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 ℝM 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(√M)). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to at least 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.