PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming (1109.4499v1)

Abstract: Suppose we wish to recover a signal x in C^n from m intensity measurements of the form |<x,z_i>|^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program---a trace-norm minimization problem; this holds with large probability provided that m is on the order of n log n, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis a vis additive noise.

• The paper introduces a convex relaxation method that leverages semidefinite programming to recover signals exactly from intensity-only measurements with high probability.
• It demonstrates that the approach remains stable under noise, ensuring minimal recovery error in practical, noisy environments.
• The method requires on the order of n log n measurements, making it both efficient and broadly applicable in fields like crystallography and optics.

Analyzing PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

The paper "PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming" presents a significant advancement in the domain of signal processing and convex optimization for phase retrieval. Authored by Emmanuel J. Candès, Thomas Strohmer, and Vladislav Voroninski, it rigorously investigates the recovery of signals from intensity measurements where the phase information is unknown, a problem extensively studied in fields such as X-ray crystallography and optics.

Problem Setting

The fundamental problem addressed is the recovery of a signal $x \in \mathbb{C}^n$ from $m$ intensity measurements of the form $|\langle x, z_i \rangle|^2$, where the vectors $z_i$ are sampled independently and uniformly at random on the unit sphere. This setting inherently loses phase information, rendering traditional linear recovery methods inapplicable.

Key Contributions

1. Convex Relaxation via Semidefinite Programming (SDP): The authors propose solving the signal recovery problem through a convex relaxation. By formulating the recovery as a trace-norm minimization problem, they leverage semidefinite programming to find the solution. Specifically, they show that solving a convex program can recover the signal exactly (up to a global phase factor) with high probability, provided $m$ is on the order of $n \log n$.
2. Stability with Noise: The methodology is shown to be robust to additive noise. This is a critical feature since real-world measurements are seldom noise-free. The paper provides proof that when data are contaminated by a small amount of noise, the recovery error remains small, ensuring the practical applicability of the method.

Methodology and Theoretical Results

The core of the proposed method is the application of convex programming to a traditionally combinatorial problem. Key theoretical constructs and results include:

• Rank Minimization and Trace-Norm Relaxation: The signal recovery problem is cast as a rank minimization problem over positive semidefinite matrices. A trace-norm relaxation transforms this into a tractable SDP. This theoretical underpinning ensures that the recovery of the signal can be guaranteed with high probability.
• RIP-1 Properties: The authors establish Restricted Isometry Property (RIP)-like conditions for the linear operator $\mathcal{A}$ mapping Hermitian matrices to vectors. This is crucial for analyzing the performance of the recovery algorithm.
• Dual Certificates: The existence of a dual certificate is shown to be a sufficient condition for the uniqueness of the solution to the trace-minimization problem. The construction of such certificates is discussed in detail, ensuring the solvability and correctness of the proposed SDP approach.

Numerical Results

Extensive numerical simulations underline the practical feasibility of PhaseLift. Performance is evaluated under varying signal-to-noise ratios (SNRs) and different numbers of measurements. The results demonstrate that PhaseLift maintains robust performance in the presence of noise and achieves exact recovery with $m$ on the order of $n \log n$.

Practical and Theoretical Implications

The practical implications of this research are profound:

• Broad Applicability: Theoretical guarantees of recovery and stability open the door for widespread application in areas requiring phase retrieval, including optical imaging, quantum state tomography, and beyond.
• Efficiency: Utilizing convex optimization allows for leveraging advanced SDP solvers, making the approach computationally viable even for higher-dimensional problems.

Future Directions

The paper sets a foundational framework for future research. Several potential directions include:

• Analysis of Structured Random Models: Investigating other types of measurement vectors, particularly those with structured randomness, could yield further insights and multiplicative benefits in specific applications.
• Advanced Noise Models: Considering stochastic noise models could lead to sharper error bounds and better practical algorithms tailored to specific types of measurement noise.
• Algorithm Improvements: Future work could focus on improving the efficiency and scalability of the SDP solver used in PhaseLift, enabling application to even larger problem instances.

In conclusion, the PhaseLift method and the theoretical guarantees surrounding its exact and stable recovery from magnitude measurements represent a significant step forward in signal processing. By demonstrating that a combinatorial problem can be addressed through a convex approach, the authors provide a robust, practical tool for a variety of applications in science and engineering.