A spectral method for stable bispectrum inversion with application to multireference alignment

This thesis aims to develop an alignment-free method to estimate the underlying signal from a large number of noisy randomly shifted observations, called multi-reference alignment problem. Specifically, we make use of invariant features including mean, power spectrum, and the bispectrum of the signal from the observations. We propose a new algorithm using spectral decomposition of the bispectrum phase matrix for this specific problem. 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 noisy cases, we will select one eigenvector with largest spectral gap to estimate the original signal. Such spectral method is robust to noise and empirically comparable to iterative phase synchronization and optimization on phase manifold for noise variance sigma squared less than or equal to 0.32. It can be also be used as a stable and efficient initialization technique for local non-convex optimization for bispectrum inversion. Using 12-fold symmetric property of bispectrum, we are able to increase our computational efficiency by roughly ten times.U of I Onlyundergraduate senior thesis not recommended for open acces