Using a Complex Autoencoder as a Generative Prior for Phase Retrieval

Abstract

In various areas such as applied physics and engineering, one attempts to extract certain information from an object of interest (for example, the structure of a crystal or a planetary object). In some of these applications, including optics, acoustics and crystallography, one can only measure the object intensities, while the phase information is lost or cannot be easily estimated. The inverse problem of reconstructing phase from these magnitude-only measurements is known as phase retrieval. This well-known problem is generally ill-posed due to the lack of uniqueness guarantees and the lack of stability of the solution. Recently, promising results have come from incorporating prior knowledge in the form of neural networks into the problem.

In this thesis, we use a generative model to incorporate prior knowledge into the phase retrieval problem. We focus on finite-dimensional phase retrieval from masked Fourier measurements of vector signals. A generative prior is usually an embedding map from some lower-dimensional space (latent space) to the space of the admissible object. The latent space should capture the essential features of the data. A generative prior reduces the set of admissible signals, which can improve the reconstruction accuracy and reduce the amount of necessary measurements in heavy noise regimes. However, this comes at the cost of introducing a model bias. Building upon recent results by Aslan et al. (2025), where principal component analysis (PCA) is used as regularization in the least-squares solution of the phase retrieval problem, our main approach is to examine the use of a complex-valued autoencoder model as a prior, serving as a non-linear generalization of PCA.

We examine the error and model bias of the reconstruction under varying levels of measurement noise and model discrepancy, in comparison to the use of PCA, and find that the choice of activation function of the autoencoder has a strong effect on its performance with phase retrieval. Restricting the signal space to signals with positive real and imaginary parts, we observed trends comparable to those in reconstructions with the linear model. To our knowledge, the literature on the use of complex autoencoders as a prior for phase retrieval with vector signals is limited. This thesis contributes to further understanding of machine learning in phase retrieval, highlighting some of the challenges and points of attention when incorporating a complex-valued neural network.