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GraHTP: A Provable Newton-like Algorithm for Sparse Phase Retrieval

Licheng Dai, Xiliang Lu, Juntao You

TL;DR

This work tackles sparse phase retrieval with complex measurements by introducing GraHTP, a provable Newton-like nonconvex algorithm that couples a projected gradient step with hard-thresholding and a Gauss-Newton subspace refinement. The method achieves contraction in a local basin and, after entering a small neighborhood of the true signal, exhibits quadratic convergence with a per-iteration cost of $O(mn+s^2m)$ and sample complexity $m\ge C s\log(n/s)$. Theoretical results cover noiseless and noisy models, establish finite-step/convergent behavior, and rely on spectral initialization to provide a suitable starting point. Empirical results across real-valued, complex-valued, and partial-DFT measurements show GraHTP outperforms state-of-the-art approaches in both recovery accuracy and computational efficiency.

Abstract

This paper investigates the sparse phase retrieval problem, which aims to recover a sparse signal from a system of quadratic measurements. In this work, we propose a novel non-convex algorithm, termed Gradient Hard Thresholding Pursuit (GraHTP), for sparse phase retrieval with complex sensing vectors. GraHTP is theoretically provable and exhibits high efficiency, achieving a quadratic convergence rate after a finite number of iterations, while maintaining low computational complexity per iteration. Numerical experiments further demonstrate GraHTP's superior performance compared to state-of-the-art algorithms.

GraHTP: A Provable Newton-like Algorithm for Sparse Phase Retrieval

TL;DR

This work tackles sparse phase retrieval with complex measurements by introducing GraHTP, a provable Newton-like nonconvex algorithm that couples a projected gradient step with hard-thresholding and a Gauss-Newton subspace refinement. The method achieves contraction in a local basin and, after entering a small neighborhood of the true signal, exhibits quadratic convergence with a per-iteration cost of and sample complexity . Theoretical results cover noiseless and noisy models, establish finite-step/convergent behavior, and rely on spectral initialization to provide a suitable starting point. Empirical results across real-valued, complex-valued, and partial-DFT measurements show GraHTP outperforms state-of-the-art approaches in both recovery accuracy and computational efficiency.

Abstract

This paper investigates the sparse phase retrieval problem, which aims to recover a sparse signal from a system of quadratic measurements. In this work, we propose a novel non-convex algorithm, termed Gradient Hard Thresholding Pursuit (GraHTP), for sparse phase retrieval with complex sensing vectors. GraHTP is theoretically provable and exhibits high efficiency, achieving a quadratic convergence rate after a finite number of iterations, while maintaining low computational complexity per iteration. Numerical experiments further demonstrate GraHTP's superior performance compared to state-of-the-art algorithms.
Paper Structure (17 sections, 8 theorems, 87 equations, 9 figures, 1 algorithm)

This paper contains 17 sections, 8 theorems, 87 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work and Our Contributions
  3. Notation
  4. Algorithms
  5. The Proposed Algorithm
  6. Theoretical Results
  7. Initialization
  8. Numerical Experiments
  9. Real-valued Signal Case
  10. Complex-valued Signal Case
  11. Partial Discrete Fourier Transform Matrix
  12. Appendix
  13. Key Lemmas
  14. Proof of Part a.) of \ref{['local convergence']}
  15. Proof of Part b.) of \ref{['local convergence']}
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\bm{x}^\dag \in {\mathbb R}^n$ be any s-sparse signal. Consider $m$ noiseless measurements $y_j=\left|\left\langle\bm{a}_j, \bm{x}^\dag\right\rangle\right|^2$ from i.i.d. $\bm{a}_j\sim {\mathcal{CN}}(\bm{0},\bm{I})$, $j=1,2\cdots,m$. Then, there exist positive constants $\mu_1, \mu_2, \delta, provided $\mu^k \in \left(\frac{\mu_1}{\left\|{\bm{x}^\dag}\right\|_2^2}, \frac{\mu_2}{\left\|{\bm{

Figures (9)

  • Figure 1: Relative error versus number of iterations ((a)-(d)) and relative error versus running time ((e)-(h)) for CoPRAM, ThWF, SPARTA, HTP and our algorithm GraHTP, with fixed signal dimension $n = 3000$ and sample size $m = 2000$. The results represent the average of $100$ independent trial runs.
  • Figure 2: Running time for successful recovery versus signal dimension for CoPRAM, ThWF, SPARTA, HTP and our algorithm GraHTP, with fixed sample size $m = 2120$ and sparsity $20$. All results were obtained by averaging 100 independent experiments with those fail trials filtered out.
  • Figure 3: Phase transition for algorithm CoPRAM, ThWF, SPARTA, HTP and our algorithm GraHTP, the results were obtained by averaging 100 independent experiments.
  • Figure 4: Phase transition for algorithm CoPRAM, ThWF, SPARTA, HTP and our algorithm GraHTP, white block means $100\%$ successful recovery, black block means $0\%$ successful recovery and grey block means the rate of successful recovery between $0\%$ and $100\%$. The results were obtained by averaging 100 independent experiments.
  • Figure 5: 1-D signal reconstruction of algorithm CoPRAM, ThWF, SPARTA, HTP and our algorithm GraHTP. PSNR represents peak signal-to-noise ratio and Time(s) is the running time in seconds.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 1 more