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

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 the bispectrum contains the phase information of the signal, reliable algorithms for bispectrum inversion are useful in many applications. We propose a new algorithm using spectral decomposition of the bispectrum phase matrix for this task. For clean signals, we show that the eigenvectors of the bispectrum phase 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 nonconvex optimization for bispectrum inversion.