  We study super-resolution multi-reference alignment, the problem of
estimating a signal from many circularly shifted, down-sampled, and noisy
observations. We focus on the low SNR regime, and show that a signal in
$\mathbb{R}^M$ is uniquely determined when the number $L$ of samples per
observation is of the order of the square root of the signal's length
$(L=O(\sqrt{M}))$. Phrased more informally, one can square the resolution. This
result holds if the number of observations is proportional to at least
1/SNR$^3$. In contrast, with fewer observations recovery is impossible even
when the observations are not down-sampled ($L=M$). The analysis combines tools
from statistical signal processing and invariant theory. We design an
expectation-maximization algorithm and demonstrate that it can super-resolve
the signal in challenging SNR regimes.
