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

arXiv: https://arxiv.org/abs/2312.07839 · HTML: https://mlsanity.com/paper/2312.07839/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 39 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 5 of 5)

- [Multireference Alignment using Semidefinite Programming](https://mlsanity.com/paper/1308.5256/) (2013)
- [Likelihood landscape and maximum likelihood estimation for the discrete orbit recovery model](https://mlsanity.com/paper/2004.00041/) (2020)
- [Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM](https://mlsanity.com/paper/2107.01305/) (2021)
- [Sparse Multi-Reference Alignment : Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem](https://mlsanity.com/paper/2106.12996/) (2021)
- [Sparse multi-reference alignment: sample complexity and computational hardness](https://mlsanity.com/paper/2109.11656/) (2021)

## Abstract (arXiv)

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.