# Smoothed Robust Phase Retrieval

arXiv: https://arxiv.org/abs/2409.01570 · HTML: https://mlsanity.com/paper/2409.01570/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 38 references that are in the map.
- First posted on arXiv 3 Sep 2024.

## Builds on (most cited first, 19 of 19)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Subgradient methods for sharp weakly convex functions](https://mlsanity.com/paper/1803.02461/) (2018)
- [Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow](https://mlsanity.com/paper/1605.08285/) (2016)
- [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017)
- [Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow](https://mlsanity.com/paper/1506.03382/) (2015)
- [Smoothed Quantile Regression with Large-Scale Inference](https://mlsanity.com/paper/2012.05187/) (2020)
- [The nonsmooth landscape of phase retrieval](https://mlsanity.com/paper/1711.03247/) (2017)
- [Smoothing quantile regressions](https://mlsanity.com/paper/1905.08535/) (2019)
- [High-Dimensional Quantile Regression: Convolution Smoothing and Concave Regularization](https://mlsanity.com/paper/2109.05640/) (2021)
- [Holographic image reconstruction with phase recovery and autofocusing using recurrent neural networks](https://mlsanity.com/paper/2102.12281/) (2021)
- [Solving phase retrieval with random initial guess is nearly as good as by spectral initialization](https://mlsanity.com/paper/2101.03540/) (2021)
- [A Unified Algorithm for Penalized Convolution Smoothed Quantile Regression](https://mlsanity.com/paper/2205.02432/) (2022)
- [A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrieval](https://mlsanity.com/paper/2304.12522/) (2023)

## 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.