# Optimal rates of estimation for multi-reference alignment

## TL;DR

This paper establishes the optimal rates for adaptive estimation in the multi-reference alignment model, revealing the fundamental limits and the role of moment tensors in signal processing applications.

## Contribution

It introduces a novel analysis of the multi-reference alignment problem, deriving matching upper and lower bounds based on local KL divergence control and moment tensors.

## Key findings

- Derived optimal estimation rates depending on signal-to-noise ratio
- Established tight bounds for the multi-reference alignment model
- Highlighted the importance of moment tensors in estimation accuracy

## Abstract

In this paper, we establish optimal rates of adaptive estimation of a vector in the multi-reference alignment model, a problem with important applications in fields such as signal processing, image processing, and computer vision, among others. We describe how this model can be viewed as a multivariate Gaussian mixture model under the constraint that the centers belong to the orbit of a group. This enables us to derive matching upper and lower bounds that feature an interesting dependence on the signal-to-noise ratio of the model. Both upper and lower bounds are articulated around a tight local control of Kullback-Leibler divergences that showcases the central role of moment tensors in this problem.