In this paper, we study the problem of recovering a ground truth high dimensional piecewise linear curve $C^*(t):[0, 1]\to\mathbb{R}^d$ from a high noise Gaussian point cloud with covariance $\sigma^2I$ centered around the curve. We establish that the sample complexity of recovering $C^*$ from data scales with order at least $\sigma^6$. We then show that recovery of a piecewise linear curve from the third moment is locally well-posed, and hence $O(\sigma^6)$ samples is also sufficient for recovery. We propose methods to recover a curve from data based on a fitting to the third moment tensor with a careful initialization strategy and conduct some numerical experiments verifying the ability of our methods to recover curves. All code for our numerical experiments is publicly available on GitHub.

翻译：本文研究从高噪声高斯点云中恢复真实高维分段线性曲线$C^*(t):[0, 1]\to\mathbb{R}^d$的问题，该点云以曲线为中心、协方差为$\sigma^2I$。我们证明从数据中恢复$C^*$的样本复杂度至少以$\sigma^6$量级增长。随后指出基于三阶矩的分段线性曲线恢复在局部是适定的，因此$O(\sigma^6)$个样本也足以实现恢复。我们提出了通过拟合三阶矩张量来从数据中恢复曲线的方法，并采用精细的初始化策略，同时进行了数值实验以验证所提方法恢复曲线的能力。所有数值实验代码已在GitHub上公开。