Dan Edidin: The beltway problem for the orthogonal group

Abstract.

The classical beltway problem asks for the recovery of a collection of points on the circle from their set of pairwise distances. Motivated by cryo-EM we consider a generalization, which is the recovery of the O(n)-orbit of a \(\delta\)-function supported at a finite collection of points in \(R^n\) from its auto-correlation. When the points are radially collision free, meaning that their magnitudes are distinct, then it is easy to show that the second moment uniquely determines the orbit. However, when some of the support points have the same magnitude the problem is more difficult. We prove that if the magnitude of at least one point is distinct from the others is distinct then, for generic configurations, the O(n) orbit is recoverable from the second moment. We also give an algorithm for recovering the points in this case. This is based on joint work with Arun Suresh.