With or Without Replacement? Improving Confidence in Fourier Imaging

TL;DR

This paper proposes a reweighted sampling method without replacement to enhance the accuracy of debiased estimators for uncertainty quantification in Fourier imaging, especially in structured measurement scenarios like MRI.

Contribution

It introduces a novel reweighted sampling approach that improves debiased estimators for UQ in Fourier imaging, bridging the gap between theory and practical applications.

Findings

Reweighted sampling enhances estimator performance in Fourier imaging.

The method improves confidence interval accuracy in MRI scenarios.

Sampling without replacement reduces the remainder term impact.

Abstract

Over the last few years, debiased estimators have been proposed in order to establish rigorous confidence intervals for high-dimensional problems in machine learning and data science. The core argument is that the error of these estimators with respect to the ground truth can be expressed as a Gaussian variable plus a remainder term that vanishes as long as the dimension of the problem is sufficiently high. Thus, uncertainty quantification (UQ) can be performed exploiting the Gaussian model. Empirically, however, the remainder term cannot be neglected in many realistic situations of moderately-sized dimensions, in particular in certain structured measurement scenarios such as Magnetic Resonance Imaging (MRI). This, in turn, can downgrade the advantage of the UQ methods as compared to non-UQ approaches such as the standard LASSO. In this paper, we present a method to improve the debiased estimator by sampling without replacement. Our approach leverages recent results of ours on the structure of the random nature of certain sampling schemes showing how a transition between sampling with and without replacement can lead to a weighted reconstruction scheme with improved performance for the standard LASSO. In this paper, we illustrate how this reweighted sampling idea can also improve the debiased estimator and, consequently, provide a better method for UQ in Fourier imaging.