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Adaptive Algorithms for Robust Phase Retrieval

Zhong Zheng, Necdet Serhat Aybat, Shiqian Ma, Lingzhou Xue

TL;DR

This work tackles robust phase retrieval by formulating it as a nonsmooth nonconvex composite problem and introducing two adaptive first-order methods, AdaSubGrad and AdaIPL, that use quantile-based residual information to set step sizes. The authors establish a statistical foundation showing that key residual statistics scale with a distance-like measure, enabling local linear convergence under sharpness assumptions without requiring precise knowledge of problem constants. The methods demonstrate robust performance to sparse corruptions and show competitive or superior results in synthetic and image-recovery experiments compared to fixed-step alternatives. Overall, the work contributes practical, hyperparameter-robust algorithms for RPR with solid theoretical guarantees and strong empirical validation, potentially impacting imaging applications where measurements are imperfect.

Abstract

This paper considers the robust phase retrieval, which can be cast as a nonsmooth and nonconvex composite optimization problem. We propose two first-order algorithms with adaptive step sizes: the subgradient algorithm (AdaSubGrad) and the inexact proximal linear algorithm (AdaIPL). Our contribution lies in the novel design of adaptive step sizes based on quantiles of the absolute residuals. Local linear convergences of both algorithms are analyzed under different regimes for the hyper-parameters. Numerical experiments on synthetic datasets and image recovery also demonstrate that our methods are competitive against the existing methods in the literature utilizing predetermined (possibly impractical) step sizes, such as the subgradient methods and the inexact proximal linear method.

Adaptive Algorithms for Robust Phase Retrieval

TL;DR

This work tackles robust phase retrieval by formulating it as a nonsmooth nonconvex composite problem and introducing two adaptive first-order methods, AdaSubGrad and AdaIPL, that use quantile-based residual information to set step sizes. The authors establish a statistical foundation showing that key residual statistics scale with a distance-like measure, enabling local linear convergence under sharpness assumptions without requiring precise knowledge of problem constants. The methods demonstrate robust performance to sparse corruptions and show competitive or superior results in synthetic and image-recovery experiments compared to fixed-step alternatives. Overall, the work contributes practical, hyperparameter-robust algorithms for RPR with solid theoretical guarantees and strong empirical validation, potentially impacting imaging applications where measurements are imperfect.

Abstract

This paper considers the robust phase retrieval, which can be cast as a nonsmooth and nonconvex composite optimization problem. We propose two first-order algorithms with adaptive step sizes: the subgradient algorithm (AdaSubGrad) and the inexact proximal linear algorithm (AdaIPL). Our contribution lies in the novel design of adaptive step sizes based on quantiles of the absolute residuals. Local linear convergences of both algorithms are analyzed under different regimes for the hyper-parameters. Numerical experiments on synthetic datasets and image recovery also demonstrate that our methods are competitive against the existing methods in the literature utilizing predetermined (possibly impractical) step sizes, such as the subgradient methods and the inexact proximal linear method.
Paper Structure (36 sections, 25 theorems, 112 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 36 sections, 25 theorems, 112 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Existing Algorithms and Challenges
  3. Proposed Algorithms: AdaSubGrad and AdaIPL
  4. AdaSubGrad
  5. AdaIPL
  6. Summary of Contributions
  7. Basic Properties of $F(\cdot)$
  8. Differentiability properties of $F(\cdot)$
  9. Data Generation Process and Key Conditions
  10. Proof of Theorem \ref{['thm_assumptions_high_prob']}
  11. $r^{\tilde{p}}(x) = {\Theta}(\tilde{\Delta}(x))$ with high probability
  12. $F(x) - F(x_\star) = {\Theta}(\tilde{\Delta}(x))$ with high probability
  13. Convergence Rate of AdaSubGrad
  14. Term 1: $\|x^{k}-\hat{x}\|_2^{2}$
  15. Term 2: $\|\xi^k\|$
  16. ...and 21 more sections

Key Result

Lemma 1.2

(LAC) and (HAC) imply (LAC-exact) and (HAC-exact), respectively.

Figures (1)

  • Figure 1: Comparisons on Synthetic Datasets. The left and right subfigures show the results for $p_{\rm fail} = 0.1$ and $0.2$, respectively.

Theorems & Definitions (46)

  • Definition 1.1
  • Lemma 1.2
  • Lemma 2.1: Lemma 6 in zheng2023new, local Lipschitz continuity
  • Lemma 2.2: Absolute deviation bound
  • proof
  • Lemma 2.3: Local linear approximation duchi2019solving
  • Lemma 2.5: Lemma 7 in zheng2023new
  • Lemma 2.6
  • proof
  • Lemma 3.2
  • ...and 36 more