Codes and Expansions (CodEx) Seminar

Ben Blum-Smith (Johns Hopkins University):
Estimating a signal corrupted by a cyclic group action plus noise

In many problems of practical interest, a signal is corrupted not only by noise but also by a transformation selected randomly from some group. One example is cryo-electron microscopy, an imaging technique that generates many very noisy images of a molecule observed from unknown viewing directions—the group \(SO(3)\) of three-dimensional rotations enters because of the uncertainty in the viewing angles. Another is multi-reference alignment, where one receives a vector signal whose components have been cyclically shifted by an unknown shift. In this talk we discuss general considerations governing how hard it is to recover the signal (up to the group action, which is the best one can do), having to do with the minimum degrees of polynomials that are able to distinguish signals. We also give new bounds on this difficulty in the special case that the group in question is cyclic (but acts not necessarily by shifts). The first part of the talk is based on joint work with Afonso Bandeira, Joe Kileel, Amelia Perry, Jonathan Niles-Weed, and Alexander Wein. The second part is based on joint work with Thays Garcia, Rawin Hidalgo, and Consuelo Rodriguez.