Constructing confidence intervals for the signals in sparse phase retrieval

Abstract

In this paper, we provide a general methodology to draw statistical inferences on individual signal coordinates or linear combinations of them in sparse phase retrieval. Given an initial estimator of the targeting parameter, which is generated by some existing algorithm, we can modify it in a way that the modified version is asymptotically normal and unbiased. Then confidence intervals and hypothesis testing can be constructed based on this asymptotic normality. For conciseness, we focus on confidence intervals in this work, while a similar procedure can be adopted for hypothesis testing. Under some mild assumptions on the signal and sample size, we establish theoretical guarantees for the proposed method. These assumptions are generally weak in the sense that the dimension could exceed the sample size and many non-zero small coordinates are allowed. Furthermore, theoretical analysis reveals that the modified estimators for individual coordinates have uniformly bounded variance, and hence simultaneous inference is possible. Numerical simulations in a wide range of settings are supportive of our theoretical results.