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Motivated by the problem of determining the atomic structure ofmacromolecules using single-particle cryo-electron microscopy (cryo-EM), westudy the sample and computational complexities of the sparse multi-referencealignment (MRA) model: the problem of estimating a sparse signal from itsnoisy, circularly shifted copies. Based on its tight connection to thecrystallographic phase retrieval problem, we establish that if the number ofobservations is proportional to the square of the variance of the noise, thenthe sparse MRA problem is statistically feasible for sufficiently sparsesignals. To investigate its computational hardness, we consider three types ofcomputational frameworks: projection-based algorithms, bispectrum inversion,and convex relaxations. We show that a state-of-the-art projection-basedalgorithm achieves the optimal estimation rate, but its computationalcomplexity is exponential in the sparsity level. The bispectrum frameworkprovides a statistical-computational trade-off: it requires more observations(so its estimation rate is suboptimal), but its computational load is provablypolynomial in the signal's length. The convex relaxation approach providespolynomial time algorithms (with a large exponent) that recover sufficientlysparse signals at the optimal estimation rate. We conclude the paper bydiscussing potential statistical and algorithmic implications for cryo-EM.

Sparse multi-reference alignment: sample complexity and computational hardness