With or Without Replacement? Improving Confidence in Fourier Imaging

arXiv (Cornell University)

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.

Authors 5

1. Frederik Hoppe Aachen Chair of Mathematics of Information Processing

   RWTH Aachen University

   Affiliation as printed

   Chair of Mathematics of Information Processing , RWTH Aachen University , Germany

2. Claudio Mayrink Verdun

   Harvard University · LMU Klinikum · Ludwig-Maximilians-Universität München

   Affiliation as printed

   Department of Mathematics , LMU Munich , Germany , and

   Harvard John A. Paulson School of Engineering and Applied Sciences , Harvard University , USA

3. Felix Krahmer

   Technische Hochschule Ingolstadt · Technical University of Munich

   Affiliation as printed

   Faculty of Electrical Engineering and Information Technology , TH Ingolstadt , Germany ,

   TUM School of Computation,

4. Marion Irene Menzel

   Munich Center for Machine Learning · Technical University of Munich

   Affiliation as printed

   Information and Technology , Technical University Munich , Germany , and

   Munich Center for Machine Learning , Germany

5. Holger Rauhut
   Affiliation as printed

   GE HealthCare , Munich , Germany, and TUM School of Natural Sciences , Munich , Germany