A Fast and Provable Algorithm for Sparse Phase Retrieval

TL;DR

This paper tackles sparse phase retrieval from magnitude-only measurements by introducing a second-order Newton-type algorithm with hard thresholding. By using a dual loss strategy—employing the intensity-based loss as the objective while using the amplitude-based loss to identify the support for Newton updates—the method updates only a small free set of coordinates and solves a reduced Newton system, achieving quadratic convergence after a finite number of iterations with sample complexity $m = \Omega(s^2 \log n)$. Theoretical results establish non-asymptotic quadratic convergence in the noiseless case and linear convergence with a bounded error in the presence of noise, under Gaussian measurements; initialization via sparse spectral methods ensures the iterates remain in a favorable neighborhood. Empirically, the approach outperforms state-of-the-art first-order methods in convergence speed and robustness, demonstrating practical efficiency for high-dimensional sparse phaseless recovery. The work highlights a meaningful step toward combining second-order information with sparsity-aware updates in nonconvex phase retrieval, with implications for diffraction, imaging, and related sensing problems.

Abstract

We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we propose an innovative second-order algorithm that employs a Newton-type method with hard thresholding. This algorithm overcomes the linear convergence limitations of first-order methods while preserving their hallmark per-iteration computational efficiency. We provide theoretical guarantees that our algorithm converges to the $s$-sparse ground truth signal $\mathbf{x}^{\natural} \in \mathbb{R}^n$ (up to a global sign) at a quadratic convergence rate after at most $O(\log (\Vert\mathbf{x}^{\natural} \Vert /x_{\min}^{\natural}))$ iterations, using $Ω(s^2\log n)$ Gaussian random samples. Numerical experiments show that our algorithm achieves a significantly faster convergence rate than state-of-the-art methods.

Figures (6)

• Figure 1: Relative error versus iterations for various algorithms, with fixed signal dimension $n=5000$ and sample size $m=3000$. The results represent the average of 100 independent trial runs.
• Figure 2: Phase transition performance of various algorithms for signals of dimension $n=3000$ with sparsity levels $s=25$ and $50$. The results represent the average of 200 independent trial runs. The parameter settings for ThWF, SPARTA, CoPRAM, and HTP in experiments are consistent with those in the studies by jagatap2019samplecai2022sparse.
• Figure 3: Reconstruction of the signal with a dimension of $30{,}000$ from noisy phaseless measurements by various algorithms. Time(s) is the running time in seconds.
• Figure 4: Phase transition comparison of various algorithms applied to block-sparse signals with a dimension of $n=3000$ and sparsity levels $s=20$ and $30$. The signal generation process aligns with the experiments depicted in Figure 2 of the study by jagatap2019sample. The parameter settings for ThWF, SPARTA, CoPRAM, and HTP are consistent with those used in the studies by jagatap2019samplecai2022sparse. The results reflect the average of 200 independent trial runs. A recovery is considered successful if the relative error was less than $10^{-3}$.
• Figure 5: Comparing phase transitions among various algorithms for a signal dimension of $n=3000$, across different sparsity levels and numbers of measurements. The successful recovery rates are indicated by varying grey levels in the corresponding block. Black signifies a $0\%$ successful recovery rate, white indicates $100\%$, and grey represents values between $0\%$ and $100\%$. A signal recovery is considered successful if its relative error is less than $10^{-3}$.

Theorems & Definitions (24)

• Theorem 3.1
• Theorem 3.2
• Lemma B.1
• Lemma B.2
• Lemma B.3
• Lemma B.4
• Lemma B.5
• Lemma B.6
• Lemma B.7
• Lemma B.8