An adaptive variational model for multireference alignment with mixed noise

The multireference alignment (MRA) problem is to estimate an underlying signal from a large number of noisy circularly-shifted observations. The existing methods are under the hypothesis of a single Gaussian noise. However, the hypothesis of a single-type noise is inefficient for solving practical problems like single particle cryo-EM. In this paper, we derive an adaptive variational model by combining maximum a posteriori (MAP) estimation and the soft-max method under the assumption of Gaussian mixture noise. There are two adaptive weights for detecting cyclical shifts and types of noise separately. The existence of a minimizer is mathematically proved. We design a novel algorithm for the proposed model using the alternating direction iterative method and the augmented Lagrange method. There are some convergence analyses for the proposed algorithm under some conditions. The numerical results show that the proposed model performs better than the existing methods in that the level of one Gaussian noise is high and the other is low.