Stable Phase Retrieval with Mirror Descent

TL;DR

This work develops a stability-guaranteed phase retrieval framework for real-valued signals from noisy phaseless measurements using mirror descent with an entropy-based Bregman geometry. By proving relative smoothness of the objective and employing backtracking, the authors show deterministic convergence to non-saddle critical points, and, under Gaussian measurements with sufficient samples and high SNR, global convergence to near-optimal minimizers near the true vector up to sign. A spectral initialization further improves sample complexity, yielding local convergence around the target with noise-induced error bounds matching minimax rates. The approach extends to sub-Gaussian and, experimentally, CDP models, and is supported by extensive simulations demonstrating computational efficiency and robust reconstruction under noise.

Abstract

In this paper, we aim to reconstruct an n-dimensional real vector from m phaseless measurements corrupted by an additive noise. We extend the noiseless framework developed in [15], based on mirror descent (or Bregman gradient descent), to deal with noisy measurements and prove that the procedure is stable to (small enough) additive noise. In the deterministic case, we show that mirror descent converges to a critical point of the phase retrieval problem, and if the algorithm is well initialized and the noise is small enough, the critical point is near the true vector up to a global sign change. When the measurements are i.i.d Gaussian and the signal-to-noise ratio is large enough, we provide global convergence guarantees that ensure that with high probability, mirror descent converges to a global minimizer near the true vector (up to a global sign change), as soon as the number of measurements m is large enough. The sample complexity bound can be improved if a spectral method is used to provide a good initial guess. We complement our theoretical study with several numerical results showing that mirror descent is both a computationally and statistically efficient scheme to solve the phase retrieval problem.

Figures (5)

• Figure 1: Reconstruction of signal from Gaussian measurements. The noise mean is $\widetilde{\epsilon}$.
• Figure 2: Phase diagrams for Gaussian measurements.
• Figure 3: Reconstruction of signal from Noisy CDP. The noise mean is $\widetilde{\epsilon}$
• Figure 4: Reconstruction of an image from noisy CDP measurements.
• Figure 5: Landscape of the function $f$ as $m\to\infty$; we have $(m,n)=(200,2)$ and the true vectors are $[\pm3/4,0]$. The noise vector is generated at uniform in [-1,1] such that $\widetilde{\epsilon}\approx 5. 10^{-3}$. One clearly sees that the geometry of the landscape of $f$ is preserved and that the only minimizers of $f$ are very close to the true vectors.

Theorems & Definitions (27)

• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 3.1
• Definition 3.2: Strict saddle point
• Theorem 3.3
• Remark 3.4
• Theorem 4.1
• Corollary 4.2