Uncertainty Quantification for Sparse Fourier Recovery

MCML Authors

Felix Krahmer

Prof. Dr.

Principal Investigator

Optimization & Data Analysis

Claudio Mayrink Verdun

Dr.

* Former Member

→ Group Felix Krahmer
 Optimization & Data Analysis

Holger Rauhut

Prof. Dr.

Principal Investigator

Mathematical Data Science and Artificial Intelligence

Abstract

One of the most prominent methods for uncertainty quantification in high-dimen-sional statistics is the desparsified 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 desparsified 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 an MR image that is sparse in the standard basis provided the number of measurements satisfies n≳max{slog2slogp,slog2p} or that is sparse with respect to the Haar Wavelet basis provided a slightly larger number of measurements.

misc

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Preprint

Sep. 2023

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Authors

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

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Links

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Research Area

 A2 | Mathematical Foundations