitude of the signal, so much so that the rotations cannot be estimated reliably. We propose a simple and efficient procedure based on invariant polynomials (effectively: the Gram matrices) to recover the signal, and we assess it against fundamental limits of the problem that we derive. We show our approach adapts to the noise level and is statistically optimal (up to constants) for both the low and high noise regimes. In studying the variance of our estimator, we encounter the question of the sensivity of a type of thin Cholesky factorization, for which we provide an improved bound which may be of independent interest.
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