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What is Wrong with End-to-End Learning for Phase Retrieval?

Wenjie Zhang, Yuxiang Wan, Zhong Zhuang, Ju Sun

TL;DR

The paper addresses the problem that end-to-end learning for nonlinear inverse problems like FFPR is hampered by intrinsic forward-model symmetries that keep the forward map $\boldsymbol Y = |\mathcal{F}\boldsymbol X|^2$ invariant. It introduces a symmetry-breaking preprocessing pipeline that centers object content and canonicalizes Fourier-phase representations in phase space, yielding a representative, connected target surface for learning. Empirically, the approach yields substantial, consistent improvements across backbones (e.g., UNet, SiSPRNet) on simulated Bragg CDI FFPR tasks, boosting both MSE and symmetry-adjusted MSE (SA-MSE) on training and test sets, and enabling data-driven methods to surpass a traditional baseline. This symmetry-aware preprocessing reduces learning difficulty, improves generalization, and broadens the practical impact of data-driven FFPR methods in scientific imaging.

Abstract

For nonlinear inverse problems that are prevalent in imaging science, symmetries in the forward model are common. When data-driven deep learning approaches are used to solve such problems, these intrinsic symmetries can cause substantial learning difficulties. In this paper, we explain how such difficulties arise and, more importantly, how to overcome them by preprocessing the training set before any learning, i.e., symmetry breaking. We take far-field phase retrieval (FFPR), which is central to many areas of scientific imaging, as an example and show that symmetric breaking can substantially improve data-driven learning. We also formulate the mathematical principle of symmetry breaking.

What is Wrong with End-to-End Learning for Phase Retrieval?

TL;DR

The paper addresses the problem that end-to-end learning for nonlinear inverse problems like FFPR is hampered by intrinsic forward-model symmetries that keep the forward map invariant. It introduces a symmetry-breaking preprocessing pipeline that centers object content and canonicalizes Fourier-phase representations in phase space, yielding a representative, connected target surface for learning. Empirically, the approach yields substantial, consistent improvements across backbones (e.g., UNet, SiSPRNet) on simulated Bragg CDI FFPR tasks, boosting both MSE and symmetry-adjusted MSE (SA-MSE) on training and test sets, and enabling data-driven methods to surpass a traditional baseline. This symmetry-aware preprocessing reduces learning difficulty, improves generalization, and broadens the practical impact of data-driven FFPR methods in scientific imaging.

Abstract

For nonlinear inverse problems that are prevalent in imaging science, symmetries in the forward model are common. When data-driven deep learning approaches are used to solve such problems, these intrinsic symmetries can cause substantial learning difficulties. In this paper, we explain how such difficulties arise and, more importantly, how to overcome them by preprocessing the training set before any learning, i.e., symmetry breaking. We take far-field phase retrieval (FFPR), which is central to many areas of scientific imaging, as an example and show that symmetric breaking can substantially improve data-driven learning. We also formulate the mathematical principle of symmetry breaking.
Paper Structure (6 sections, 1 theorem, 2 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 6 sections, 1 theorem, 2 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Experiments
  4. Related work
  5. Acknowledgments
  6. Author Biography

Key Result

Proposition 1

Consider the conjugate flipping and global phase transfer symmetries only. The set $\mathcal{H}$, except for its negligible subset ${\cal{N}} = \{1 \} \times \{\omega \in \mathbb S: \textup{Im}\pqty{\omega}=0\}^{M_1 \times M_2-1}$, is a connected, smallest representative in the phase domain $\mathbb

Figures (3)

  • Figure 1: Illustration of the three intrinsic symmetries of FFPR: (i) global phase shift, (ii) 2D conjugate flipping and (iii) translation. Any composition of these three symmetries, when applied to a feasible solution $\boldsymbol X$, will lead to the same measurement$\boldsymbol Y$ in the Fourier domain.
  • Figure 2: Our proposed symmetry breaking process substantially improves end-to-end learning on the square root problem. Top row: learning results on the raw training set with ultra-dense sampling (i) and dense sampling (ii), vs. on the training set after symmetry breaking; Bottom row: illustration of the three desired properties after symmetry breaking, without which the end-to-end learning can still suffer: (iv) non-smallest, (v) non- connected, and (vi) non-representative.
  • Figure 3: Visual comparison of reconstruction results by different methods. Columns 1 and 2 are the results for UNet and SiSPRNet trained on $5000$ data points before symmetry breaking, columns 3 and 4 are the corresponding results after symmetry breaking. For better visualization of the magnitudes before symmetry breaking, we transform the numbers by the $(\cdot)^{1/4}$ function.

Theorems & Definitions (1)

  • Proposition 1