Recent advances in applications such as cryo-electron microscopy have sparked increased interest in the mathematical analysis of multi-reference alignment (MRA) problems, where the goal is to recover a hidden signal from many noisy observations. The simplest model considers observations of a 1-d hidden signal which have been randomly translated and corrupted by high additive noise. This talk generalizes this classic problem by incorporating random dilations into the data model in addition to random translations and additive noise, and explores multiple approaches to its solution based on translation invariant representations. Random dilations cause large perturbations in the high frequencies, making this a challenging model. When the dilation distribution is unknown, the power spectrum of the hidden signal can be approximated by applying a nonlinear unbiasing procedure to a wavelet-based, translation invariant representation and then solving an optimization problem. When the dilation distribution is known, a more accurate unbiasing procedure can be applied directly to the empirical Fourier invariants to obtain an unbiased estimator of the Fourier invariants of the hidden signal, and the convergence rate of the estimator can be precisely quantified in terms of the sample size and noise levels. Theoretical results are supported by extensive numerical experiments on a wide range of signals. Time permitting, we will also see how these signal processing tools can be applied in the novel context of distribution learning from biased, sparse batches.
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