Super-resolution multi-reference alignment.

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 ${ℝ}^M$ is uniquely determined when the number L of samples per observation is of the order of the square root of the signal's length ( $L=O\left(\sqrt{M}\right)$ ). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to 1/SNR³. 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.