In industrial applications, phase retrieval algorithms can be used to obtain information on optical system misalignment. Because of the specific wavelengths used, often the input data for such algorithms are affected by a high level of noise and quantized with a low bit resolution, and the traditional methods fail to restore the phase accurately. The restoration accuracy can be increased with the presented method of phase retrieval from a (single) overexposed measurement of a point-spread function. We demonstrate that under certain conditions, any projection-based phase retrieval method can be adjusted to accept the input data with the saturated pixels. The modification uses a concept of the clipped set which is able to represent and restore the information lost due to overexposure correctly. With moderate levels of overexposure, the phase restoration accuracy is increased due to the improved signal-to-noise ratio of the PSF. The presentation describes the concept of a clipped set and the procedure of calculating the projection on it and demonstrates the application on the simulated and experimental data.

We present a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. It bases on first-order proximal algorithms for minimizing a Tikhonov functional associated with the respective tensorial lifted problem with nuclear norm regularization [1]. It is well known, however, that a direct application of such algorithms involves computations in the tensor-product space, in particular, singular-value thresholding. Due to the prohibitively high dimension of the latter space, such algorithms are infeasible without appropriate modification. Thus, to overcome this limitation, we show that all computational steps can be adapted to perform on low-rank representations of the iterates, yielding feasible, memory and computationally efficient tensor-free algorithms [2]. We present and discuss the numerical performance of these methods for the two-dimensional Fourier phase retrieval problem. In particular, we show that the incorporation of smoothness constraints within the framework greatly improve image recovery results.

[1] R. Beinert, K. Bredies. Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting, Inverse Problems 35(1): 015002, 2019.

[2] R. Beinert, K. Bredies. Tensor-free proximal methods for lifted bilinear/quadratic inverse problems with applications to phase retrieval, Foundations of Computational Mathematics 21(5): 1181-1232, 2021.

Recently, the success of optimization is related to re- and over-parametrization, that are widely used - for instance - in neural networks applications. However, there is still an open question of how to find systematically what is the inductive bias hidden behind the model for a particular optimization scheme. The goal of this talk is taking a step in this direction, studying extensively many reparametrization used in the state of the art and providing a common structure to analyze the problem in a unified way. We show that gradient descent on the objective function for many reparametrization is equivalent to mirror descent on the original problem. The mirror function depends on the reparametrization and introduces an inductive bias, which plays the role of the regularizer. Our theoretical results provide asymptotic behavior and convergence results.