Constructing Confidence Intervals for the Signals in Sparse Phase Retrieval

TL;DR

This paper introduces a methodology for constructing confidence intervals for individual signals in sparse phase retrieval, allowing for valid statistical inference even in high-dimensional settings with many small signals.

Contribution

The authors develop a general approach to modify existing estimators to achieve asymptotic normality and unbiasedness, enabling confidence interval construction in sparse phase retrieval.

Findings

The method provides asymptotically normal and unbiased estimators.

Confidence intervals are valid even when dimension exceeds sample size.

Numerical simulations support theoretical guarantees.

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 for the targeting parameter (some simple function of the signal), 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 testings 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 testings. 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 interval estimation is possible. Numerical simulations in a wide range of settings are supportive of our theoretical results.