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High-Dimensional Confidence Regions in Sparse MRI

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

TL;DR

This work extends uncertainty quantification for high-dimensional sparse regression to MRI by developing confidence regions based on the desparsified LASSO with subsampled Fourier measurements. Conditioning on the sampling operator $F_Ω$ yields asymptotic normality of the debiased estimator, with a data regime $n \gtrsim \max\{ s_0 \log^2 s_0 \log p, s_0 \log^2 p \}$ ensuring vanishing bias. The approach provides pixelwise confidence intervals with radii shrinking at the optimal $1/\sqrt{n}$ rate, leveraging RIP-like properties of normalized subsampled Fourier matrices and consistent noise estimation. Practically, this enables statistically valid uncertainty quantification in MR reconstruction pipelines, with potential extensions to multi-parameter MRI and dictionary-based sparsity.

Abstract

One of the most promising solutions for uncertainty quantification in high-dimensional statistics is the debiased LASSO that relies on unconstrained $\ell_1$-minimization. The initial works focused on real Gaussian designs as a toy model for this problem. However, in medical imaging applications, such as compressive sensing for MRI, the measurement system is represented by a (subsampled) complex Fourier matrix. The purpose of this work is to extend the method to the MRI case in order to construct confidence intervals for each pixel of an MR image. We show that a sufficient amount of data is $n \gtrsim \max\{ s_0\log^2 s_0\log p, s_0 \log^2 p \}$.

High-Dimensional Confidence Regions in Sparse MRI

TL;DR

This work extends uncertainty quantification for high-dimensional sparse regression to MRI by developing confidence regions based on the desparsified LASSO with subsampled Fourier measurements. Conditioning on the sampling operator yields asymptotic normality of the debiased estimator, with a data regime ensuring vanishing bias. The approach provides pixelwise confidence intervals with radii shrinking at the optimal rate, leveraging RIP-like properties of normalized subsampled Fourier matrices and consistent noise estimation. Practically, this enables statistically valid uncertainty quantification in MR reconstruction pipelines, with potential extensions to multi-parameter MRI and dictionary-based sparsity.

Abstract

One of the most promising solutions for uncertainty quantification in high-dimensional statistics is the debiased LASSO that relies on unconstrained -minimization. The initial works focused on real Gaussian designs as a toy model for this problem. However, in medical imaging applications, such as compressive sensing for MRI, the measurement system is represented by a (subsampled) complex Fourier matrix. The purpose of this work is to extend the method to the MRI case in order to construct confidence intervals for each pixel of an MR image. We show that a sufficient amount of data is .
Paper Structure (12 sections, 1 theorem, 14 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 1 theorem, 14 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and related works
  3. Sparse regression
  4. The desparsified LASSO
  5. Subsampled Fourier matrices
  6. Confidence regions in the subsampled Fourier case
  7. Asymptotic normality of debiased LASSO in the subsampled Fourier case
  8. $\ell_2$-consistency of LASSO
  9. Main theoretical result
  10. Noise estimation
  11. Numerical Experiments
  12. Conclusion and future work

Key Result

Theorem 1

journal2022 Let $\frac{1}{\sqrt{n}}F_{\Omega}$ be a normalized subsampled Fourier matrix with $n \gtrsim s_0\log^2 s_0\log p$ rows. Let further $\lambda\geq 2\lambda_0:= 2\frac{\sigma\sqrt{K}}{\sqrt{n}}(2+\sqrt{12\log p})$. Then, the following decomposition holds where the debiased LASSO $\hat{\beta}^u$ is defined in eq:debiased_lasso and $W\mid F_{\Omega}\sim\mathcal{N}(0,\sigma^2\hat{\Sigma})$.

Figures (2)

  • Figure 1: Original non-sparse MR angiography; single brain slice with vessels lighting up; image intensities in arbitrary units [u.a.].
  • Figure 2: Confidence intervals based on the debiased LASSO for the pixels with the largest magnitude sorted in descending order.

Theorems & Definitions (1)

  • Theorem 1