Heterogeneous multireference alignment: a single pass approach

TL;DR

This paper introduces a single-pass algorithm for heterogeneous multireference alignment that estimates multiple signals from noisy, cyclically shifted observations without identifying individual shifts or classes, useful in cryo-electron microscopy.

Contribution

It presents a novel low-complexity method that estimates multiple signals directly from invariant features, bypassing the need for shift or class estimation.

Findings

Accurately estimates multiple signals from noisy data.

Works effectively without estimating shifts or classes.

Potential to resolve up to roughly √L signals for signal length L.

Abstract

Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where $K$ signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the $K$ signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the $K$ signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals $K$ that can be resolved as a function of the signal length $L$ is on the order of $\sqrt{L}$.