Recovering a three-dimensional signal from noisy observations that have undergone unknown rotations is a fundamental problem in cryo-electron microscopy. Each image is a projection of the molecule from an unknown orientation. The standard assumption is that rotations are drawn uniformly from SO(3) — every orientation equally likely. Non-uniform distributions seem harder: the signal is sampled unevenly, with some orientations overrepresented and others missed.
Bendory and colleagues (arXiv:2602.20590) show the non-uniform case is actually easier. When the rotation distribution is non-uniform, the first and second moments of the observations are sufficient to uniquely determine both the signal and the distribution. The sample complexity in high-noise regimes scales quadratically rather than cubically with noise variance — a full polynomial improvement over the uniform case.
The mechanism: non-uniformity breaks symmetries in the moment equations that create ambiguities under uniform sampling. With uniform rotations, many distinct signals produce identical second moments. With non-uniform rotations, the asymmetry in the sampling lifts these degeneracies. The distribution itself is information that the moments encode. The algorithm solves well-conditioned linear systems iteratively — the non-uniformity makes the linear systems better conditioned.
The result inverts the expected difficulty ordering. Uniform sampling appears clean; non-uniform sampling appears messy. But the messiness carries structure that the uniformity lacks. The non-uniform distribution is both an unknown to be estimated and a resource that enables estimation.
The general observation: asymmetry in the observation process can be information, not noise. A non-uniform distribution of measurements that seems to complicate reconstruction can actually simplify it by breaking symmetries that create ambiguity under uniform conditions.