Performance bound of the intensity-based model for noisy phase retrieval

The aim of noisy phase retrieval is to estimate a signal 𝐱_0∈ℂ^d from m noisy intensity measurements b_j=|⟨𝐚_j,𝐱_0 ⟩|^2+η_j, j=1,…,m, where 𝐚_j ∈ℂ^d are known measurement vectors and η=(η_1,…,η_m)^⊤∈ℝ^m is a noise vector. A commonly used model for estimating 𝐱_0 is the intensity-based model 𝐱:=_𝐱∈ℂ^d∑_j=1^m (|⟨𝐚_j,𝐱⟩|^2-b_j )^2. Although one has already developed many efficient algorithms to solve the intensity-based model, there are very few results about its estimation performance. In this paper, we focus on the estimation performance of the intensity-based model and prove that the error bound satisfies min_θ∈ℝ𝐱-e^iθ𝐱_0_2 ≲min{√(η_2)/m^1/4, η_2/𝐱_0_2 ·√(m)} under the assumption of m ≳ d and 𝐚_j, j=1,…,m, being Gaussian random vectors. We also show that the error bound is sharp. For the case where 𝐱_0 is a s-sparse signal, we present a similar result under the assumption of m ≳ s log (ed/s). To the best of our knowledge, our results are the first theoretical guarantees for the intensity-based model and its sparse version. Our proofs employ Mendelson's small ball method which can deliver an effective lower bound on a nonnegative empirical process.