Wavelet invariants for statistically robust multi-reference alignment

TL;DR

This paper introduces a wavelet-based signal representation that is translation invariant and robust to noise and dilations, enabling improved power spectrum estimation in multi-reference alignment problems.

Contribution

It presents a novel nonlinear wavelet representation that uniquely defines the power spectrum and includes an unbiasing method for noise and dilation effects.

Findings

The wavelet representation is translation invariant and robust to noise.

The unbiasing procedure effectively estimates the power spectrum.

Numerical experiments confirm robustness and accuracy.

Abstract

We propose a nonlinear, wavelet based signal representation that is translation invariant and robust to both additive noise and random dilations. Motivated by the multi-reference alignment problem and generalizations thereof, we analyze the statistical properties of this representation given a large number of independent corruptions of a target signal. We prove the nonlinear wavelet based representation uniquely defines the power spectrum but allows for an unbiasing procedure that cannot be directly applied to the power spectrum. After unbiasing the representation to remove the effects of the additive noise and random dilations, we recover an approximation of the power spectrum by solving a convex optimization problem, and thus reduce to a phase retrieval problem. Extensive numerical experiments demonstrate the statistical robustness of this approximation procedure.