The Sample Complexity of Sparse Multireference Alignment and Single-Particle Cryo-Electron Microscopy

.Multireference alignment (MRA) is the problem of recovering a signal from its multiple noisy copies, each acted upon by a random group element. MRA is mainly motivated by single-particle cryo–electron microscopy (cryo-EM) that has recently joined X-ray crystallography as one of the two leading technologies to reconstruct biological molecular structures. Previous papers have shown that, in the high-noise regime, the sample complexity of MRA and cryo-EM is \(n=ω (σ²ᵈ)\), where \(n\) is the number of observations, \(σ^2\) is the variance of the noise, and \(d\) is the lowest-order moment of the observations that uniquely determines the signal. In particular, it was shown that, in many cases, \(d=3\) for generic signals, and thus, the sample complexity is \(n=ω (σ^6)\). In this paper, we analyze the second moment of the MRA and cryo-EM models. First, we show that, in both models, the second moment determines the signal up to a set of unitary matrices whose dimension is governed by the decomposition of the space of signals into irreducible representations of the group. Second, we derive sparsity conditions under which a signal can be recovered from the second moment, implying sample complexity of \(n=ω (σ^4)\). Notably, we show that the sample complexity of cryo-EM is \(n=ω (σ^4)\) if at most one-third of the coefficients representing the molecular structure are nonzero; this bound is near-optimal. The analysis is based on tools from representation theory and algebraic geometry. We also derive bounds on recovering a sparse signal from its power spectrum, which is the main computational problem of X-ray crystallography.