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Smoothed Robust Phase Retrieval

arXiv 2409.01570 ↗First posted 3 Sep 2024stat.ML cs.LG eess.SPcited by 0

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Abstract (arXiv)

The phase retrieval problem in the presence of noise aims to recover the signal vector of interest from a set of quadratic measurements with infrequent but arbitrary corruptions, and it plays an important role in many scientific applications. However, the essential geometric structure of the nonconvex robust phase retrieval based on the ℓ1-loss is largely unknown to study spurious local solutions, even under the ideal noiseless setting, and its intrinsic nonsmooth nature also impacts the efficiency of optimization algorithms. This paper introduces the smoothed robust phase retrieval (SRPR) based on a family of convolution-type smoothed loss functions. Theoretically, we prove that the SRPR enjoys a benign geometric structure with high probability: (1) under the noiseless situation, the SRPR has no spurious local solutions, and the target signals are global solutions, and (2) under the infrequent but arbitrary corruptions, we characterize the stationary points of the SRPR and prove its benign landscape, which is the first landscape analysis of phase retrieval with corruption in the literature. Moreover, we prove the local linear convergence rate of gradient descent for solving the SRPR under the noiseless situation. Experiments on both simulated datasets and image recovery are provided to demonstrate the numerical performance of the SRPR.

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References · 19

  1. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 526
  2. 2017How to Escape Saddle Points Efficientlycited by 284
  3. 2014Phase Retrieval via Wirtinger Flow: Theory and Algorithmscited by 123
  4. 2016A Geometric Analysis of Phase Retrievalcited by 92
  5. 2016The Landscape of Empirical Risk for Non-convex Lossescited by 86
  6. 2018Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrievalcited by 71
  7. 2015Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systemscited by 50
  8. 2018Subgradient methods for sharp weakly convex functionscited by 33
  9. 2016Solving Systems of Random Quadratic Equations via Truncated Amplitude Flowcited by 28
  10. 2017Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrievalcited by 24
  11. 2015Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flowcited by 23
  12. 2020Smoothed Quantile Regression with Large-Scale Inferencecited by 23
7 more
  1. 2017The nonsmooth landscape of phase retrievalcited by 16
  2. 2019Smoothing quantile regressionscited by 16
  3. 2021High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularizationcited by 14
  4. 2021Holographic image reconstruction with phase recovery and autofocusing using recurrent neural networkscited by 10
  5. 2021Solving phase retrieval with random initial guess is nearly as good as by spectral initializationcited by 9
  6. 2022A Unified Algorithm for Penalized Convolution Smoothed Quantile Regressioncited by 3
  7. 2023A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrievalcited by 1

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