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Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns

Meng Huang, Jinming Wen, Ran Zhang

Abstract

The PhaseLift algorithm is an effective convex method for solving the phase retrieval problem from Fourier measurements with coded diffraction patterns (CDP). While exact reconstruction guarantees are well-established in the noiseless case, the stability of recovery under noise remains less well understood. In particular, when the measurements are corrupted by an additive noise vector $\vw \in \R^m$, existing recovery bounds scale on the order of $\norm{\vw}$, which is conjectured to be suboptimal. More recently, Soltanolkotabi conjectured that the optimal PhaseLift recovery bound should scale with the average noise magnitude, that is, on the order of $\norm{\vw}/\sqrt m$. However, establishing this theoretically is considerably more challenging and has remained an open problem. In this paper, we focus on this conjecture and prove that under adversarial noise, the recovery error of PhaseLift is bounded by $O\xkh{ \sqrt{\frac{\norm{\vw}\log n }{\sqrt m}}}\norm{\vx_0}$. Here, $\vx_0 \in \C^n$ is the signals we aim to recover. Moreover, for mean-zero sub-Gaussian noise vector $\vw \in \R^m$, a upper error bound and its corresponding minimax lower bound are also provided. Our results represent a significant step toward Soltanolkotabi's conjecture, offering new insights into the stability of PhaseLift under noisy CDP measurements.

Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns

Abstract

The PhaseLift algorithm is an effective convex method for solving the phase retrieval problem from Fourier measurements with coded diffraction patterns (CDP). While exact reconstruction guarantees are well-established in the noiseless case, the stability of recovery under noise remains less well understood. In particular, when the measurements are corrupted by an additive noise vector , existing recovery bounds scale on the order of , which is conjectured to be suboptimal. More recently, Soltanolkotabi conjectured that the optimal PhaseLift recovery bound should scale with the average noise magnitude, that is, on the order of . However, establishing this theoretically is considerably more challenging and has remained an open problem. In this paper, we focus on this conjecture and prove that under adversarial noise, the recovery error of PhaseLift is bounded by . Here, is the signals we aim to recover. Moreover, for mean-zero sub-Gaussian noise vector , a upper error bound and its corresponding minimax lower bound are also provided. Our results represent a significant step toward Soltanolkotabi's conjecture, offering new insights into the stability of PhaseLift under noisy CDP measurements.

Paper Structure

This paper contains 20 sections, 10 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Motivation
  4. Related work
  5. Phase retrieval based on random measurements
  6. Phase retrieval from structured measurements
  7. Our contributions
  8. Notations
  9. Organization
  10. Preliminaries
  11. Robust injectivity and uniform upper bound
  12. Approximate dual certificates
  13. Proof of Main Result
  14. Proof of Theorem \ref{['mainresult']}
  15. Proof of Theorem \ref{['lowbound']}
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let ${\bm x}_0 \in {\mathbb C}^n$ and $\omega \ge 1$. Suppose that the masks $\left\{{\bm d}_l\right\}_{l=1}^L$ satisfy Assumption assump:mask, and the number of masks $L$ satisfies $L \ge C_0 \log^2 n$ for some constant $C_0 > 0$ depending only on $M$ and $\nu$. For any noise vector ${\bm w} \in {\ Here, ${\bm X}_0 = {\bm x}_0 {\bm x}_0^*$, and $C > 0$ is a constant only depends on $M, \nu$.

Figures (1)

  • Figure 1: The ratio $\rho_m$ versus the number of masks $L$ under adversarial noises with $n=128$.

Theorems & Definitions (23)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • Lemma 2.1
  • ...and 13 more