Estimation in the group action channel

TL;DR

This paper investigates the limits of estimating signals affected by random group transformations and noise, establishing bounds on estimation accuracy and conditions for consistency of maximum likelihood estimators in high-noise scenarios.

Contribution

It provides a lower bound on the mean square error for estimating the signal's orbit and identifies conditions under which the maximum likelihood estimator is consistent.

Findings

MSE is bounded away from zero when N/σ^{2d} is bounded.

MLE is consistent if N/σ^{2d} diverges.

Establishes fundamental limits in high-noise group action channels.

Abstract

We analyze the problem of estimating a signal from multiple measurements on a $\mbox{group action channel}$ that linearly transforms a signal by a random group action followed by a fixed projection and additive Gaussian noise. This channel is motivated by applications such as multi-reference alignment and cryo-electron microscopy. We focus on the large noise regime prevalent in these applications. We give a lower bound on the mean square error (MSE) of any asymptotically unbiased estimator of the signal's orbit in terms of the signal's moment tensors, which implies that the MSE is bounded away from 0 when $N/\sigma^{2d}$ is bounded from above, where $N$ is the number of observations, $\sigma$ is the noise standard deviation, and $d$ is the so-called $\mbox{moment order cutoff}$. In contrast, the maximum likelihood estimator is shown to be consistent if $N /\sigma^{2d}$ diverges.