Uncertainty Quantification for Sparse Fourier Recovery

MCML Authors

Felix Krahmer

Prof. Dr.

Principal Investigator

* Former Principal Investigator

Holger Rauhut

Prof. Dr.

Core PI

Mathematical Data Science and Artificial Intelligence

Abstract

One of the most prominent methods for uncertainty quantification in high-dimen-sional statistics is the debiased LASSO that relies on unconstrained ℓ1-minimization. The majority of initial works focused on real (sub-)Gaussian designs. However, in many applications, such as magnetic resonance imaging (MRI), the measurement process possesses a certain structure due to the nature of the problem. The measurement operator in MRI can be described by a subsampled Fourier matrix. The purpose of this work is to extend the uncertainty quantification process using the debiased LASSO to design matrices originating from a bounded orthonormal system, which naturally generalizes the subsampled Fourier case and also allows for the treatment of the case where the sparsity basis is not the standard basis. In particular, we construct honest confidence intervals for every pixel of a p-dimensional MR image that is s0 sparse in the standard basis provided the number of measurements n satisfies n ≳ max {s0log2s0log p, s0log2p}, or that is sparse with respect to the Haar wavelet basis provided a slightly larger number of measurements.

article HKV+27

---

Applied and Computational Harmonic Analysis

87.101942. Feb. 2027.

---

Authors

F. Hoppe • F. Krahmer • C. M. Verdun • M. I. Menzel • H. Rauhut

---

Links

DOI

---

Research Area

 A2 | Mathematical Foundations