# The Lasso with general Gaussian designs with applications to hypothesis testing

arXiv: https://arxiv.org/abs/2007.13716 · HTML: https://mlsanity.com/paper/2007.13716/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2020.

## Built on by (most cited first, 30 of 40)

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## Abstract (arXiv)

The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates p is of the same order or larger than the number of observations n. Classical asymptotic normality theory does not apply to this model due to two fundamental reasons: (1) The regularized risk is non-smooth; (2) The distance between the estimator widehatboldsymbolθ and the true parameters vector boldsymbolθ^* cannot be neglected. As a consequence, standard perturbative arguments that are the traditional basis for asymptotic normality fail. On the other hand, the Lasso estimator can be precisely characterized in the regime in which both n and p are large and n/p is of order one. This characterization was first obtained in the case of Gaussian designs with i.i.d. covariates: here we generalize it to Gaussian correlated designs with non-singular covariance structure. This is expressed in terms of a simpler ``fixed-design'' model. We establish non-asymptotic bounds on the distance between the distribution of various quantities in the two models, which hold uniformly over signals boldsymbolθ^* in a suitable sparsity class and over values of the regularization parameter. As an application, we study the distribution of the debiased Lasso and show that a degrees-of-freedom correction is necessary for computing valid confidence intervals.