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Smoothed Robust Phase Retrieval

Zhong Zheng, Lingzhou Xue

TL;DR

The smoothed robust phase retrieval (SRPR) based on a family of convolution-type smoothed loss functions is introduced and it is proved that the SRPR enjoys a benign geometric structure with high probability under the noiseless situation.

Abstract

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 $\ell_1$-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.

Smoothed Robust Phase Retrieval

TL;DR

The smoothed robust phase retrieval (SRPR) based on a family of convolution-type smoothed loss functions is introduced and it is proved that the SRPR enjoys a benign geometric structure with high probability under the noiseless situation.

Abstract

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 -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.
Paper Structure (19 sections, 18 theorems, 57 equations, 8 figures, 1 table)

This paper contains 19 sections, 18 theorems, 57 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. A General Framework
  4. Convolution-type Smoothing
  5. Properties of the Smoothed Objective Function
  6. Algorithms and Rate of Convergence
  7. Overview of Benign Landscape
  8. Landscape Analysis in the Noiseless Case
  9. Notations
  10. Population Landscape
  11. Empirical Landscape
  12. Empirical Landscape in the Presence of Noise
  13. Benign Landscape When Sparse Corruptions Exists
  14. Sketch of Proof
  15. Numerical Experiments
  16. ...and 4 more sections

Key Result

Lemma 1

(Generalized Sharpness) Under Assumptions ass_data, ass_kernel and ass_direct, when $p_{\text{fail}} = \gamma = 0$ and $n\geq c p$, with probability at least $1-C\exp(-c'n)$, we have where $\Delta(x) = \min\{\|x-x_\star\|_2,\|x+x_\star\|_2\}$, and $c,c',C,\lambda_s$ are constants independent of $\delta$.

Figures (8)

  • Figure 1: An illustration of the noiseless population landscape
  • Figure 2: Noiseless Empirical Landscape
  • Figure 3: Empirical Landscape with Corruptions
  • Figure 4: Comparison of success rate and CPU time for the noiseless simulated datasets
  • Figure 5: Decreasing pattern of $F_\delta(x)$ with random initialization for the noiseless simulated dataset
  • ...and 3 more figures

Theorems & Definitions (18)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • ...and 8 more