  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.
