# Multireference Alignment without Alignment

Tuesday, October 31, 2017 - 1:25pm - 2:25pm

Lind 305

Nicolas Boumal (Princeton University)

Consider the problem of estimating a signal x in R^n from numerous noisy observations of it, where each observation has been cyclically shifted in an unknown way. In the absence of shifts, one could simply average the observations. Thus, one natural approach is to try first to align (undo the shifts), then average. Unfortunately, beyond a certain noise level, aligning is impossible.

I will show how the signal x can be estimated from the observations directly, without alignment. The technique relies on so-called invariant features: moments of the data that are invariant under cyclic shifts. Recovering the signal from these moments is a non-convex problem and I'll discuss some ideas that allow to solve it in practice.

Going a few steps further, we will get into a heterogeneous variant of this problem and a 2D version of it. At that point, I will show the connections with cryo electron microscopy---the motivating application for this work.

Nicolas Boumal obtained his PhD in applied mathematics from the Université catholique de Louvain, Belgium, in 2014. He was a postdoc at Inria in Paris until early 2016, then joined the mathematics department of Princeton University as an instructor. His research interests revolve broadly around optimization, with a focus on Riemannian optimization for which he develops the toolbox Manopt (www.manopt.org).

I will show how the signal x can be estimated from the observations directly, without alignment. The technique relies on so-called invariant features: moments of the data that are invariant under cyclic shifts. Recovering the signal from these moments is a non-convex problem and I'll discuss some ideas that allow to solve it in practice.

Going a few steps further, we will get into a heterogeneous variant of this problem and a 2D version of it. At that point, I will show the connections with cryo electron microscopy---the motivating application for this work.

Nicolas Boumal obtained his PhD in applied mathematics from the Université catholique de Louvain, Belgium, in 2014. He was a postdoc at Inria in Paris until early 2016, then joined the mathematics department of Princeton University as an instructor. His research interests revolve broadly around optimization, with a focus on Riemannian optimization for which he develops the toolbox Manopt (www.manopt.org).