2025/10/01 by Huang, Gao, Li, Song, Needell, Deanna · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2510.00551
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 pmbx exceeds a certain energy threshold, both estimators achieve a signal-independent, minimax optimal error rate O(√((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 O(‖\pmbx‖1/42⋅((n)/(m))1/4), which decreases as the energy of signal \pmbx 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 O( \frac‖ ξ‖Lq‖ \pmbx ‖2 ⋅ √((n)/(m)) ) in the high-energy regime, while the NCVX-LS estimator further achieves the minimax optimal rate O( √‖ξ‖Lq⋅ ((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.