[Paper Review] Median-Truncated Nonconvex Approach for Phase Retrieval with Outliers

Huishuai Zhang, Yuejie Chi+1 more|arXiv (Cornell University)|Mar 11, 2016

Advanced X-ray Imaging Techniques3 citations

TL;DR

This paper proposes median-truncated nonconvex algorithms—median-TWF and median-RWF—for robust phase retrieval under adversarial outliers. By using the sample median to guide pruning of corrupted measurements during initialization and gradient descent, the methods achieve provable recovery of the signal from near-optimal sample sizes even when up to a constant fraction of measurements are arbitrarily corrupted, with linear convergence and stability under dense bounded noise.

ABSTRACT

This paper investigates the phase retrieval problem, which aims to recover a signal from the magnitudes of its linear measurements. We develop statistically and computationally efficient algorithms for the situation when the measurements are corrupted by sparse outliers that can take arbitrary values. We propose a novel approach to robustify the gradient descent algorithm by using the sample median as a guide for pruning spurious samples in initialization and local search. Adopting the Poisson loss and the reshaped quadratic loss respectively, we obtain two algorithms termed median-TWF and median-RWF, both of which provably recover the signal from a near-optimal number of measurements when the measurement vectors are composed of i.i.d. Gaussian entries, up to a logarithmic factor, even when a constant fraction of the measurements are adversarially corrupted. We further show that both algorithms are stable in the presence of additional dense bounded noise. Our analysis is accomplished by developing non-trivial concentration results of median-related quantities, which may be of independent interest. We provide numerical experiments to demonstrate the effectiveness of our approach.

Motivation & Objective

• To develop statistically and computationally efficient phase retrieval algorithms robust to sparse, arbitrarily large outliers.
• To address the limitation of existing Wirtinger flow-type methods, which fail under adversarial outliers despite strong performance in clean settings.
• To achieve provable recovery with a near-optimal number of measurements (up to logarithmic factors) when measurement vectors are i.i.d. Gaussian.
• To ensure stability under additional dense bounded noise, extending applicability to realistic noisy environments.
• To establish theoretical guarantees via novel concentration inequalities for median-related statistics.

Proposed method

• Uses the sample median of measurement magnitudes as a robust guide to identify and prune spurious, outlier-corrupted samples during initialization and local search.
• Adapts the truncated Wirtinger flow (TWF) and reshaped Wirtinger flow (RWF) frameworks by incorporating median-based truncation to improve robustness.
• Employs the Poisson loss and reshaped quadratic loss functions in two variants: median-TWF and median-RWF, respectively.
• Applies a median-based thresholding rule to exclude measurements whose magnitudes deviate significantly from the median, reducing influence of outliers.
• Uses a Lipschitz approximation of indicator functions to control concentration of median-related quantities in high-dimensional analysis.
• Establishes uniform convergence bounds over the unit sphere using $5$-nets and sub-Gaussian tail inequalities, enabling high-probability guarantees.

Experimental results

Research questions

• RQ1 Can gradient-based phase retrieval methods be made robust to adversarial outliers that take arbitrary values, without requiring prior knowledge of their locations or magnitudes?
• RQ2 Can robustness be achieved with near-optimal sample complexity—on the order of $n$—even when a constant fraction of measurements are corrupted?
• RQ3 Does the use of the sample median as a pruning mechanism improve convergence and stability compared to standard nonconvex solvers like TWF or RWF?
• RQ4 Can the proposed algorithms maintain linear convergence rates and stability under additional dense bounded noise?
• RQ5 Are novel concentration inequalities for median-based statistics sufficient to support theoretical guarantees in high-dimensional, non-i.i.d. settings?

Key findings

• The median-TWF and median-RWF algorithms provably recover the true signal from $m = \mathcal{O}(n \log n)$ measurements when measurement vectors are i.i.d. Gaussian, even with up to a constant fraction of adversarial outliers.
• Both algorithms achieve linear convergence rates and require $\mathcal{O}(mn\log(1/\epsilon))$ floating-point operations to reach $\epsilon$-accuracy.
• The methods are stable under additional dense bounded noise, maintaining robust performance in noisy environments.
• Theoretical analysis relies on new concentration results for median-related quantities, which may be of independent interest beyond phase retrieval.
• Numerical experiments demonstrate the effectiveness of the median-truncation strategy in suppressing outlier effects and improving recovery accuracy compared to baseline methods.