  Multireference alignment (MRA) problem is to estimate an underlying signal
from a large number of noisy circularly-shifted observations. The existing
methods are always proposed 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 focus on the
MRA problem under the assumption of Gaussian mixture noise. We derive an
adaptive variational model by combining maximum a posteriori (MAP) estimation
and soft-max method. There are two adaptive weights which are for detecting
cyclical shifts and types of noise. Furthermore, we provide a statistical
interpretation of our model by using expectation-maximization(EM) algorithm.
The existence of a minimizer is mathematically proved. The numerical results
show that the proposed model has a more impressive performance than the
existing methods when one Gaussian noise is large and the other is small.
