Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
149 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Phase Retrieval via Wirtinger Flow: Theory and Algorithms (1407.1065v3)

Published 3 Jul 2014 in cs.IT, math.FA, math.IT, math.NA, math.OC, math.ST, and stat.TH

Abstract: We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x of Cn about which we have phaseless samples of the form y_r = |< a_r,x >|2, r = 1,2,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a non-convex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of non-convex optimization schemes that may have implications for computational problems beyond phase retrieval.

Citations (1,248)

Summary

  • The paper introduces Wirtinger Flow (WF), a novel non-convex optimization method that guarantees geometric convergence for exact phase retrieval.
  • It employs a spectral initialization combined with iterative gradient descent using Wirtinger derivatives to refine complex signal estimates.
  • Empirical results demonstrate that WF recovers signals accurately with as few as 4.5n measurements for Gaussian models and six patterns for CDP, highlighting its efficiency.

Phase Retrieval via Wirtinger Flow: Theory and Algorithms

This paper addresses the phase retrieval problem, which is paramount in various disciplines such as X-ray crystallography, microscopy, and quantum mechanics. The objective is to recover a complex-valued signal from the magnitude of its linear measurements, formulated as yr=ar,x2y_r = |\langle a_r, x \rangle|^2 for r=1,,mr = 1, \ldots, m, without phase information. Traditional methods either rely on convex relaxation, which can be computationally expensive and impractical for high-dimensional data, or heuristic approaches with limited theoretical guarantees.

The authors propose a novel algorithm named Wirtinger Flow (WF), which operates through non-convex optimization. The WF algorithm includes two main components: an initialization step using a spectral method and an iterative update rule similar to gradient descent. The initialization estimates the solution by finding the principal eigenvector of ryrarar\sum_r y_r a_r a_r^*, and the iterative updates refine this estimate by minimizing a quadratic loss function via gradient descent using Wirtinger derivatives.

Algorithm and Theoretical Contributions

The WF algorithm's key advantage lies in its rigorous theoretical analysis, proving that it can retrieve the exact phase information from near-minimal measurements. The convergence to the true solution occurs at a geometric rate, ensuring both computational and data efficiency. The theoretical framework applies to two important models: the Gaussian model, where sampling vectors are i.i.d. complex Gaussians, and the coded diffraction pattern (CDP) model common in optics and microscopy.

In addition to proving exact recovery results, the paper provides conditions under which the proposed algorithm performs robustly. The initialization ensures that the initial guess is sufficiently close to the true signal, and the gradient descent steps are carefully crafted to maintain and improve this proximity throughout the iterations.

Numerical Results and Implications

Empirical results demonstrate the efficacy of the WF algorithm. For Gaussian measurements, about $4.5n$ phaseless measurements suffice for accurate recovery. For the CDP model, six patterns typically ensure successful recovery. The algorithm’s practical performance is verified on natural images and 3D projections of molecular structures, achieving high precision within a reasonable computational time frame. For example, the WF algorithm successfully reconstructs large images with relative errors of approximately 101410^{-14}, showcasing its potential for real-world applications.

Implications for Future AI Developments

The WF algorithm's success in phase retrieval suggests that non-convex optimization can be effectively employed for other challenging inverse problems in AI and signal processing. The insights gained from the algorithm's design and analysis could inspire novel approaches for solving high-dimensional optimization problems without resorting to prohibitive SDP-based methods. Furthermore, the theoretical contributions regarding the convergence properties of the WF algorithm can be extended to other non-convex optimization problems, potentially leading to more efficient algorithms with strong performance guarantees.

Conclusion

"Phase Retrieval via Wirtinger Flow: Theory and Algorithms" provides a significant advancement in phase retrieval methodologies by introducing a theoretically sound and computationally efficient algorithm. The WF algorithm bridges the gap between heuristic methods and convex relaxation techniques, offering a robust alternative with provable convergence and practicality for high-dimensional data. Future research may extend this approach to other complex optimization problems, contributing to the broader AI and machine learning fields.