One of the most prominent methods for uncertainty quantification in high-dimen-sional statistics is the desparsified LASSO that relies on unconstrained $\ell_1$-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 $n \gtrsim\max\{ s\log^2 s\log p, s \log^2 p \}$ or that is sparse with respect to the Haar Wavelet basis provided a slightly larger number of measurements.

翻译：高维统计中不确定性量化的最显著方法之一是依赖于无约束$\ell_1$最小化的去偏LASSO。早期研究主要关注实值（次）高斯设计。然而，在许多应用场景中，例如磁共振成像，测量过程因问题本质而具有特定结构。MRI中的测量算子可通过子采样傅里叶矩阵描述。本文旨在将基于去偏LASSO的不确定性量化方法扩展至源自有界正交系统的设计矩阵，该框架自然推广了子采样傅里叶情形，并允许处理稀疏基非标准基的情况。具体而言，我们为标准基下稀疏的MR图像每个像素构造了诚实置信区间，前提是测量次数满足$n \gtrsim\max\{ s\log^2 s\log p, s \log^2 p \}$；或对哈尔小波基稀疏的图像，在测量次数略大的条件下同样适用。