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Stable Phase Retrieval: Optimal Rates in Poisson and Heavy-tailed Models

Gao Huang, Song Li, Deanna Needell

TL;DR

The paper tackles stable phase retrieval under realistic noise models by analyzing two estimators—NCVX-LS and CVX-LS—within a unified framework built on multiplier inequalities and small-ball methods. It proves minimax-optimal error rates in the high-energy regime for both Poisson and heavy-tailed noises, and demonstrates energy-adaptive rates for the Poisson model in the low-energy regime, with a near-optimal rate for heavy-tailed noise as well. A key innovation is interpreting Poisson noise as sub-exponential at high energy and heavy-tailed at low energy, enabling uniform, signal-energy adaptive guarantees across models. The framework is extended to sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution, underscoring its broad applicability to stable, robust inverse problems in sensing and imaging.

Abstract

We investigate stable recovery guarantees for phase retrieval under two realistic and challenging noise models: the Poisson model and the heavy-tailed model. Our analysis covers both nonconvex least squares (NCVX-LS) and convex least squares (CVX-LS) estimators. For the Poisson model, we demonstrate that in the high-energy regime where the true signal $pmb{x}$ exceeds a certain energy threshold, both estimators achieve a signal-independent, minimax optimal error rate $\mathcal{O}(\sqrt{\frac{n}{m}})$, with $n$ denoting the signal dimension and $m$ the number of sampling vectors. In contrast, in the low-energy regime, the NCVX-LS estimator attains an error rate of $\mathcal{O}(\|\pmb{x}\|^{1/4}_2\cdot(\frac{n}{m})^{1/4})$, which decreases as the energy of signal $\pmb{x}$ diminishes and remains nearly optimal with respect to the oversampling ratio. This demonstrates a signal-energy-adaptive behavior in the Poisson setting. For the heavy-tailed model with noise having a finite $q$-th moment ($q>2$), both estimators attain the minimax optimal error rate $\mathcal{O}( \frac{\| ξ\|_{L_q}}{\| \pmb{x} \|_2} \cdot \sqrt{\frac{n}{m}} )$ in the high-energy regime, while the NCVX-LS estimator further achieves the minimax optimal rate $\mathcal{O}( \sqrt{\|ξ\|_{L_q}}\cdot (\frac{n}{m})^{1/4} )$ in the low-energy regime. Our analysis builds on two key ideas: the use of multiplier inequalities to handle noise that may exhibit dependence on the sampling vectors, and a novel interpretation of Poisson noise as sub-exponential in the high-energy regime yet heavy-tailed in the low-energy regime. These insights form the foundation of a unified analytical framework, which we further apply to a range of related problems, including sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution.

Stable Phase Retrieval: Optimal Rates in Poisson and Heavy-tailed Models

TL;DR

The paper tackles stable phase retrieval under realistic noise models by analyzing two estimators—NCVX-LS and CVX-LS—within a unified framework built on multiplier inequalities and small-ball methods. It proves minimax-optimal error rates in the high-energy regime for both Poisson and heavy-tailed noises, and demonstrates energy-adaptive rates for the Poisson model in the low-energy regime, with a near-optimal rate for heavy-tailed noise as well. A key innovation is interpreting Poisson noise as sub-exponential at high energy and heavy-tailed at low energy, enabling uniform, signal-energy adaptive guarantees across models. The framework is extended to sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution, underscoring its broad applicability to stable, robust inverse problems in sensing and imaging.

Abstract

We investigate stable recovery guarantees for phase retrieval under two realistic and challenging noise models: the Poisson model and the heavy-tailed model. Our analysis covers both nonconvex least squares (NCVX-LS) and convex least squares (CVX-LS) estimators. For the Poisson model, we demonstrate that in the high-energy regime where the true signal exceeds a certain energy threshold, both estimators achieve a signal-independent, minimax optimal error rate , with denoting the signal dimension and the number of sampling vectors. In contrast, in the low-energy regime, the NCVX-LS estimator attains an error rate of , which decreases as the energy of signal diminishes and remains nearly optimal with respect to the oversampling ratio. This demonstrates a signal-energy-adaptive behavior in the Poisson setting. For the heavy-tailed model with noise having a finite -th moment (), both estimators attain the minimax optimal error rate in the high-energy regime, while the NCVX-LS estimator further achieves the minimax optimal rate in the low-energy regime. Our analysis builds on two key ideas: the use of multiplier inequalities to handle noise that may exhibit dependence on the sampling vectors, and a novel interpretation of Poisson noise as sub-exponential in the high-energy regime yet heavy-tailed in the low-energy regime. These insights form the foundation of a unified analytical framework, which we further apply to a range of related problems, including sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution.

Paper Structure

This paper contains 56 sections, 40 theorems, 242 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prior Art and Bottlenecks
  3. Poisson Model
  4. Heavy-tailed Model
  5. Stable Phase Retrieval
  6. Contributions of This Paper
  7. Notation and Outline
  8. Problem Setup
  9. Main Results
  10. Poisson Model
  11. Heavy-tailed Model
  12. Towards An Architecture
  13. Multiplier Inequalities
  14. Upper Bounds for NUBC
  15. Multiplier Processes
  16. ...and 41 more sections

Key Result

Theorem 1

Suppose that sampling vectors $\left\{\pmb{\varphi}_k\right\}_{k=1}^{m}$ satisfy Assumption sample, and that the Poisson model poisson follows the distribution specified in Assumption noise0 (a). Then there exist some universal constants $L,c,C_1,C_2,C_3 > 0$ dependent only on $K$ and $\mu$ such tha For the CVX-LS estimator, one has By finding the largest eigenvector with largest eigenvalue of $\

Figures (6)

  • Figure 1: Poisson: NCVX-LS with $m/n$.
  • Figure 2: Poisson: CVX-LS with $m/n$.
  • Figure 3: Poisson: NCVX-LS with $\sqrt{\left\lVert\pmb{x}\right\lVert_2}$.
  • Figure 4: Hevay-tail: NCVX-LS with $m/n$.
  • Figure 5: Hevay-tail: CVX-LS with $m/n$.
  • ...and 1 more figures

Theorems & Definitions (73)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • proof
  • Proposition 2: krahmer2020complex
  • proof
  • ...and 63 more