Provable Phase Retrieval with Mirror Descent

TL;DR

The paper tackles real-phase retrieval from magnitude-only measurements by introducing a mirror-descent algorithm built on a carefully chosen entropy-based kernel $\psi$, enabling relative smoothness rather than global Lipschitz conditions. It proves deterministic convergence to non-saddle critical points and, under local relative strong convexity, local linear convergence; in the random-model regime, it establishes high-probability recovery guarantees for Gaussian measurements and local linear convergence for CDP measurements, with spectral initialization helping in the latter. The results show that with sufficient measurements, the method achieves global recovery (up to sign) for Gaussian data with a dimension-free convergence rate, and strong, fast local convergence for CDP measurements, while remaining computationally efficient at $O(n)$ per iteration. Numerical experiments on 1D signals and 2D surface roughness illustrate the method’s practicality and its competitive performance against established phase-retrieval algorithms, supporting its potential for precision optics and related inverse problems.

Abstract

In this paper, we consider the problem of phase retrieval, which consists of recovering an $n$-dimensional real vector from the magnitude of its $m$ linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the \iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements $m$ is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.

Figures (5)

• Figure 1: Reconstruction of a 1D signal by mirror descent from Gaussian measurements.
• Figure 2: Reconstruction of a 1D signal by mirror descent from CDP measurements.
• Figure 3: Roughness surface profile reconstruction by solving the phase retrieval problem from the CDP measurement model using mirror descent with uniform random initialization.
• Figure 4: Phase diagrams of mirror descent (MD) with spectral and uniform random initialization. (a) Gaussian measurements. (b) CDP measurements.
• Figure 5: Comparison of mirror descent to other methods in the literature. Each plot shows the empirical probability of success based on 100 random trials for two different measurement models (Gaussian and CDP) and a varied number of measurements.

Theorems & Definitions (31)

• Remark 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Lemma 2.7
• Proposition 2.8
• Definition 2.9
• Definition 2.10