PhaseMax: Convex Phase Retrieval via Basis Pursuit (1610.07531v3)

Abstract: We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a "non-lifting" relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.

• The paper presents PhaseMax, a non-lifting convex formulation that directly solves phase retrieval in the original signal space.
• It bridges phase retrieval and sparse recovery by demonstrating that the dual problem of PhaseMax is equivalent to Basis Pursuit.
• Numerical results and theoretical analysis show that PhaseMax offers robust performance and scalability, even in the presence of noise.

Overview of "PhaseMax: Convex Phase Retrieval via Basis Pursuit"

Phase retrieval is a critical problem in signal processing that involves recovering a signal from magnitude-only measurements. In the paper titled "PhaseMax: Convex Phase Retrieval via Basis Pursuit" by Tom Goldstein and Christoph Studer, the authors introduce a novel approach to phase retrieval using a convex optimization framework named PhaseMax. This method deviates significantly from traditional convex methods, which typically rely on semidefinite relaxation and lifting techniques. Instead, PhaseMax is a non-lifting relaxation that operates within the original signal dimension, offering computational efficiency and scalability advantages.

Key Contributions and Methodology

1. Non-Lifting Convex Formulation: The authors formulate the phase retrieval problem as a non-lifting convex optimization problem, PhaseMax, as an alternative to semidefinite programming approaches like PhaseLift. PhaseMax directly works within the original signal space, allowing it to be solved efficiently using existing Basis Pursuit algorithms.
2. Dual Problem Connection: The paper establishes that the dual of the PhaseMax problem is Basis Pursuit, traditionally used for sparse signal recovery. This insight bridges the gap between phase retrieval and sparse recovery, enabling the use of established algorithms in a new problem domain.
3. Theoretical Guarantees: The authors provide comprehensive theoretical analysis, including sharp lower bounds on the success probability of PhaseMax across various random measurement ensembles. They also explore the impact of measurement noise on solution accuracy, demonstrating robustness to noise perturbations.
4. Implications for Implementation: PhaseMax's formulation as a convex problem offers stability and the potential for integration with additional signal constraints, such as sparsity or total variation. This feature could be particularly advantageous in applications where prior knowledge about the signal can be leveraged.
5. Numerical and Empirical Evaluation: Through extensive numerical results, the paper demonstrates PhaseMax's effectiveness in practice, showing its potential competitiveness with non-convex methods, especially given its theoretical guarantees.

Implications and Future Directions

The development of PhaseMax presents several theoretical and practical implications for phase retrieval and related fields:

• Scalability and Performance: The ability to solve phase retrieval problems without lifting expands the scale of problems that can be efficiently tackled, which is crucial for applications in imaging and signal processing.
• Further Research and Optimization: Future research could focus on optimizing the computational implementation of PhaseMax to leverage its full potential in real-time and resource-constrained environments. This might include parallelization or specialized hardware acceleration.
• Extensions to Other Constraints: Given its convex nature, PhaseMax could be extended to accommodate various signal constraints and applications beyond standard phase retrieval, such as in quantum mechanics or coherent diffraction imaging.
• Comparative Analysis with Non-Convex Methods: While PhaseMax offers solid theoretical guarantees, comparative studies with non-convex approaches that might require fewer measurements or provide faster convergence could illuminate scenarios where each method excels.

In summary, PhaseMax represents a significant advancement in the field of phase retrieval, combining theoretical robustness with practical efficiency. By recasting phase retrieval as a Basis Pursuit problem, the authors not only enhance computational feasibility but also open new pathways for innovation in signal processing techniques that require phase information recovery.