Unbiasing Procedures for Scale-invariant Multi-reference Alignment

TL;DR

This paper introduces a novel unbiasing method for the scale-invariant multi-reference alignment problem, enabling recovery of signals from noisy, randomly transformed observations by estimating the power spectrum with quantifiable accuracy.

Contribution

It proposes a data-driven, nonlinear unbiasing procedure for power spectrum estimation in a generalized alignment problem, including a method to learn unknown dilation parameters.

Findings

The estimator's error decreases with increased sample size and lower noise levels.

The method accurately recovers signals in numerical experiments across various scenarios.

Theoretical convergence rates are established for the proposed estimator.

Abstract

This article discusses a generalization of the 1-dimensional multi-reference alignment problem. The goal is to recover a hidden signal from many noisy observations, where each noisy observation includes a random translation and random dilation of the hidden signal, as well as high additive noise. We propose a method that recovers the power spectrum of the hidden signal by applying a data-driven, nonlinear unbiasing procedure, and thus the hidden signal is obtained up to an unknown phase. An unbiased estimator of the power spectrum is defined, whose error depends on the sample size and noise levels, and we precisely quantify the convergence rate of the proposed estimator. The unbiasing procedure relies on knowledge of the dilation distribution, and we implement an optimization procedure to learn the dilation variance when this parameter is unknown. Our theoretical work is supported by extensive numerical experiments on a wide range of signals.