A Spectral Method for Stable Bispectrum Inversion with Application to Multireference Alignment

TL;DR

This paper introduces a spectral algorithm for stable bispectrum inversion that accurately estimates signals from noisy, randomly shifted observations, with applications in multireference alignment.

Contribution

A novel spectral decomposition method for bispectrum inversion that is robust to noise and useful for signal estimation in alignment problems.

Findings

Eigenvectors of the normalized bispectrum matrix correspond to true signal phases.

The spectral method is robust to noise and provides stable initialization.

Applicable to large datasets with noisy, randomly shifted observations.

Abstract

We focus on an alignment-free method to estimate the underlying signal from a large number of noisy randomly shifted observations. Specifically, we estimate the mean, power spectrum, and bispectrum of the signal from the observations. Since bispectrum contains the phase information of the signal, reliable algorithms for bispectrum inversion is useful in many applications. We propose a new algorithm using spectral decomposition of the normalized bispectrum matrix for this task. For clean signals, we show that the eigenvectors of the normalized bispectrum matrix correspond to the true phases of the signal and its shifted copies. In addition, the spectral method is robust to noise. It can be used as a stable and efficient initialization technique for local non-convex optimization for bispectrum inversion.