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