ShapeFit：从损坏的成对方向精确恢复位置
ShapeFit: Exact location recovery from corrupted pairwise directions
摘要：
令$t_1,\ldots,t_n \in \mathbb{R}^d$并考虑位置恢复问题：给定成对方向观测值的子集$\{(t_i - t_j) / \|t_i - t_j\|_2\}_{i<j \in [n] \times [n]}$，其中这些观测值的恒定部分被任意损坏，找到$\{t_i\}_{i=1}^n$直到全局平移和缩放。我们提出了一种新的算法的位置恢复问题，它由一个简单的凸规划$dn$实变量。我们证明了这个程序恢复一组$n$ i.i.d.如果观测值由\erdosrenyi图给出，则$d$足够大，并且假设涉及任何特定位置的观测值中最多有一个常数部分被逆向破坏。我们还证明，该计划完全恢复高斯位置$d=3$，如果损坏的观察在每个位置的分数是，多对数因子，最多是一个常数。这两个恢复定理是基于一组确定性的条件，我们证明是足够的精确恢复。
英文摘要：
Let $t_1,\ldots,t_n \in \mathbb{R}^d$ and consider the location recovery problem: given a subset of pairwise direction observations $\{(t_i - t_j) / \|t_i - t_j\|_2\}_{i<j \in [n] \times [n]}$, where a constant fraction of these observations are arbitrarily corrupted, find $\{t_i\}_{i=1}^n$ up to a global translation and scale. We propose a novel algorithm for the location recovery problem, which consists of a simple convex program over $dn$ real variables. We prove that this program recovers a set of $n$ i.i.d. Gaussian locations exactly and with high probability if the observations are given by an \erdosrenyi graph, $d$ is large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. We also prove that the program exactly recovers Gaussian locations for $d=3$ if the fraction of corrupted observations at each location is, up to poly-logarithmic factors, at most a constant. Both of these recovery theorems are based on a set of deterministic conditions that we prove are sufficient for exact recovery.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
