Codes and Expansions (CodEx) Seminar

Tamir Bendory (Tel Aviv University):
Estimation over groups and the unitary matrix retrieval problem

This talk addresses the problem of recovering a signal from multiple noisy copies, each translated by a random group element. This estimation problem is primarily inspired by single-particle cryo-electron microscopy (cryo-EM), a leading technique for reconstructing biological molecular structures. I will begin by analyzing the second moment of this statistical model, showing that it determines the signal up to a group of unitary matrices, whose dimension is governed by the decomposition of the signal space into irreducible representations. Then, I will present semi-algebraic conditions that enable retrieval of the missing unitary matrices, implying that the sample complexity is proportional to the square of the noise variance. Finally, I will introduce a new computational framework for cryo-EM, combining a sparse molecular representation with projection-based techniques commonly used in X-ray crystallography phase retrieval.