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Affine Phase Retrieval via Second-Order Methods

Bing Gao

TL;DR

This work introduces Newton and Gauss-Newton methods for affine phase retrieval, showing that the problem is strongly convex under signal-dependent conditions and enabling global quadratic convergence in noiseless settings. By employing a resampling scheme, the authors establish quadratic convergence for both Gaussian and admissible CDP measurement models, with sample complexities $m\ge C n \log n$ and $L\ge C \log^3 n$, respectively, and extendability to the complex domain. Numerical experiments validate the theory, revealing rapid convergence and exact recovery with relatively few measurements, while maintaining competitive computational efficiency against first-order methods. The results suggest that exploiting the affine structure yields robust, scalable second-order optimization for phase retrieval tasks with structured measurements.

Abstract

In this paper, we study the affine phase retrieval problem, which aims to recover signals from the magnitudes of affine measurements. We develop second-order optimization methods based on Newton and Gauss-Newton iterations and establish that, under specific a priori conditions, the problem exhibits strong convexity. Theoretically, we prove that the Newton method with resampling achieves global quadratic convergence in the noiseless setting for both Gaussian measurements and admissible coded diffraction patterns (CDPs). Furthermore, we demonstrate that the same theoretical framework naturally extends to the Gauss-Newton method, implying its quadratic convergence. To validate our theoretical findings, we conduct extensive numerical experiments. The results confirm the quadratic convergence of second-order methods, while their computational efficiency remains comparable to that of first-order methods. Additionally, our experiments demonstrate that second-order methods achieve exact recovery with relatively few measurements, highlighting their practical feasibility and robustness.

Affine Phase Retrieval via Second-Order Methods

TL;DR

This work introduces Newton and Gauss-Newton methods for affine phase retrieval, showing that the problem is strongly convex under signal-dependent conditions and enabling global quadratic convergence in noiseless settings. By employing a resampling scheme, the authors establish quadratic convergence for both Gaussian and admissible CDP measurement models, with sample complexities and , respectively, and extendability to the complex domain. Numerical experiments validate the theory, revealing rapid convergence and exact recovery with relatively few measurements, while maintaining competitive computational efficiency against first-order methods. The results suggest that exploiting the affine structure yields robust, scalable second-order optimization for phase retrieval tasks with structured measurements.

Abstract

In this paper, we study the affine phase retrieval problem, which aims to recover signals from the magnitudes of affine measurements. We develop second-order optimization methods based on Newton and Gauss-Newton iterations and establish that, under specific a priori conditions, the problem exhibits strong convexity. Theoretically, we prove that the Newton method with resampling achieves global quadratic convergence in the noiseless setting for both Gaussian measurements and admissible coded diffraction patterns (CDPs). Furthermore, we demonstrate that the same theoretical framework naturally extends to the Gauss-Newton method, implying its quadratic convergence. To validate our theoretical findings, we conduct extensive numerical experiments. The results confirm the quadratic convergence of second-order methods, while their computational efficiency remains comparable to that of first-order methods. Additionally, our experiments demonstrate that second-order methods achieve exact recovery with relatively few measurements, highlighting their practical feasibility and robustness.
Paper Structure (23 sections, 10 theorems, 109 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 23 sections, 10 theorems, 109 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Phase Retrieval and Affine Phase Retrieval
  3. Targets and Motivations
  4. Contributions
  5. Notations
  6. Organization
  7. Newton and Gauss-Newton Iteration
  8. Complex Gradient and Hessian
  9. Newton and Gauss-Newton Iteration
  10. The Newton and Gauss-Newton Method with Re-sampling
  11. Convergence Property of the Newton/Gauss-Newton Method with Re-sampling
  12. Two crucial Lemmas
  13. The main Theorem
  14. Numerical Experiments
  15. Effect of Resampling
  16. ...and 8 more sections

Key Result

Lemma 3.1

Assume ${\mathbf z}_k, {\mathbf x}\in{\mathbb C}^n$ and let the measurement vectors ${\mathbf a}_j\in{\mathbb C}^n$, $j=1,\ldots, m$ be distributed according to either the Gaussian model or the admissible CDPs model. The measurements are independent of both ${\mathbf z}_k$ and ${\mathbf x}$. Additio holds with probability at least $1-c_1\exp(-\gamma_\epsilon n)-c_2/n^2$ for the Gaussian model and

Figures (4)

  • Figure 1: Convergence of Newton and Gauss-Newton methods with/without resampling: record the relative error of each iteration step. (a) ${\mathbf x}$ is randomly generated and (b) ${\mathbf x}={\mathbf a}_1$.
  • Figure 2: Accuracy under noise: record the relative error of Newton method for different values of $b=t\|{\mathbf x}\|^2$. (a) Gaussian model with $m=4n$. (b) Admissible CDPs model with $L=8$.
  • Figure 3: Convergence experiments: relative error versus iteration count. Here $n = 512$ and $m=4n$ for the Gaussian model: (a) noiseless case, (b) Gaussian noise case. For the admissible CDPs model with $L=8$: (c) noiseless case, (d) Gaussian noise case.
  • Figure 4: Success rate versus measurement count. (a) Gaussian model with $m/n=1:0.1:5$. (b) Admissible CDPs model with $L\in[3,10]$.

Theorems & Definitions (22)

  • Remark 1.1
  • Definition 1.1: Gaussian model
  • Definition 1.2: Admissible coded diffraction patterns (CDPs) model
  • Lemma 3.1: Strong convexity
  • proof
  • Remark 3.1
  • Lemma 3.2: Lipschitz continuity
  • proof
  • Theorem 3.1
  • proof
  • ...and 12 more