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PtyGenography: using generative models for regularization of the phase retrieval problem

Selin Aslan, Tristan van Leeuwen, Allard Mosk, Palina Salanevich

TL;DR

The paper addresses phase retrieval from magnitude-only measurements and analyzes how generative priors can regularize the inverse problem at the cost of reconstruction bias. It derives error bounds for conventional and generative formulations and introduces a unified variational objective that blends data fidelity with a generative regularizer, enabling adaptive trade-offs between bias and variance. Numerical experiments on masked Fourier measurements with a PCA-trained generator show that the combined approach often outperforms the individual formulations across noise regimes, while pure generative priors can introduce bias on out-of-distribution signals. This work provides a practical framework for robust phase retrieval using data-driven priors and highlights avenues for adaptive regularization and theoretical tightening of bounds.

Abstract

In phase retrieval and similar inverse problems, the stability of solutions across different noise levels is crucial for applications. One approach to promote it is using signal priors in a form of a generative model as a regularization, at the expense of introducing a bias in the reconstruction. In this paper, we explore and compare the reconstruction properties of classical and generative inverse problem formulations. We propose a new unified reconstruction approach that mitigates overfitting to the generative model for varying noise levels.

PtyGenography: using generative models for regularization of the phase retrieval problem

TL;DR

The paper addresses phase retrieval from magnitude-only measurements and analyzes how generative priors can regularize the inverse problem at the cost of reconstruction bias. It derives error bounds for conventional and generative formulations and introduces a unified variational objective that blends data fidelity with a generative regularizer, enabling adaptive trade-offs between bias and variance. Numerical experiments on masked Fourier measurements with a PCA-trained generator show that the combined approach often outperforms the individual formulations across noise regimes, while pure generative priors can introduce bias on out-of-distribution signals. This work provides a practical framework for robust phase retrieval using data-driven priors and highlights avenues for adaptive regularization and theoretical tightening of bounds.

Abstract

In phase retrieval and similar inverse problems, the stability of solutions across different noise levels is crucial for applications. One approach to promote it is using signal priors in a form of a generative model as a regularization, at the expense of introducing a bias in the reconstruction. In this paper, we explore and compare the reconstruction properties of classical and generative inverse problem formulations. We propose a new unified reconstruction approach that mitigates overfitting to the generative model for varying noise levels.

Paper Structure

This paper contains 8 sections, 3 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Reconstruction approach
  3. Characterizing the reconstruction error
  4. Detecting bias
  5. Towards a unified approach
  6. Numerical results
  7. Experiment
  8. Conclusion and discussion

Key Result

Lemma 1

Let ${\bf \tilde{f}} = \mathop{\mathrm{arg\,min}}\limits_{{\bf f}} \|\mathcal{A}({\bf f})-{\bf y}\|_2$ with ${\bf y} = \mathcal{A}({\bf f_0}) + \boldsymbol{\varepsilon}$. Then the reconstruction error is given by

Figures (3)

  • Figure 1: Samples of the data set on which the generative model was trained ($n=64$). The top row displays the real part while the bottom row displays the imaginary part.
  • Figure 2: Samples generated by the generative model ($k=30$) The top row displays the real part while the bottom row displays the imaginary part.
  • Figure 3: Relative reconstruction error for the three methods for varying signal-to-noise ratio levels, tested on both in-distribution and out-of-distribution scenarios.

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Remark 3