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Super-resolution multi-reference alignment

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Document pages: 25 pages

Abstract: We study super-resolution multi-reference alignment, the problem ofestimating a signal from many circularly shifted, down-sampled, and noisyobservations. 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 perobservation 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. Thisresult holds if the number of observations is proportional to at least1 SNR$^3$. In contrast, with fewer observations recovery is impossible evenwhen the observations are not down-sampled ($L=M$). The analysis combines toolsfrom statistical signal processing and invariant theory. We design anexpectation-maximization algorithm and demonstrate that it can super-resolvethe signal in challenging SNR regimes.

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