Provable Method of Moments for Cryo-EM

Abstract

We study the recovery of a three-dimensional volume in single-particle cryo-electron microscopy (cryo-EM) from noisy tomographic projections acquired at unknown viewing rotations, drawn independently and uniformly from $\mathrm{SO}(3)$. We show that, in a spherical-shell representation satisfying an explicit shell-separation condition, the first three moments generically determine the underlying real volume up to its intrinsic global $\mathrm{O}(3)$ ambiguity. The proof is constructive and yields a frequency-marching procedure that successively recovers the volume. Consequently, consistent recovery in the high-noise regime has sample complexity $n=\omega ({\sigma }^6)$, where $n$ is the number of observations and ${\sigma }^2$ is the noise variance. Numerical experiments validate the constructive recovery scheme under finite sampling and additive noise, and indicate that direct third-moment optimization can succeed beyond the structural assumptions required by the constructive proof.