Euclidean distance geometry and the orthogonal beltway problem

The orthogonal beltway problem is the problem of recovering the $\mathrm{O}(n)$-orbit of a $\delta$-function supported at a finite number of points in \r^n from its auto-correlation or, equivalently, second moment. It was introduced as a generalization of the classical beltway problem in X-ray crystallography and was motivated by cryo-electron microscopy. In this paper we prove that if $m > n$, then the $\mathrm{O}(n)$-orbit of generic binary signal supported at $m$ points where at least $\ell$ of them have equal magnitude can be recovered from its auto-correlation. We also provide a connection to Euclidean distance geometry and prove, as a corollary of our main theorem, that if $m > n$, then the $\mathrm{O}(n)$-orbit of a generic collection of $m$ points on the sphere $S^{n-1}$ can be recovered from their unlabeled interpoint distances. We take advantage of the parallels to Euclidean distance geometry and develop a polynomial-time reconstruction algorithm for recovering the $\O(n)$-orbits of binary $\delta$-functions from their second-moment data when at least one of the points has distinct magnitude. In $\mathbb{R}^3$, the complexity of our algorithm is bounded from above by $O(m^8)$ but we show that in practice the complexity is much lower. We also demonstrate that the algorithm is robust to low levels of noise. Finally, we extend our algorithm to successfully perform recovery when all the support vectors lie on a common sphere, and in this case we match the time complexity of $O(m^8)$.