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Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval

Meng Huang

TL;DR

The paper tackles complex phase retrieval by developing a minimal-norm Gauss-Newton method that operates in a constrained subspace to counteract GN matrix singularity. It leverages leave-one-out techniques to prove that, with $m \ge C n \log^3 n$, the GN trajectory remains in a Region of Incoherence and Contraction (RIC) and converges asymptotically quadratically without any sample splitting. The main result provides a high-probability bound on the iterates: ${\rm dist}({\bm z}_{k+1},{\bm x}^\sharp) \le \frac{2}{\|{\bm x}^\sharp\|} {\rm dist}^2({\bm z}_k,{\bm x}^\sharp) + {\varepsilon_0}{\rm dist}({\bm z}_k,{\bm x}^\sharp)$ with ${\varepsilon_0}=c_1(\frac{n\log^3 m}{m})^{1/4}$, ensuring quadratic convergence as $m\to\infty$. Numerical experiments demonstrate strong, fast recovery under noiseless and noisy Poisson/Gaussian settings and show superior performance on a natural CDP-based image task. The results advance practical phase retrieval by eliminating sample splitting while achieving fast, provable quadratic convergence in the complex domain.

Abstract

In this paper, we introduce a Gauss-Newton method for solving the complex phase retrieval problem. In contrast to the real-valued setting, the Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus, non-invertible. To address this, we utilize a Gauss-Newton step that moves orthogonally to certain trivial directions. We establish that this modified Gauss-Newton step has a closed-form solution, which corresponds precisely to the minimal-norm solution of the associated least squares problem. Additionally, using the leave-one-out technique, we demonstrate that $m\ge O( n\log^3 n)$ independent complex Gaussian random measurements ensures that the entire trajectory of the Gauss-Newton iterations remains confined within a specific region of incoherence and contraction with high probability. This finding allows us to establish the asymptotic quadratic convergence rate of the Gauss-Newton method without the need of sample splitting.

Asymptotic quadratic convergence of the Gauss-Newton method for complex phase retrieval

TL;DR

The paper tackles complex phase retrieval by developing a minimal-norm Gauss-Newton method that operates in a constrained subspace to counteract GN matrix singularity. It leverages leave-one-out techniques to prove that, with , the GN trajectory remains in a Region of Incoherence and Contraction (RIC) and converges asymptotically quadratically without any sample splitting. The main result provides a high-probability bound on the iterates: with , ensuring quadratic convergence as . Numerical experiments demonstrate strong, fast recovery under noiseless and noisy Poisson/Gaussian settings and show superior performance on a natural CDP-based image task. The results advance practical phase retrieval by eliminating sample splitting while achieving fast, provable quadratic convergence in the complex domain.

Abstract

In this paper, we introduce a Gauss-Newton method for solving the complex phase retrieval problem. In contrast to the real-valued setting, the Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus, non-invertible. To address this, we utilize a Gauss-Newton step that moves orthogonally to certain trivial directions. We establish that this modified Gauss-Newton step has a closed-form solution, which corresponds precisely to the minimal-norm solution of the associated least squares problem. Additionally, using the leave-one-out technique, we demonstrate that independent complex Gaussian random measurements ensures that the entire trajectory of the Gauss-Newton iterations remains confined within a specific region of incoherence and contraction with high probability. This finding allows us to establish the asymptotic quadratic convergence rate of the Gauss-Newton method without the need of sample splitting.
Paper Structure (30 sections, 18 theorems, 207 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 30 sections, 18 theorems, 207 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Related work
  4. Our Contributions
  5. Notations
  6. Basic notations
  7. Wirtinger calculus
  8. Moore-Penrose pseudoinverse
  9. Organization
  10. The Gauss-Newton Method
  11. Main results
  12. Numerical Experiments
  13. Recovery of signals with noiseless measurements
  14. Robusteness to Poisson and Gaussian noises
  15. Recovery of Natural Image
  16. ...and 15 more sections

Key Result

Lemma 2.1

Suppose that the sample complexity obeys $m\ge C_0n\log^3 n$ for some sufficiently large constant $C_0>0$. With probability at least $1-O(m^{-10})$ holds simultaneously for all ${\bm z} \in {\mathbb C}^n$ obeying Here, $C_1>0$ is a universal constant.

Figures (5)

  • Figure 1: Relative error versus iteration count for Gauss-Newton, WF, TWF, and TAF methods.: (a) The complex Gaussian case; (b) The CDP case.
  • Figure 2: The empirical success rate for different $m/n$ based on $100$ random trails. (a) Success rate for complex Gaussian case, (b) Success rate for CDP case.
  • Figure 3: Relative error versus number of iterations for Gauss-Newton, WF, TWF, and TAF methods under noisy measurements: (a) Gaussian noises; (b) Poisson noises.
  • Figure 4: SNR versus relative MSE on a dB-scale under the noisy measurements: (a) Gaussian noises; (b) Poisson noises.
  • Figure 5: The Milky Way Galaxy image: The Gauss-Newton method with $L=18$ takes $22$ iterations, computation time is $168.8$ s, relative error is $4.56 \times 10^{-15}$.

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 4.1
  • Example 4.2
  • ...and 27 more