Lasso Universality Under Linearly Dependent Covariates in the Sparse Regime

Throughout the last decade, Gaussian universality has been widely studied for high-dimensional estimation problems. Most of the literature focuses on i.i.d. sensing matrices or accounts for special forms of dependence, such as block dependence or other specific row/column dependencies. More general simultaneous row and column mixing has not yet been fully studied. In this paper, we focus on that setting. We prove a Gaussian universality theorem for the lasso in the sparse regime, where the non- Gaussian covariates have linearly dependent rows and columns. To the best of our knowledge, our setting permits a broader simultaneous row and column dependence structure than those treated in much of the prior universality literature. Numerical illustrations for various sparse profiles support the universality claims of this paper.