Minimax-optimal estimation for sparse multi-reference alignment with collision-free signals

The Multi-Reference Alignment (MRA) problem aims at the recovery of an unknown signal from repeated observations under the latent action of a group of cyclic isometries, in the presence of additive noise of high intensity σ. It is a more tractable version of the celebrated cryo EM model. In the crucial high noise regime, it is known that its sample complexity scales as σ6. Recent investigations have shown that for the practically significant setting of sparse signals, the sample complexity of the maximum likelihood estimator asymptotically scales with the noise level as σ4. In this work, we investigate minimax optimality for signal estimation under the MRA model for so-called collision-free signals. In particular, this signal class covers the setting of generic signals of dilute sparsity (wherein the support size s=O(L1/3), where L is the ambient dimension. We demonstrate that the minimax optimal rate of estimation in for the sparse MRA problem in this setting is σ2/√n, where n is the sample size. In particular, this widely generalizes the sample complexity asymptotics for the restricted MLE in this setting, establishing it as the statistically optimal estimator. Finally, we demonstrate a concentration inequality for the restricted MLE on its deviations from the ground truth.