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With or Without Replacement? Improving Confidence in Fourier Imaging

Frederik Hoppe, Claudio Mayrink Verdun, Felix Krahmer, Marion I. Menzel, Holger Rauhut

TL;DR

Addresses uncertainty quantification in high-dimensional Fourier imaging where the ground truth is sparse in a non-canonical basis; introduces a reweighted sampling without replacement scheme that converts the debiasing problem into a replacement-like setting via counts $\gamma_i$ and preconditioners, preserving RIP guarantees. In the Haar domain, the debiased estimator has a Gaussian term $W^z$ with reduced variance and a smaller remainder $R^z$, yielding sharper confidence intervals. Numerical MRI-like experiments show substantial improvements in estimator error and coverage of confidence regions, highlighting practical gains for medical imaging.

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.

With or Without Replacement? Improving Confidence in Fourier Imaging

TL;DR

Addresses uncertainty quantification in high-dimensional Fourier imaging where the ground truth is sparse in a non-canonical basis; introduces a reweighted sampling without replacement scheme that converts the debiasing problem into a replacement-like setting via counts and preconditioners, preserving RIP guarantees. In the Haar domain, the debiased estimator has a Gaussian term with reduced variance and a smaller remainder , yielding sharper confidence intervals. Numerical MRI-like experiments show substantial improvements in estimator error and coverage of confidence regions, highlighting practical gains for medical imaging.

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.
Paper Structure (8 sections, 2 theorems, 22 equations, 3 figures, 2 tables)

This paper contains 8 sections, 2 theorems, 22 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Debiased LASSO
  3. Sampling schemes
  4. Improving the Debiased LASSO's Confidence
  5. Standard Debiasing
  6. Reweighting Sampling Without Replacement Debiasing
  7. Numerical Experiments
  8. Conclusion

Key Result

Theorem 1

krahmer2013stable Let $\Phi=\{\varphi_j\}_{j=1}^N$ and $\Psi =\{\psi_k\}_{k=1}^N$ be orthonormal bases of $\mathbb{C}^N$. Assume the local coherence of $\Phi$ with respect to $\Psi$ is pointwise bounded by the function $\kappa$, that is $\sup\limits_{1\leq k\leq N} |\langle \varphi_j, \psi_k\rangle| and choose $m$ (possibly not distinct) indices $j \in \Omega \subset [N]$ i.i.d. from the probabili

Figures (3)

  • Figure 1: Quantitative comparison between the two methods: The blue markers are the values for the reweighting approach, and the green for the straightforward approach. The x-axis represents the dependency on $\lambda$ as a multiply of $\lambda_0$. The y-axis shows the $\ell_2$-norm of the following quantities: $\hat{x}-x^0$ (circle), $\hat{x}^u-x^0$ (plus) and $R^x$ (star). Note that the same values apply for $z$ since the Haar transform is an isometry w.r.t. the $\ell_2$-norm.
  • Figure 2: Modified Shepp-Logan phantom. The marked red line shows the pixel for which Figure \ref{['fig:conf_int']} displays the confidence intervals.
  • Figure 3: Confidence intervals (blue errorbars) with debiased LASSO (blue circles) for underlying image pixels (red pluses) along the marked red line in Figure \ref{['fig:marked_line']} shown for one realization of the subsampling and noise. Here, we chose $\lambda = 15 \lambda_0$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof