  Motivated by cutting-edge applications like cryo-electron microscopy
(cryo-EM), the Multi-Reference Alignment (MRA) model entails the learning of an
unknown signal from repeated measurements of its images under the latent action
of a group of isometries and additive noise of magnitude $\sigma$. Despite
significant interest, a clear picture for understanding rates of estimation in
this model has emerged only recently, particularly in the high-noise regime
$\sigma \gg 1$ that is highly relevant in applications. Recent investigations
have revealed a remarkable asymptotic sample complexity of order $\sigma^6$ for
certain signals whose Fourier transforms have full support, in stark contrast
to the traditional $\sigma^2$ that arise in regular models. Often prohibitively
large in practice, these results have prompted the investigation of variations
around the MRA model where better sample complexity may be achieved. In this
paper, we show that sparse signals exhibit an intermediate $\sigma^4$ sample
complexity even in the classical MRA model. Further, we characterise the
dependence of the estimation rate on the support size $s$ as $O_p(1)$ and
$O_p(s^{3.5})$ in the dilute and moderate regimes of sparsity respectively. Our
techniques have implications for the problem of crystallographic phase
retrieval, indicating a certain local uniqueness for the recovery of sparse
signals from their power spectrum. Our results explore and exploit connections
of the MRA estimation problem with two classical topics in applied mathematics:
the beltway problem from combinatorial optimization, and uniform uncertainty
principles from harmonic analysis. Our techniques include a certain enhanced
form of the probabilistic method, which might be of general interest in its own
right.
