[Paper Review] On DC based Methods for Phase Retrieval

Meng Huang, Ming‐Jun Lai+2 more|arXiv (Cornell University)|Oct 22, 2018

Advanced X-ray Imaging Techniques3 citations

TL;DR

This paper proposes a novel DC (difference of convex) optimization framework for phase retrieval, decomposing the non-convex least-squares objective into a difference of convex functions and solving it via accelerated proximal methods. The approach achieves linear convergence to critical points and outperforms Wirtinger Flow and Gauss-Newton methods, especially with $ m \approx 2n $ measurements, and enables sparse signal recovery at $ m \approx n $ using $ \ell_1 $ regularization.

ABSTRACT

In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization sub-problems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm term is added into the minimization mentioned above. The subproblem is approximated by a proximal gradient method which is solved by the shrinkage-threshold technique directly without iterations. In addition, a modified Attouch-Peypouquet technique is used to accelerate the iterative computation. These lead to more effective algorithms than the Wirtinger flow (WF) algorithm and the Gauss-Newton (GN) algorithm and etc.. A convergence analysis of both DC based algorithms shows that the iterative solutions is convergent linearly to a critical point and will be closer to a global minimizer than the given initial starting point. Our study is a deterministic analysis while the study for the Wirtinger flow (WF) algorithm and its variants, the Gauss-Newton (GN) algorithm, the trust region algorithm is based on the probability analysis. In particular, the DC based algorithms are able to retrieve solutions using a number $m$ of measurements which is about twice of the number $n$ of entries in the solution with high frequency of successes. When $m\approx n$, the $\ell_1$ DC based algorithm is able to retrieve sparse signals.

Motivation & Objective

• Address the challenge of solving non-convex phase retrieval problems with improved convergence and robustness compared to existing stochastic or probabilistic methods.
• Develop a deterministic, convex-analysis-based framework to minimize the nonlinear least-squares objective in phase retrieval.
• Enable sparse phase retrieval by incorporating $ \ell_1 $ regularization into the DC optimization framework.
• Provide theoretical convergence guarantees for the proposed DC-based algorithms, showing linear convergence to critical points.
• Analyze the nonexistence of exact solutions for arbitrary measurement vectors and bound the number of distinct solutions when they exist.

Proposed method

• Formulate the phase retrieval problem as minimizing a difference of two convex functions (DC) by splitting the standard nonlinear least-squares functional into $ f = g - h $, where $ g $ and $ h $ are convex.
• Solve the resulting DC subproblems using Nesterov’s accelerated gradient descent or the Barzilai-Borwein (BB) method for non-sparse cases.
• For sparse phase retrieval, augment the objective with an $ \ell_1 $-norm penalty and solve each subproblem via a proximal gradient method using soft-thresholding without iteration.
• Integrate a modified Attouch-Peypouquet technique to accelerate convergence of the iterative scheme.
• Use a real-valued parametrization $ \mathbf{z} = \mathbf{x} + i\mathbf{y} $ to handle complex signals and derive the Hessian of the objective function for convergence analysis.
• Establish that the Hessian is positive semi-definite at global minimizers, with a null space direction corresponding to the unimodular phase ambiguity.

Experimental results

Research questions

• RQ1 Can a deterministic, DC-based optimization framework outperform probabilistic algorithms like Wirtinger Flow in phase retrieval?
• RQ2 What is the minimal number of measurements $ m $ required for successful recovery using the proposed DC method, especially in the sparse case?
• RQ3 How does the inclusion of $ \ell_1 $ regularization affect the performance and recovery threshold of the DC algorithm?
• RQ4 Under what conditions does the phase retrieval problem have no solution for a given set of measurements?
• RQ5 What is the upper bound on the number of distinct solutions when a solution exists for a given measurement set?

Key findings

• The proposed DC-based algorithms converge linearly to a critical point, with iterates improving monotonically from the initial guess.
• The method outperforms Wirtinger Flow and Gauss-Newton algorithms in terms of convergence speed and success rate, particularly in deterministic settings.
• For general phase retrieval, the algorithm achieves high success rates with $ m \approx 2n $ measurements, matching known theoretical lower bounds.
• For sparse signals, the $ \ell_1 $-regularized DC algorithm recovers the signal with $ m \approx n $ measurements, demonstrating compressed sensing-like behavior.
• The paper proves that the Hessian of the objective function is positive semi-definite at global minimizers, with a null space direction corresponding to the unimodular phase ambiguity.
• The paper establishes that for arbitrary measurement values, a solution to the phase retrieval problem may not exist, and when it does, the number of distinct solutions is bounded above.