We investigate both the theoretical and algorithmic aspects of
likelihood-based methods for recovering a complex-valued signal from multiple
sets of measurements, referred to as looks, affected by speckle
(multiplicative) noise. Our theoretical contributions include establishing the
first existing theoretical upper bound on the Mean Squared Error (MSE) of the
maximum likelihood estimator under the deep image prior hypothesis. Our
theoretical results capture the dependence of MSE upon the number of parameters
in the deep image prior, the number of looks, the signal dimension, and the
number of measurements per look. On the algorithmic side, we introduce the
concept of bagged Deep Image Priors (Bagged-DIP) and integrate them with
projected gradient descent. Furthermore, we show how employing Newton-Schulz
algorithm for calculating matrix inverses within the iterations of PGD reduces
the computational complexity of the algorithm. We will show that this method
achieves the state-of-the-art performance.

使用最大似然估计方法和深度图像先验，研究了从受散斑（乘性）噪声影响的多组测量中恢复复值信号的理论和算法方面。

袋装深度图像先验恢复存在斑点噪声的图像

Bagged Deep Image Prior for Recovering Images in the Presence of Speckle  Noise

We study the problem of recovering the phase from magnitude measurements;
specifically, we wish to reconstruct a complex-valued signal x of C^n about
which we have phaseless samples of the form y_r = |< a_r,x >|^2, r = 1,2,...,m
(knowledge of the phase of these samples would yield a linear system). This
paper develops a non-convex formulation of the phase retrieval problem as well
as a concrete solution algorithm. In a nutshell, this algorithm starts with a
careful initialization obtained by means of a spectral method, and then refines
this initial estimate by iteratively applying novel update rules, which have
low computational complexity, much like in a gradient descent scheme. The main
contribution is that this algorithm is shown to rigorously allow the exact
retrieval of phase information from a nearly minimal number of random
measurements. Indeed, the sequence of successive iterates provably converges to
the solution at a geometric rate so that the proposed scheme is efficient both
in terms of computational and data resources. In theory, a variation on this
scheme leads to a near-linear time algorithm for a physically realizable model
based on coded diffraction patterns. We illustrate the effectiveness of our
methods with various experiments on image data. Underlying our analysis are
insights for the analysis of non-convex optimization schemes that may have
implications for computational problems beyond phase retrieval.

本文提出一种非凸公式的相位恢复方法，通过随机数迭代更新的规则精确地重建了信号的相位信息。此算法具有低计算复杂性并在计算和数据资源方面都非常有效。