Multireference Alignment is Easier with an Aperiodic Translation Distribution

In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal's orbit by accessing multiple independent observations. Of particular interest is the sample complexity, i.e., the number of observations/samples needed in terms of the signal-to-noise ratio (the signal energy divided by the noise variance) in order to drive the mean-square error (MSE) to zero. Previous work showed that if the translations are drawn from the uniform distribution, then, in the low SNR regime, the sample complexity of the problem scales as $\omega(1/\text{SNR}^3)$. In this work, using a generalization of the Chapman--Robbins bound for orbits and expansions of the $\chi^2$ divergence at low SNR, we show that in the same regime the sample complexity for any aperiodic translation distribution scales as $\omega(1/\text{SNR}^2)$. This rate is achieved by a simple spectral algorithm. We propose two additional algorithms based on non-convex optimization and expectation-maximization. We also draw a connection between the multireference alignment problem and the spiked covariance model

Estimation in the group action channel

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.Comment: 5 pages, conferenc

Multi-target detection with rotations

We consider the multi-target detection problem of estimating a two-dimensional target image from a large noisy measurement image that contains many randomly rotated and translated copies of the target image. Motivated by single-particle cryo-electron microscopy, we focus on the low signal-to-noise regime, where it is difficult to estimate the locations and orientations of the target images in the measurement. Our approach uses autocorrelation analysis to estimate rotationally and translationally invariant features of the target image. We demonstrate that, regardless of the level of noise, our technique can be used to recover the target image when the measurement is sufficiently large.Comment: 20 pages, 5 figure