We address the problem of reconstructing a set of points on a line or a loop from their unassigned noisy pairwise distances. When the points lie on a line, the problem is known as the turnpike; when they are on a loop, it is known as the beltway. We approximate the problem by discretizing the domain and representing the $N$ points via an $N$ -hot encoding, which is a density supported on the discretized domain. We show how the distance distribution is then simply a collection of quadratic functionals of this density and propose to recover the point locations so that the estimated distance distribution matches the measured distance distribution. This can be cast as a constrained nonconvex optimization problem which we solve using projected gradient descent with a suitable spectral initializer. We derive conditions under which the proposed distance distribution matching approach locally converges to a global optimizer at a linear rate. Compared to the conventional backtracking approach, our method jointly reconstructs all the point locations and is robust to noise in the measurements. We substantiate these claims with state-of-the-art performance across a number of numerical experiments. Our method is the first practical approach to solve the large-scale noisy beltway problem where the points lie on a loop.

中文翻译：

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从距离分布重建点集

我们解决了根据未分配的有噪声的成对距离在直线或环路上重建一组点的问题。当这些点位于一条线上时，该问题称为收费公路；当它们成环时，被称为环道。我们通过离散化域并表示$ N $ 通过一个点 $ N $ -hot编码，这是离散域上支持的密度。我们将说明距离分布如何只是该密度的二次函数的集合，并建议恢复点位置，以便估算的距离分布与测得的距离分布相匹配。可以将其转换为一个受约束的非凸优化问题，我们可以使用投影梯度下降和合适的频谱初始化程序来解决该问题。我们得出条件，在该条件下，拟议的距离分布匹配方法以线性速率局部收敛至全局优化器。与传统的回溯方法相比，我们的方法可以共同重建所有点的位置，并且对测量中的噪声具有鲁棒性。我们通过许多数值实验的最新性能证实了这些主张。我们的方法是解决点位于环路上的大规模嘈杂环道问题的第一种实际方法。