Tight Bounds for Multi-Reference Alignment

How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown circular shifts? This simple problem has surprisingly broad applications, in fields from structural biology to aircraft radar imaging. When the signal-to-noise ratio is high, it is possible to infer the relative cyclic shifts of the observations and thereby realign them, but these approaches break down in the presence of large noise.

In this talk, we will tackle the low SNR regime and show that the optimal rates of estimation depend strongly on particular properties of the Fourier spectrum of the signal. We also show much better rates for generic signals.

Based on joint work with Afonso Bandeira (NYU), Amelia Perry, and Philippe Rigollet.