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Robust Sparse Phase Retrieval: Statistical Guarantee, Optimality Theory and Convergent Algorithm

Jun Fan, Ailing Yan, Xianchao Xiu, Wanquan Liu

TL;DR

This work addresses robust sparse phase retrieval in real and complex settings by formulating a regularized Huber loss with an $\ell_{1/2}$-norm penalty, yielding the objective $F(\mathbf{x})=f(\mathbf{x})+\lambda\|\mathbf{x}\|_{1/2}^{1/2}$. It establishes statistical guarantees (consistency) for the real-valued case, and a fixed-point optimality characterization in the complex domain via Wirtinger derivatives, complemented by a convergent majorization-minimization algorithm with proven global convergence and a linear rate under mild conditions. The MM scheme reduces to half-thresholding updates with an Armijo line search to ensure descent, and its convergence relies on KL-based arguments and Lipschitz-like properties of the Wirtinger gradient. Numerical experiments across synthetic signals and images show superior robustness to various noise models, particularly in the complex-valued setting, with favorable running times. The results offer a theoretically grounded, scalable approach to robust PR that extends to high-dimensional regimes and motivates future work on non-asymptotic error bounds and deeper integration with learning-based methods.

Abstract

Phase retrieval (PR) is a popular research topic in signal processing and machine learning. However, its performance degrades significantly when the measurements are corrupted by noise or outliers. To address this limitation, we propose a novel robust sparse PR method that covers both real- and complex-valued cases. The core is to leverage the Huber function to measure the loss and adopt the $\ell_{1/2}$-norm regularization to realize feature selection, thereby improving the robustness of PR. In theory, we establish statistical guarantees for such robustness and derive necessary optimality conditions for global minimizers. Particularly, for the complex-valued case, we provide a fixed point inclusion property inspired by Wirtinger derivatives. Furthermore, we develop an efficient optimization algorithm by integrating the gradient descent method into a majorization-minimization (MM) framework. It is rigorously proved that the whole generated sequence is convergent and also has a linear convergence rate under mild conditions, which has not been investigated before. Numerical examples under different types of noise validate the robustness and effectiveness of our proposed method.

Robust Sparse Phase Retrieval: Statistical Guarantee, Optimality Theory and Convergent Algorithm

TL;DR

This work addresses robust sparse phase retrieval in real and complex settings by formulating a regularized Huber loss with an -norm penalty, yielding the objective . It establishes statistical guarantees (consistency) for the real-valued case, and a fixed-point optimality characterization in the complex domain via Wirtinger derivatives, complemented by a convergent majorization-minimization algorithm with proven global convergence and a linear rate under mild conditions. The MM scheme reduces to half-thresholding updates with an Armijo line search to ensure descent, and its convergence relies on KL-based arguments and Lipschitz-like properties of the Wirtinger gradient. Numerical experiments across synthetic signals and images show superior robustness to various noise models, particularly in the complex-valued setting, with favorable running times. The results offer a theoretically grounded, scalable approach to robust PR that extends to high-dimensional regimes and motivates future work on non-asymptotic error bounds and deeper integration with learning-based methods.

Abstract

Phase retrieval (PR) is a popular research topic in signal processing and machine learning. However, its performance degrades significantly when the measurements are corrupted by noise or outliers. To address this limitation, we propose a novel robust sparse PR method that covers both real- and complex-valued cases. The core is to leverage the Huber function to measure the loss and adopt the -norm regularization to realize feature selection, thereby improving the robustness of PR. In theory, we establish statistical guarantees for such robustness and derive necessary optimality conditions for global minimizers. Particularly, for the complex-valued case, we provide a fixed point inclusion property inspired by Wirtinger derivatives. Furthermore, we develop an efficient optimization algorithm by integrating the gradient descent method into a majorization-minimization (MM) framework. It is rigorously proved that the whole generated sequence is convergent and also has a linear convergence rate under mild conditions, which has not been investigated before. Numerical examples under different types of noise validate the robustness and effectiveness of our proposed method.

Paper Structure

This paper contains 23 sections, 15 theorems, 189 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. Our Contributions
  4. Outline
  5. Preliminaries
  6. Notations
  7. Wirtinger Derivative
  8. Subdifferential
  9. Kurdyka-Lojasiewicz Property
  10. Statistical Guarantee
  11. Optimality Theory
  12. Convergent Algorithm
  13. Majorization-Minimization
  14. Convergence Analysis
  15. Convergence Rate
  16. ...and 8 more sections

Key Result

Lemma 2.1

(Lemma A.2 in sun2018geometric) Let $\varphi(\mathbf{z}): \mathbb{C}^d\to\mathbb{R}$ hold the continuous first-order Witinger derivative. For any $\mathbf{z}_0\in\mathbb{C}^d$, it follows that

Figures (3)

  • Figure 1: Success rates versus $n/p$ for signal examples: (a)-(c) are the results for real-valued cases, (d)-(f) are the results for complex-valued cases.
  • Figure 2: Relative errors versus iterations with $n=6p$ for signal examples: (a)-(c) are the results for real-valued cases, (d)-(f) are the results for complex-valued cases.
  • Figure 3: Tested images: (a) HI with the pixel size $72\times60$, (b) Star with the pixel size $40\times40$, (c) Cell with the pixel size $96\times80$.

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 4.1
  • ...and 19 more