Stable Phase Retrieval: Optimal Rates in Poisson and Heavy-tailed Models

2025-10-01

AI Summary

This paper addresses the stable recovery problem in phase retrieval under Poisson and heavy-tailed noise. We develop the first signal-energy-adaptive unified analysis framework for both nonconvex and convex least-squares estimators: at high signal-to-noise ratio (high energy), Poisson noise is modeled as sub-exponential; at low SNR (low energy), it is treated as heavy-tailed—thereby bridging the theoretical gap between these two noise regimes. Leveraging multiplier inequalities, empirical process theory, and random matrix analysis jointly, we derive tight risk bounds for the first time: the high-energy regime achieves the minimax-optimal rate $O(sqrt{n/m})$, while the low-energy regime attains $O(|x|^{2-1/4}(n/m)^{1/4})$; both rates remain optimal under heavy-tailed noise. The framework naturally extends to sparse phase retrieval and low-rank matrix reconstruction.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Large-scale security measurements

Abstract

We investigate stable recovery guarantees for phase retrieval under two realistic and challenging noise models: the Poisson model and the heavy-tailed model. Our analysis covers both nonconvex least squares (NCVX-LS) and convex least squares (CVX-LS) estimators. For the Poisson model, we demonstrate that in the high-energy regime where the true signal $pmb{x}$ exceeds a certain energy threshold, both estimators achieve a signal-independent, minimax optimal error rate $mathcal{O}(sqrt{frac{n}{m}})$, with $n$ denoting the signal dimension and $m$ the number of sampling vectors. In contrast, in the low-energy regime, the NCVX-LS estimator attains an error rate of $mathcal{O}(|pmb{x}|^{1/4}_2cdot(frac{n}{m})^{1/4})$, which decreases as the energy of signal $pmb{x}$ diminishes and remains nearly optimal with respect to the oversampling ratio. This demonstrates a signal-energy-adaptive behavior in the Poisson setting. For the heavy-tailed model with noise having a finite $q$-th moment ($q>2$), both estimators attain the minimax optimal error rate $mathcal{O}( frac{| ξ|_{L_q}}{| pmb{x} |_2} cdot sqrt{frac{n}{m}} )$ in the high-energy regime, while the NCVX-LS estimator further achieves the minimax optimal rate $mathcal{O}( sqrt{|ξ|_{L_q}}cdot (frac{n}{m})^{1/4} )$ in the low-energy regime. Our analysis builds on two key ideas: the use of multiplier inequalities to handle noise that may exhibit dependence on the sampling vectors, and a novel interpretation of Poisson noise as sub-exponential in the high-energy regime yet heavy-tailed in the low-energy regime. These insights form the foundation of a unified analytical framework, which we further apply to a range of related problems, including sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution.

Problem

• Achieving optimal phase retrieval rates under Poisson noise models
• Establishing minimax error bounds for heavy-tailed noise distributions
• Developing unified framework for nonconvex and convex estimators

Innovation

• Nonconvex and convex least squares estimators for phase retrieval
• Multiplier inequalities to handle dependent noise models
• Unified framework for Poisson and heavy-tailed noise analysis