# Lasso Universality Under Linearly Dependent Covariates in the Sparse Regime

arXiv: https://arxiv.org/abs/2608.08390 · HTML: https://mlsanity.com/paper/2608.08390/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 56 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [Graph-based Approximate Message Passing Iterations](https://mlsanity.com/paper/2109.11905/) (2021)
- [Gaussian Universality of Perceptrons with Random Labels](https://mlsanity.com/paper/2205.13303/) (2022)
- [Universality laws for randomized dimension reduction, with applications](https://mlsanity.com/paper/1511.09433/) (2015)
- [When does Gaussian equivalence fail and how to fix it: Non-universal behavior of random features with quadratic scaling](https://mlsanity.com/paper/2512.03325/) (2025)
- [Universality in Transfer Learning for Linear Models](https://mlsanity.com/paper/2410.02164/) (2024)
- [Asymptotics of Linear Regression with Linearly Dependent Data](https://mlsanity.com/paper/2412.03702/) (2024)
- [Long-time dynamics and universality of nonconvex gradient descent](https://mlsanity.com/paper/2509.11426/) (2025)
- [High-Dimensional Statistics: Reflections on Progress and Open Problems](https://mlsanity.com/paper/2605.05076/) (2026)
- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)

## Built on by (most cited first, 2 of 2)

- [Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Design](https://mlsanity.com/paper/2609.33087/) (2026)
- [Multi-Agent Deferred-Decision Trajectory Optimization with Anytime-Valid Target Retention for Moving Candidates](https://mlsanity.com/paper/2610.17003/) (2026)

## Abstract (arXiv)

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.